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Energy level broadening in an N-particle system : a solvable model with a hierarchy of interactions Lukac, Eugene G. 1975

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ENERGY LEVEL BROADENING IN AN N-PARTICLE SYSTEM SOLVABLE MODEL WITH A HIERARCHY OF INTERACTIONS by EUGENE G. LUKAC B.S., Lowell Technological Institute, 1968 M.S., University of Connecticut, 1970 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1975 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my writ ten permission. Department of Ph"v<;.rr<; The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date August 20, 1975 - i i -ABSTRACT A one-dimensional chain of particles interacting through harmonic forces i s used in a theoretical study of properties of the strength function which are of current interest in nuclear physics. The work was motivated by the observation of intermediate structure in the low-energy 1 2C+ 1 2C total reaction cross section. A nested hierarchy of harmonic os-c i l l a t o r systems is constructed to parallel the hierarchy of three stages through which the 1 2C+ 1 2C reaction is envisioned to proceed. The hierarchy consists of a "single particle" system in which one group of 12 particles interacts with another group of 12 particles through an average potential, a "doorway" system i n which six groups of 4 particles interact with each other via another average potential, and a "compound" system in which a l l particles are grouped together. The construction and usefulness of the model is discussed with reference to the choice of potentials, the elimination of spurious states, and the possibility of actually obtaining the wavefunctions. Overlap integrals of wavefunctions, in terms of which the strength function i s de-fined, are shown to be doable although they are, in general, N-dimensional. For each step of the hierarchy the strength function is determined, and by analyzing the fluctuations and the effect of the density of states i t s shape is compared to the widely used Lorentzian shape. Group theory i s used to study the effect of degeneracy on the overlap integrals. Without the inclusion of quartet clustering in the last step of the hierarchy, the ratio of widths of the strength function in the hierarchy is found not to correspond to the ratio obtained in the 1 2C+ 1 2C experiments. Attention i s drawn to the consequences of using a Lorentzian strength function for cross section extrapolations. - i i i -TABLE OF CONTENTS Chapter 1. INTRODUCTION 1 1-1. Evolution of a Quantum Mechanical State 2 1-2. The Role of Overlaps i n Nuclear Physics 3 1-3. Calculation of the Strength Function from Nuclear Properties: Early Attempts 10 1-4. Experimental Status of Intermediate Structure in Heavy Ion Reactions 12 1-5. Stellar Reaction Rates 15 1-6. The Alpha Particle Doorway States 17 1-7. Statement of the Problem 22 Chapter 2. FEATURES OF THE MODEL 26 Chapter 3. THE NORMAL COORDINATES 29 3-1. Meaning and Existence of Normal Coordinates 29 3-2. Linear Chain with Periodic Boundary Conditions 31 3-3. Linear Chain with Free Ends 33 3- 4. Remarks 35 Chapter 4. PROTOTYPE CALCULATIONS WITH FOUR PARTICLES 37 4- 1. Compound System 38 4-2. Single Particle System 43 4-3. Overlap of the Ground State Wavef unctions ^ 4-4. Comparison of the Energies 53 4-5. Moments of the Strength Function 55 4-6. The Model Interactions as an Average Over Exact Interactions 60 - i v -Chapter 5. MODEL INTERACTION FOR SYSTEMS WITH N PARTICLES 65 Chapter 6. OVERLAP INTEGRALS FOR A SYSTEM WITH 24 PARTICLES 69 6-1. Single Particle System in Its Ground State 69 6-2. Single Particle System in an Excited State 75 Chapter 7. THE DENSITY OF STATES 8 2 Chapter 8. THE SHAPE OF THE STRENGTH FUNCTION 85 Chapter 9. A HIERARCHY OF SYSTEMS 93 9-1. The Hierarchy 93 9-2. Spreading Widths for Excited States 97 Chapter 10. SUMMARY AND DISCUSSION 101 APPENDIX A. NUCLEAR POTENTIALS AND HARMONIC OSCILLATOR POTENTIALS .. 109 A-1. Approximation of a Potential by a Harmonic Oscillator Potential 1 0 9 A-2. Construction of a Nucleon-Nucleon Potential m APPENDIX B. DIFFICULTIES WITH THE EXCLUSION PRINCIPLE 114 APPENDIX C. PREDICTION OF NULL OVERLAP INTEGRALS 120 C-l. Structure of the Overlap Integral 120 C-2. Group Theoretical Prediction of Null Overlap Integrals .. 123 C-3. The Invariance Groups of the Single Particle and the Doorway Systems 125 C-4. The Representations of the Common Invariance Group G .... 129 C-5. Classification of the Eigenfunctions 140 BIBLIOGRAPHY 160 -v-LIST OF TABLES Table Page 9-1. The normal mode frequencies of the three systems in the harmonic oscillator hierarchy 96 C-I. Traces of the (reducible) representation D(H), and some of the irreducible representations d^(H) 134 C-II. Classification of fundamental wavefunctions according to energy and to representation of G 148 - v i -LIST OF FIGURES Figure Page 1-1. The Lorentzian shape of <C2 >/D 9 1-2. A typical absorption cross section, normalized to a black nucleus cross section, as predicted by the optical model, and observed experimentally with low energy resolution ... 12 1-3. Typical intermediate structure in a cross section normalized to the black nucleus cross section 14 1-4. Schematic i l l u s t r a t i o n of the integrand 1(E) necessary for the calculation of the reaction rate for carbon burning at 0.75 x 10 9 °K 16 1-5. The three steps in the 1 2C+ 1 2C reaction according to the picture of Michaud and Vogt (1972) 19 1-6. The linear harmonic oscillator arrangements which correspond to the three stages of the 1 2C+ 1 2C reaction illustrated i n Figure 1-5 a) , b), and c) respectively 24 4-1. Compound system with 4 particles 38 4- 2. Single particle system with 4 particles 43 5- 1. The maximum overlap, C2_, between the ground states of the single particle and compound systems, and the correspond-ing single particle spring constant ratio, k'/k, as functions of the number of particles, N 67 6- 1. The number, m, of 2-quanta compound states having non-zero overlap with the single particle ground state as a function of the compound state energy, E. 72 - v i i -Figure Page 6-2. The average overlap squared, >> between 2-quanta compound states and the single particle ground state as a function of compound state energy, 72 6-3. The number, m, of 4-quanta compound states having non-zero overlap with the single particle ground state as a function of the compound state energy, E^ 73 6-4. The average overlap squared, ^ C 2, >, between 4-quanta compound states and the single particle ground state as a function of compound state energy, E^ 73 6-5. The strength, £c 2/AE, of the single particle ground state in the compound states with 0, 2, and. 4 quanta as a function of compound state energy, E^ 74 6-6. The strengths, £c 2/AE, of the n}2(2) single particle state in the compound states with 2, 4, and 6 quanta as a function of compound energy, E^ 78 6-7. Comparison of the strength S( 2) of the ni2(2) single particle state in 4-quanta compound states with the strength s ( 2 ) ' of the single particle ground state in 2-quanta compound states 79 8-1. The distribution of C2/D about 10 local averages 87 8-2. The strengths of Figure 6-6 compared to a Lorentzian strength function with parameters r = 0.0918 -ftw and E = 2.493 -ftu) obtained through a maximum likelihood o procedure 90 - v i i i -Figure Page 9-1. The strength, S__d, of the n 2(4) single particle state i n 4-quanta doorway states, and the strength, of the ^ ( 4 ) doorway state in 4-quanta compound states 98 9-2. A possible modification of the compound system whereby quartet clustering is reproduced by taking k>>k' 100 B-l. A simple linear chain of particles 114 C-1. The single particle system 126 C-2. The doorway system 128 - i x -ACKNOWLEDGMENT It is a real pleasure for me to express my appreciation to my supervisor, Professor Erich W. Vogt, for his guidance and encourage-ment throughout the course of my work at the University of B r i t i s h Columbia. His energy and enthusiasm have been a contagious inspiration not only in physics, but extending far beyond i t into l i f e ' s other realms. The cheerful assistance of Dr. Dan L i t v i n with group theoretical questions greatly f a c i l i t a t e d my task. I am indebted also to Robert Esch for many useful discussions and comments during the i n i t i a l stages of this work. Special thanks are due to Barbara Mitchell, whose a b i l i t y to translate scribble into typescript has made the production of this manuscript much less painful than i t would have been otherwise. I was fortunate also in obtaining important typing assistance from Diane Boyd. Perhaps the most important contribution, however, has been made wittingly or unwittingly by those scores of people, known or unknown, who by their attitudes, greetings, words, and smiles, whether for an instant or through an extended period, have made my days pleasanter, my vision higher, my heart warmer. To them I owe my a b i l i t y to enjoy my work, and to them I wish to dedicate this thesis. -1-CHAPTER 1 INTRODUCTION How does the state of a quantum mechanical system evolve when i t is disturbed? This question, as old as quantum mechanics i t s e l f , has been asked by physicists in almost every branch of physics. In the mid 1950's i t confronted nuclear physicists as the key to accounting for the spectacular successes of the optical model for nuclear reactions and the shell model for nuclear structure. Yet, the intr a c t a b i l i t y of a r e a l i s t i c many-nucleon calculation forced the question to be abandoned after some a r t i f i c i a l , though elegant, mathematical modeling. Recently, however, the observation of nuclear reactions proceeding in a hierarchy of steps not only provides a new setting for the question, but adds to i t a new urgency due to the consequent uncertainty i n stel l a r reaction rates. In this introductory chapter the opening question w i l l be given a quantitative formulation in Section 1-1. Section 1-2, showing how the question arises i n nuclear physics via i t s role i n the j u s t i f i c a t i o n of the optical model, w i l l be followed by Section 1-3 with a discussion of previously attempted solutions. The new setting for the problem i s revealed by reviewing in Section 1-4 the recent experimental observations, indicating i n Section 1-5 their importance to astrophysical calculations, and presenting i n Section 1-6 a theoretical framework within which the observations may be understood. The foregoing, serving as a h i s t o r i c a l introduction, leads in Section 1-7, to the statement of the problem to which this thesis addresses i t s e l f together with the assumptions adopted for i t s solution. -2-1-1 Evolution of a Quantum Mechanical State Wanted i n this section i s a quantitative meaning for the question, "How does the state of a quantum mechanical system evolve when i t i s disturbed?" Begin by considering a system of N particles in an eigenstate <J>_ with energy E^ of the Hamiltonian H'. Now disturb the system by permitting among the N particles a new interaction which w i l l be em-bodied i n a new Hamiltonian H. The two Hamiltonians H' and H may, i n practice, be different approximations to the true Hamiltonian for the system. The new Hamiltonian H w i l l have eigenstates xn a n d eigenvalues E^, which w i l l be assumed, for simplicity, to be discrete and non degenerate. Since the states of an observable form a complete set, one may write, for example, a state $ as an expansion i n terms of states X Q» [1.1] 0 = L C m « Xn where the expansion coefficient, C m „ =^ 0* % d? y i-s called "the overlap between the state <j>_ and the state Xn"> °r» simply, "the overlap". The square of the overlap, C^_, can then be interpreted as the probability of finding the system i n a particular state xn i f the system had been originally prepared in the state <j)_. In this sense the state <J> evolves with probability C 2 into the states x of the new Hamiltonian. Thus, a mn n plot of C2_ versus some identifying feature of the state xn» for example the energy E , gives a quantitative description of how a state <}>_ evolves when i t i s disturbed. In terms of the energy, one says that the system of N particles i n i t i a l l y having total energy E^ i s found, after the Hamiltonian i s changed from H' to H, to have energy E_ with probability C2^. (The changing of the Hamiltonian rescinds the obligation to conserve energy.) -3-Thus, the energy level is said to spread, or broaden i t s e l f , into the range of energies E for which C 2 is non negligible. n mn 1-2 The Role of Overlaps in Nuclear Physics This section w i l l show how the overlaps defined i n Section 1-1 enter into the understanding of nuclear, reactions. In 1954 Feshbach, Porter, and Weisskopf (1954) addressed them-selves to the great richness of data for neutron scattering off nuclear targets from across the entire range of atomic weights, A. Although, even when averaged over resonances, there was a large variation in cross section across the range of A, the variation was small between target nuclei neighboring in A. The gradualness of this variation from one nucleus to the next led to the idea that the neutron cross sections did not depend on the detailed interaction between the neutron and the target, but, rather, that i t depended, for the most part, on some gross features of the target nucleus - for example the nuclear radius which changes very l i t t l e between neighboring nuclei. Furthermore, the then recent accomplishments of the shell model in nuclear structure calculations i n -dicated that a nucleon moves independently inside a nucleus, and experiences the presence of the remaining nucleons only through a potential well typifying their average effect. Thus i t was proposed by Feshbach, Porter and Weisskopf to represent the interaction of the i n -coming neutron with the particles of the target nucleus by means of a single, simple potential describing the interaction of the neutron with the nucleus as a whole. They were careful to point out, however, that this simplification of the interaction cannot hope to predict the details of the cross section, but only the overall behavior after one averages over the resonances of the compound nucleus. Thus, one begins with the -4-Breit-Wigner expression for the absorption cross section near a single s-wave resonance [1.2] °l (abO = i , \ .  Pl where k is the wavenumber of the incident particle of energy E, E is the c A resonance energy of the compound nucleus,,T is interpreted as the width AC for decay of the compound nucleus level X into the entrance channel c, and = £ r x c ' ' following Vogt (1962), the cross section in Eq. [1.2] is c' averaged by integrating over i t s energy dependence, and dividing by the average compound nucleus level spacing to obtain [1.3] <o:(abs)> = <^> It i s customary to write <C^^c/> = ^P (Y;[ c^ w n e r e t n e penetration factor is proportional to the probability that the incident particle enter the interaction region when hindered by the Coulomb and angular momentum barriers, and the reduced width yj[ depends only on the internal nuclear properties. If, i n addition, one defines the strength function [1.4] then Do [1.5] (<rjai*)) - ' Hill k which contains the important result that the giant resonances in the average cross section reflect the energy dependence of the strength -5-function. This expression for the average absorption cross section, de-rived here from isolated compound nuclear resonances, can be shown (Lane and Thomas 1958) to provide a good approximation even when these resonances overlap. The simple potential (the optical potential) proposed by Feshbach, Porter, and Weisskopf (1954) was a complex square well of radius R_, namely, [1.6] = 0 for r > R0 where the real part U_ represents the average interaction of the incident particle with the target nucleus, and the imaginary part W_ i s responsible for the absorption of particles out of the incident beam into the compound nucleus. This role of W_ can be easily seen by writing the time dependent Schrbdinger equation for the incident particle inside the nuclear radius using potential [1.6]. The resulting time dependence of the single particle wavefunction is exp[-i(E-iW_)t/n] so that the mean lifetime of the state would be T = *n/W , while E i s one of the energy levels (discrete or continuous) of the real potential U . However, the f i n i t e lifetime imposes on the energy levels, because of the uncertainty principle,a width T = -h/t = Wq. This width is in addition to, and, usually, larger than the widths:T_ of the resonances of the real part of the potential. The mean free path of a particle in a state of lifetime T i s vT , which for a low incident energy nucleon incident on a typical well (U_ = -50 MeV, W_ = 6.6 MeV) is about 10 fm,. comparable to a nuclear diameter. That a nucleon can traverse a nucleus without encountering the effects of the remaining nucleons was to be expected from the va l i d i t y of the shell model, that heavy ion scattering w i l l also be describable by -6-the optical model speaks for the unexpected persistence of clusters inside nuclei. Calculation of the s-wave absorption cross section due to the potential [1.6] i n the v i c i n i t y of a single resonance p, again following Vogt (1962), yields, [1.7] cr(abO - ^ W < m k R, (eref + w , 1 where m and k are the mass and wave number, respectively, of the incident particle, and i s i t s resonant energy. It i s this cross section that represents the average compound nucleus cross section found in Eq. [1.5]. Comparing the s-wave part of Eq. [1.5] (P = kR £) with Eq. [1.7] one ob-tains for the strength function [ 1 ' 8 1 S T " TT (a -ay +w* This simple approach, the nuclear optical model, had a surprising success in describing the trends of the neutron cross sections across the periodic table, and to this extent one accepts Eq. [1.8] as providing the connection between the strength function and the optical model parameters. However, i t would, at the same time, be desirable to have an understanding in terms of detailed nuclear properties of why the strength function has the shape given in Eq. [1.8]. It was in response to this question that Lane, Thomas, and Wigner wrote their famous paper of 1955. To understand the work of Lane, Thomas, and Wigner (1955), i t is convenient to begin with the following definitions. Let be the eigen-states of the f u l l Hamiltonian, H, for N particles which obey a boundary condition at the origin and at the nuclear radius R. Let ^ u be the c p • -7-eigenstates of the N particles partitioned into 2 clumps. Their Hamiltonian, H', contains the interaction of the 2 clumps with each other through a potential V' which is the average of the 2-body potentials , [i.9] V' - ( I 1 v c A The u_ are the radial eigenstates of this potential satisfying the same boundary conditions as and the IJI contain a l l the remaining variables and quantum numbers for the N particles. Since both and ^ cu_ form a complete set of states for N particles inside the nuclear radius, one can write [1.10] where the coefficients C, are the overlaps described in Section 1-1. X;cp Now, although the reduced widths appearing t h r o u g h . i n the Breit-Wigner expression [1.2] are treated as parameters i n the theory of resonance reactions, the YJ^/n* i n fact, express formally the probabil-i t y per unit time that the entrance channel particles in the state be found on the surface of the compound nucleus whose wavefunction i s x^> after the effects of the Coulomb and angular momentum barriers have been absorbed into the penetrabilities P . Thus, the reduced width amplitudes Y^c are defined in terms of the integral of X^l* over a l l the compound nuclear coordinates except the separation of the channel constituents which is kept at R_. -8-where M is the channel reduced mass, c Putting Eq. [1.10] into Eq. [1.11] gives So far this i s an exact result. Lane, Thomas, and Wigner pointed out, however, that the average separation d between the single particle levels Ep i s much larger than the average separation D between the compound nucleus levels E^. Further, i n contrast to the "strong-coupling", or "black nucleus" model, the optical model predicts (correctly) giant resonances at the single particle energies E^. Thus in the mixture [1.10] of single particle states into compound states each one of the c a n be associated with a particular u . In other words, the coefficients C, * P A;cp are such that for each value of X, there is only one value of the index p for which C. contributes appreciably to the expansion [1.10]. This A;cp interpretation of the giant resonances permits the reduced widths, from Eq. [1.12], to be written u.13] x: - c y t p where the f i r s t factor on the right side i s the single particle reduced width, and the C 2 indicates how the coupling of the entrance channel X;cp into the compound nucleus reduces the single particle reduced width into the actual reduced width Y ? . 'Xc Now, since for the square well Up(R) = 2/R, one may compare the average of the reduced widths in Eq. [1.13] with the square well strength function [1.8] to find -9-[1.14] I w which i s the well known Lorentzian shape illustrated i n Figure 1-1, <C l>/D e r - E Fig. 1-1. - The Lorentzian shape of ( C 2 />/D . ^;cp The generality of the above discussion i s not restricted by i t s i l l u s t r a -tion i n terms of square wells. Various "diffuse edge" potentials have been used successfully (Woods and Saxon 1954), and, furthermore, Michaud, Scherk, and Vogt (1970) have shown that the effects of such potentials may be reproduced by an appropriately constructed square well. To recapitulate, then, the optical model leads to a strength function which, i n general, i s a sum of Lorentzians. Lane, Thomas, and Wigner indicated that one of these terms, <C2 y/D with a given p, w i l l A ;cp dominate this sum over p. Further, they assigned the Lorentzian shape of the <C2 >/D to the behavior of the C 2 rather than to the effect of A;cp' *;cp the spacing D. It is to be noted that the Lorentzian shape in Eq. [1.14] for <C2^ /D, which leads to the optical model strength function, arose from Lane, Thomas, and Wigner's assignment of the giant resonance structure to the behavior of the C 2 . To put i t differently, i t i s because the X;cp squared overlaps, C 2.^, have a giant resonance behavior that the optical model had been successful i n reproducing the giant resonances i n the data. This leads immediately to the question "Can the giant resonance behavior assigned by Lane, Thomas, and Wigner to the Cj[.Cp be connected to nuclear properties?", which gave birth to the next generation of research a c t i v i t i e s . It i s these efforts which w i l l be described i n Section 1-3. 1-3 Calculation of the Strength Function from Nuclear Properties: Early Attempts To calculate the overlaps C^.Cp between the compound nuclear state and the composite nuclear state involving N particles each moving i n a 3-dimensional world i s an extremely d i f f i c u l t problem, and, in fact, impossible given the present day understanding of nuclear forces and the capabilities of present day computers. One would l i k e , ideally, to obtain the compound states x^  by solving [1.15] HX, - B% %x and to obtain the single particle states ^ u by solving c p [i.i6] a' %*f - ecf ^ p where the Hamiltonians H and H', and the boundary conditions are those described i n Section 1-2. The straightforward calculation of C, re-X;cp quires, then, the evaluation of the 3N-dimensional integral u.i7] cKCf - \ rx J v . -11-Faced with the unfeasibility of the direct procedure just described, Wigner (1955), (1956), and Bloch (1957) approached the problem by expressing (C2>c_)>/D in terms of the expectation value i n the single particle states of the Green's function G(E) = (E-H) - 1, which may be ex-panded as a power series in H_, where H_ i s the residual interaction (H_ s H-H') given by the difference between the exact interaction i n H and the single particle (or average) interaction i n H'. This series converges provided the matrix of H_ in the single,particle states i s a bordered matrix with f i n i t e norm. To this end Wigner assumed the levels E to be ° cp equidistant, and the matrix elements <cp|H_|c'p')> to have the value +V or -V with equal probability i f |E__ - Ec,_,|<e, and to have the value zero i f |Ec_ - Ec,_,|>e. The model of Bloch is a slight generalization of these conditions. By keeping only the lowest order terms in 1/e the series may be summed to obtain the Lorentzian behavior of <fC2 )>/D as i n X;cp Eq. [1.14]. However, by adopting assumptions on the residual interaction H_ to cater to the mathematical t r a c t a b i l i t y of the problem, one i s l e f t with a mathematical a r t i f i c e that i s no longer able to answer the question "What i s the physical reason for the Lorentzian shape?" In the careful words of Wigner, "the model which underlies the present calculation shows only a limited similarity to the model which is believed to be correct". At this point, partly because of the d i f f i c u l t y of the problem, and partly because the optical model parameters could always be obtained by f i t s to the data, the problem was abandoned without a satisfactory reason having been found for the giant resonance behavior of the squared overlaps, C? A;cp The next three sections w i l l explore the source of the new impetus for the revival of the problem posed originally by the success of -12-the optical model; a revival necessitated now by i t s limitations. 1-4 Experimental Status of Intermediate Structure in Heavy Ion Reactions The optical model discussed in Section 1-2 predicts the existence of giant resonances at the single particle energies E^. This means that in an experiment involving the c o l l i s i o n between nucleus A and nucleus B, which is done with low energy resolution (a spread in the incident energy of a few tens of keV), one w i l l not observe i n the ab-sorption cross section the many narrow resonances of the compound nucleus A+B (typically a few keV apart around mass number 24), but a broad resonance, a few MeV in width, which i s an average over many of the compound nucleus resonances, as illustrated in Figure 1-2. The centres of the broad resonances are associated with the levels, E^, of the average potential through which particle A interacts with particle B. II II Hi l l Fig. 1-2. - A typical absorption cross section, normalized to a black nucleus cross section, as predicted by the optical model, and observed experimentally with low energy resolution. The markings on the energy axis indicate the positions of some of the compound nuclear levels which would be observed with much higher resolution. The years following the introduction of the optical model for neutron scattering saw, as described above, i t s successive refinement to -13-include diffuse edge potentials as well as spin-orbit interactions, and it s surprisingly successful extension to the analysis of heavy ion scattering experiments (Shapiro 1962, Hodgson 1967). The f i r s t puzzle in the development of the optical model began emerging i n 1960 with the observation at low energies (~ 6 MeV i n the centre of mass) by Almqvist, Bromley, and Kuehner (1960) of 3 resonances in the 1 2C+ 1 2C reaction having a width and spacing of a few hundred keV. Ten years later the puzzle became consolidated with the discovery, upon extension of the measurements, that the same resonant structure was continued down to the lowest energies (~ 4 MeV) where s t a t i s t i c a l uncertainties are small enough to make i t v i s i b l e (Patterson, Winkler, and Zaidins 1969, Mazarakis and Stephens 1972). What was the nature of this puzzle? It was precisely the width and spacing of these resonances that made them so intriguing. An order of magnitude larger than the compound nucleus resonances, their widths and spacings were yet one order of magnitude smaller than those one had learned to expect from the optical model. Specifically, the spacing of the single particle resonances, determined primarily by the reduced mass and r a d i i of the interacting nuclei, i s in this case approximately 8 MeV; the width, determined by the imaginary part of the optical potential, i s 1 - 2 MeV. In contrast, the new resonances had a spacing of 0.3 MeV and a width of 0.1 - 0.2 MeV. On the other hand, the compound nucleus levels available i n 21fMg, though overlapping at the 20 MeV excitation energy probed by the 1 2C+ 1 2C experiment, can be estimated to be separated by .02 MeV and to have a width of about 0.12 MeV (Almqvist et a l . 1964, Vogt et a l . 1964, Voit et a l . 1974). The new resonances were, thus, given the name "intermediate resonances" or "intermediate structure". In Figure 1-3 they can be seen encompassing many compound nucleus levels, and, in turn, -14-being encompassed and modulated by the giant resonance of the optical model. 1 1 I I I I M i l l llilllllll|lllllilllllili|illlilHlill[|i:;IH| > £ Fig. 1-3. - Typical intermediate structure in a cross section normalized to the black nucleus cross section. The markings on the energy axis are some of the compound nucleus levels. The dashed line i s the optical model cross section, which i s , in fact, an average over the inter-mediate structure. Although the procrustean exercise of adapting the optical model to f i t the f i r s t three observed intermediate resonances was imaginatively attempted, i t eventually became clear (Michaud and Vogt 1972) that no reasonable optical model potential for the 1 2C+ 1 2C system could give rise to a l l the observed structure. The data, in particular the spacings of the intermediate resonances, suggested that they were due to the excita-tion of some simple internal degrees of freedom. It w i l l be seen in Section 1-6 how this remark leads to a natural interpretation of inter-mediate resonances in terms of a hierarchy of structures for the 24 nucleons involved in the 1 2C+ 1 2C reaction. Since the f i r s t appearance of intermediate structure i n the 1 2C+ 1 2C system i t has been sought systematically in other heavy ion -15-reactions (Hanson et a l . 1974) though, at, present, only 1 2C+ 1 6 0 and ct+20Ne besides 1 2C+ 1 2C (Voit et a l . 1974) exhibit the necessary correlation of resonances in a l l exit channels that distinguish intermediate structure from s t a t i s t i c a l fluctuations. These reactions, however, are just the ones where one would expect to see intermediate structure according to the framework of Michaud and Vogt, to be described i n Section 1-6, which was f i r s t advanced to account for i t i n the 1 2C+ 1 2C system. The description of the framework of Michaud and Vogt w i l l be postponed i n order to f i r s t underscore in Section 1-5 the importance of a detailed understanding of the 1 2C+ 1 2C reaction for astrophysical calcula-tions. 1-5 Stellar Reaction Rates After the exhaustion of i t s helium supply i n the production of 1 2C, 1 6 0 , 2 0Ne, and some 21+Mg, a star may achieve, either quiescently or explosively, temperatures favorable for the heavy ion reaction 1 2C+ 1 2C. Lack of detailed models for hydrostatic carbon-burning stars leads to estimates (Baudet and Salpeter 1969, Arnett and Truran 1969, Arnett 1969, Barnes 1971) in the range 0.3<Tg<1.2 for the most l i k e l y temperature at which this occurs (Tg = 10 - 9T°K). To calculate the reaction rate, < W X per cm3 per sec, one averages the product of the cross section a(E) and the relative speed v using the Maxwell-BoItzman distribution N^(v) for the number of pairs per cm3 having relative speed v at the temperature T to obtain (Reeves 1968) oo [1.18] < c r v ) ' [ M ( < r ( C ) . • -16-For the case of the 1 2C+ 1 2C reaction at 0.75 x 10 9 °K the nature of the integrand, 1(E) = N(E) v(E) cr(E), of Eq. [1.18] i s illustrated in Figure 1-4. Fig. 1-4. - Schematic i l l u s t r a t i o n of the integrand 1(E) necessary for the calculation of the reaction rate for carbon burning at 0.75 x 10 9 °K. After Reeves (1968) It i s seen from Figure 1-4 that the integrand 1(E) has a maximum, the "Gamow peak", which is centered at 2 MeV, and has a width of 0.8 MeV. Thus the Gamow peak occurs i n the t a i l of the Maxwell distribution, and i n a region where the Coulomb barrier has already caused the cross section to plummet from ~103mb at 10 MeV to ~10~5mb at 2.5 MeV, the lowest energy at which the cross section has been measured. Thus, the calculation of astrophysical reaction rates requires knowledge of the cross section at energies where i t i s too small to be accessible to laboratory measurements. It is here that the optical model must ri s e from i t s descriptive role to the position of a tool for extra-polation. Although i t is possible to extrapolate the cross section to lower energies by obtaining a phenomenological f i t to the experimental data at higher energies, such a process i s sensitive to the assumptions made regarding the dominant physical features below 2.5 MeV, leading, at -17-present, to an uncertainty factor of 10 in the reaction rate (Michaud and Vogt 1972). This uncertainty demands a revival of the search (last taken up by Bloch (1957) as described in Section 1-3) for those physical features in a nuclear reaction responsible for the giant resonances i n the cross section. Further, since, as explained i n Section 1-4, a giant resonance represents an average over the intermediate resonances, i t i s necessary, in turn, to expose as well the physical features giving rise to the inter-mediate structure. It is thus that Wigner and Bloch's endeavor to c l a r i f y the origin of the giant resonances acquires, thanks to the richness of the 1 2C+ 1 2C system, the dual task of understanding both the giant resonances and the intermediate resonances. It was i n this context that in 1972 Michaud and Vogt opened a door to future approaches by introducing an a-particle framework which permits a qualitative understanding of the intermediate structure in the 1 2C+ 1 2C reaction. An outline of this description w i l l now be given i n Section 1-6. 1-6 The Alpha-Particle Doorway States Intermediate structure i n nuclear cross sections i s an observed phenomenon. Its interpretation centres on the concept of a "doorway state". This idea was already implicitly contained in a qualitative dis-cussion of nuclear reactions given by Weisskopf (1961). The expression "doorway state" was coined by Block and Feshbach (1963) who f i r s t used i t to interpret the systematic deviations of the neutron strength function around the optical model predictions. A doorway state refers to a simple nuclear excitation (2 particle - 1 hole i n the treatment of Block and Feshbach) conceived as the only way the entrance channel (1 particle -0 hole) can begin to excite the compound nucleus by means of 2-body forces. -18-Later work by Feshbach and collaborators (Feshbach, Kerman, and Lemmer 1967, Kennedy and Sh r i l l s 1968, Feshbach 1973a, Feshbach 1973b) extended the concept to permit more general types of doorway states. Michaud and Vogt (1972) were the f i r s t to conceive of a-particle doorway states as appropriate for the 1 2C+ 1 2C reaction as follows. After the carbon nuclei enter the region of nuclear interaction the'reaction i s envisioned as occurring i n three steps. In the f i r s t step the two carbon nuclei interact through some average C - C potential, V1^ (C-C), whose resonances give rise to the large "single particle" structure i n the cross section. The large width indicates that, as one would expect, the carbon nuclei are not able to maintain their identity for very long i n the presence of the residual interaction which is due to the forces the three alpha particles in one carbon exert on those of the other. These a - a interactions occurring through an average potential V^(a-a) soon break up each of the 2 carbons into 3 alpha particles. (It i s easier to remove an a-particle than a nucleon from carbon - the former costs 7.4 MeV, the latter 18.8 MeV). The increased number of degrees of freedom gives rise to the narrower spacing of the intermediate resonances, and the high st a b i l i t y of alpha particles suggests i t s role in determining their nar-rower width compared to single particle resonances. Thus the doorway states give rise to the intermediate structure as the resonances of a system of 6 alpha particles interacting through the potential Vjj (a-a). Eventually, however, even the alpha particles w i l l succumb to the nuclear forces since they are not yet eigenstates of the f u l l Hamiltonian for 24 nucleons interacting pair wise through potentials (N-N). The residual interaction due to V^j(N-N) acting between members of different alpha particles tears them apart to form the compound nuclear system 2l|Mg whose excited states are the closely spaced compound nucleus resonances. These -19-three steps in the reaction are illustrated p i c t o r i a l l y in Figure 1-5. single particle system doorway system compound system Fig. 1-5. - The three steps in the 1 ZC+ 1 2C reaction according to the picture of Michaud and Vogt (1972). The type of potential which dominates-the behavior at each stage i s indicated. The depiction of 1 2C as three a-particles derives from the known clusterization properties of 1 2C (Igo et a l . 1963) as well as the success of a-particle modeling for i t s ground state (Ikeda et a l . 1972, Mendez-Moreno et a l . 1974). The above interpretation of the 1 ZC+ 1 2C reaction proceeding into the compound nucleus 2ItMg through a-particle doorway states received striking confirmation from the experiments of Voit et a l . (1973) who found that quartet states i n 2 0Ne are preferentially populated i n the reaction 1 2C( 1 2C,ct) 2 0Ne at bombarding energies corresponding to maxima in the -20-intermediate structure. This same inconsistency with s t a t i s t i c a l model predictions for this reaction had been noticed earlier by Middleton et a l . (1971) at slightly higher energies. It i s possible, as Michaud and Vogt have done, to envisage their picture of the 1 2C+ 1 2C reaction as proceeding through a natural nested hierarchy of systems - the single particle system, the alpha particle doorway system, and the compound nucleus system - and to express the states of each system as the eigenstates of a corresponding hierarchy of nested Hamiltonians. These three Hamiltonians are given, respectively, by I L P L 9 ] H " = ^ Z v ; • Zv : ; (c -c ) [1.20] H = 1% [1.21] where m„ Is the mass of a carbon nucleus, m i s the mass of an alpha C ' a particle, and m^  is the mass of a nucleon. At each stage an average potential i s expressible as an average over the next most exact one. Thus, [1.22] -21-where <j> is the ground state wavefunction of the -tth alpha particle a i (with nucleons 1-4) and ^ a is the ground state wavefunction of the jth alpha particle (with nucleons 5-8); and [1.23] v:;(c-c) = < * tt v_(—01 pt.*) where <br is the ground state wavefunction of the -Ith carbon nucleus C i (with a's 1-3) and <j> is the ground state wavefunction of the jth 3 carbon nucleus (with a's 4-6). The residual interaction, H^, spreading single particle states into doorway states i s [i.24] H'r = R' - H" and the residual interaction, H_, spreading doorway states into compound states i s [1.25] /TV ~ H ~ If' This framework accounts qualitatively not only for a l l the characteristics of the low energy 1 2C+ 1 2C reaction (giant, intermediate, and compound resonances, their widths and spacings, and the branching ratios i n the exit channel), but also for a l l the other cases where intermediate structure has been seen in heavy ion reactions (Voit et a l . 1974). Its only shortcoming seems to be the present impossibility of quantifying the description of the systems specified by Hamiltonians [1.19] through [1.21] using nuclear forces for 24 particles i n 3 dimen-sions. However the success of this scheme, which is based simply on the -22-hierarchial grouping of the 24 nucleons involved, compels one to believe that a treatment which is inattentive to some of the complexities of real nuclei, but which preserves the distinctive groupings of the model of Michaud and Vogt, might be capable of quantitatively reproducing i t s features as well as providing a connection with the Lorentzian shape of the resonances. It is this belief which motivated the work which i s re-ported i n this thesis. 1-7 Statement of the Problem The aim of this thesis i s two-fold. On the one hand the intent is to reopen the question asked by Wigner and Bloch (see Section 1-3) re-garding the Lorentzian shape of the strength function by using a more physical model than the contrived random matrices employed previously. On the other hand i t i s sought to determine whether clustering alone as embodied in a hierarchy of interactions can reproduce the intermediate structure seen in low energy *2C+*2C scattering. These two aspects w i l l be folded into each other by calculating the spread of a single particle state into doorway states, and, i n turn, calculating the spread of doorway states into the compound states. In each case the shape of the energy dependence of the average overlap coefficients squared, <^C2X divided by the average level spacing, D, w i l l be compared to the Lorentzian shape of the optical model (see Eq. [1.14]). Motivated by the desire for some physical realism, and, yet, restricted by the requirement of mathematical so l u b i l i t y , two i n i t i a l assumptions for the N-body nuclear system w i l l be made which, though drastic, w i l l not interfere with the questions addressed. The f i r s t i s to confine the N nucleons to a one-dimensional space thus disregarding the possible effect of the a b i l i t y of the particles to get around each other -23-on the shape of the strength function. This assumption invites i t s e l f not only because of the experimentally observed one-dimensional chains of a particles in states of 1 2C (Ikeda et a l . 1972) , but also because of the repeated usefulness of one-dimensional prototypes in physics (Lieb and Mattis 1966). The second is to approximate the nucleon-nucleon force by a harmonic force chosen to best simulate i t s effects. The extent to which this can be done w i l l have to be examined, but, more importantly, i t w i l l be shown that the problem can be cast in a form which makes the results insensitive to the particular choice made. The ensuing system of N one-dimensional particles with harmonic interactions i s soluble: the eigen-functions of any Hamiltonian (single particle, doorway, compound) can be found. Further, the overlap integral, C, between any two wavefunctions, even though i t is N-dimensional, can always be evaluated, and thus one can directly study the behavior of the average overlaps, <(C2X In particular, to each of the three steps of the 1 2C+ 1 2C reaction as depicted in Figure 1-5 there corresponds an arrangement of harmonic oscillators as illustrated i n Figure 1-6. It is to be emphasized that one does not pretend that the linear chain of Figure l-6c i s a replica of the nucleus 2tfMg. The intention is simply that, to the extent that the various structures in the cross section for 1 2C+ 1 2C are due to the clusterings of Figure 1-5, the relationship between the systems of Figures l-6a, b, and c w i l l duplicate the relationship between the systems of Figures l-5a, b, and c. That i s , for the purposes of studying the successive spreading of simpler states into more complex ones leading to giant resonances, intermediate reson-ances, etc, as well as for obtaining the f i r s t quantitative evaluation of a strength function, one may avail oneself of the simplicity of harmonic -24-O^SKXKV1-© &0><ytn-O**-O OTI-OA^O^O GwvO^ -O^ -Q ty^'O-^-fytrO Om&t*0-&0 J L „ II k" a) the single particle system 0^ 0w>O^ >O O^ On-O^ O G/rK>'T>0/&<> ex^ O^ -O^ O cyjM2K»-0 OTjeCr&O'VK-C I (Yi, 1 1 y^ o- ^ I •— fl <Tpr.~ 1 1 "frflr- ' k» b) the doorway system c) the compound system Fig. 1-6. - The linear harmonic oscillator arrangements which correspond to the three stages of the 1 2C+ 1 2C reaction illustrated in Figure 1-5 a), b), and c) respectively. The Hooke's constants for each type of spring are indicated. -25-oscillators while preserving those physical features under investigation. This thesis constitutes the actualization of this tenet. With this intent, Chapter 2 discusses certain features of the model. The normal coordinates are presented in Chapter 3. Chapter 4 collects the prototype calculations which were performed f i r s t with four particles. Model interactions for N particles, treated i n Chapter 5, are followed by overlap calculations i n Chapter 6, and by a study of the density of states in Chapter 7, leading to their incorporation, i n Chapter 8, into the evaluation of the strength function. The hierarchial system of Figure 1-6 f i n a l l y makes i t s appearance in Chapter 9 where the experience of the previous four chapters is applied to calculate the wavefunctions, their overlaps from model to model, and the corresponding strength functions. Lastly, i n Chapter 10, the results are summarized to permit a discussion i n the s p i r i t of the questions raised by this intro-duction. -26-CHAPTER 2 FEATURES OF THE MODEL In constructing a physically relevant harmonic oscillator representation of nuclei attention must be paid to several features. This chapter describes b r i e f l y how some of these are handled. Further details w i l l be found in the appendixes. One asks f i r s t what choice for the parameters A and B i n the harmonic oscillator potential [2.i] V " C O = f\ t C - B H O - v w i l l give the best approximation to a nuclear potential? It w i l l be seen in what follows in this thesis, however, that the results are independent of this choice, and, thus, the question need not be pursued here. A discussion of this point i s nevertheless given in Appendix A where i t i s shown that i t i s d i f f i c u l t for a harmonic oscillator potential to reproduce the effect of a nuclear potential with a s t i f f core, but that one can construct a nuclear potential that is amenable to a harmonic oscillator approximation. Secondly, before one can confidently calculate the wave-functions for a system of particles subject to harmonic oscillator forces, i t is necessary to ascertain that no spurious states w i l l be introduced due to the motion of the centre of mass of the system. In the nuclear shell model, where wavefunctions are referred to a fixed external origin, there are introduced three extra degrees of freedom which correspond to the motion of the nucleus as a whole. The shell model wavefunctions w i l l thus contain more states than the wavefunctions of the internal motion. - 2 7 -E l l i o t t and Skyrme (1955) have shown how to deal with the spurious states in this case. For the case of one-dimensional harmonic oscillator inter-actions the problem is considerably simpler as w i l l be now shown. Consider for i l l u s t r a t i o n two particles of mass m in a harmonic oscillator potential of spring constant k fixed at the origin. In units where "R = m = k = 1, the Hamiltonian of the system i s [2.2] W 0 = - J . £ . - I £ l + • 1 rx , 1 r v v Now, i f the two particles are in a harmonic oscillator potential fixed at  their centre of mass, then, i n the same units, the Hamiltonian describing the (internal) motion of the system i s [2.3] u r - I L . i £ - v AT 1 (r,-R)" v 1/r- -RV" where R = (r^ + T<i)j2 i s the coordinate of the centre of mass. Therefore, one may write which reveals that the difference between the two Hamiltonians in question is a term describing the force on the centre of mass. E l l i o t and Skyrme (1955) give a prescription for obtaining the eigenfunctions of H. in mt terms of the eigenfunctions of H q which are the shell model eigenfunctions. However, by noticing that H may be written as [2.5] -28-and by remarking that the essential feature of H is that the potential energy term depends only on the relative coordinates, one may, for harmonic oscillator interactions, write the Hamiltonian directly in the form [2.5], and directly obtain the internal eigenfunctions. The effectiveness of this procedure w i l l be e x p l i c i t l y portrayed in Chapter 3 which deals with normal coordinates, and in Chapter 4 where prototype calculations are per-formed using a system of four particles. The Pauli exclusion principle i s a feature which one would like to take into consideration i n the model. However, i t i s argued i n Appendix B that this added realism would make the model completely un-manageable, and thus the rest of this work restricts i t s e l f to the treat-ment of unsymmetrized wavefunctions. Often in physics (see, for example, Lane, Thomas, and Wigner, 1955) the neglect of the Pauli principle has resulted i n a useful f i r s t approach. Already the present model has averted the a r t i f i c i a l i t y of the models described in Section 1-3, and, i n fact, concentrating on the arrangement of the particles in various linear chain configurations while disregarding their fermion character is not out of keeping with the intent of exploring the simplest physics which might be responsible for the intermediate structure i n heavy ion reactions de-scribed i n the introductory chapter. -29-CHAPTER 3 THE NORMAL COORDINATES This chapter deals with a set of coordinates c a l l e d normal coordinates which can be obtained by a transformation on the p a r t i c l e coordinates, and i n terms of which the c l a s s i c a l or quantum mechanical s o l u t i o n of a system with harmonic forces may be r e a d i l y obtained. Section 3-1 defines these new coordinates and establishes t h e i r existence f o r a wide cla s s of systems. Certain simple cases can be worked out ex-p l i c i t l y i n closed form and t h i s i s done i n Section 3-2 and 3-3. A few speculative remarks regarding normal coordinates i n Section 3-4 c l o s e t h i s chapter. 3-1 Meaning and Existence of Normal Coordinates The Hamiltonian appropriate to a system of N p a r t i c l e s coupled to nearest neighbors by harmonic forces i s t3.i] N - i £ + U U - ^ , ? ] where a l l the p a r t i c l e s have mass m, and the -Cth p a r t i c l e i s displaced by r ^ from i t s equilibrium p o s i t i o n . Dotted q u a n t i t i e s have been d i f f e r -entiated with respect to time. With r = r the Hamiltonian i n Eq. [3.1] n+1 n serves for a chain with free ends, and with r ., = r i i t serves f o r a n+1 1 chain with p e r i o d i c boundary conditions (which can be v i s u a l i z e d as pla c i n g the p a r t i c l e s on the circumference of a c i r c l e ) . The Hamiltonian i n Eq. 13.1] which i s designed to, and does, couple the p a r t i c l e s to each other, i s for the same reason not i n the form most conducive to a s o l u t i o n . For a system of harmonic o s c i l l a t o r s -30-the s o l u t i o n s , both c l a s s i c a l and quantum mechanical, are straightforward i f the Hamiltonian can be w r i t t e n as a sum of Hamiltonians each one de-s c r i b i n g the behavior of an i s o l a t e d e n t i t y subject only to a s i n g l e harmonic force. In the quantum mechanical case the wavefunction for the whole system i s then given by the product of wavefunctions associated with each of the Hamiltonians being added. To e f f e c t t h i s separation of the Hamiltonian one transforms from the p a r t i c l e coordinates r ^ , which may be w r i t t e n as a column matrix r_, to a new set of coordinates x_ v i a a matrix A such that r_ = A x. The new coordinates x_ are c a l l e d the "normal coordinates" f o r the system. Do such "normal coordinates", x» a c t u a l l y exist? In other words, under what conditions can one f i n d a transformation A connecting the p a r t i c l e coordinates to the normal coordinates? Consider the Hamiltonian i n Eq. [3.1], and notice that i t may be w r i t t e n as [3.2] 3* - ''"" where r_ i s the transpose of the column matrix r_, 1_ i s the N x N u n i t matrix, and the matrix U i s , f o r example f o r a chain with f r e e ends, given by [3.3] u = It can now be c l e a r l y seen that the normal coordinates are those i n which -31-the Hamiltonian H in Eq. 13.2] is "diagonal", specifically in which the matrix U is diagonal. Moreover, i t should be noticed that the matrix U i s always a real symmetric matrix not only for springs between nearest neighbors, but for any arbitrary set of springs among N particles. Re-calling (see, for eg., Holm 1964) that i f U i s a real symmetric matrix, T there exists an orthogonal matrix A such that A UA is diagonal, the ex-istence of normal coordinates has been established. Indeed, the Hamil-tonian in Eq. [3.2] becomes [3.4] H ~- £ j i T 1 * + i ^ ( A T U A ) x • Obtaining matrix A from any given U may be done by standard analytical or numerical methods. There exist cases, however, where A may be expressed in a closed form. The next sections examine some of these. 3-2 Linear Chain with Periodic Boundary Conditions For a simple linear chain of N particles interacting only with nearest neighbors through a harmonic force the Hamiltonian i s , from Eq. [3.1], given by [3.5] H = i £. L > r ^ v W C r - r ^ . V " ] For periodic boundary conditions one takes ^ n + ^ - ^\. Wanted now i s a new set of coordinates x given by a matrix, transformation r_ = A x, such that the x's are normal coordinates as defined in Section 5-1. Following Henley and Thirring (1962) one attempts a transformation in the form -32-J i / 2TTL ni\ [3.6] rA 2. \ Z i r " ) S Note that t h i s expression s a t i s f i e s the per i o d i c boundary condition vn + l = r i . Now, by s u b s t i t u t i o n of transformation [3.6] int o the Hamiltonian [3.5], and by using the formula one obtains [3.8] where the superscript * ind i c a t e s complex conjugation. This Hamiltonian, [3.8], now describes, a system of N independent harmonic o s c i l l a t o r s , each one of frequency [3.9] ' where £(= 1, 2, ... N) l a b e l s a p a r t i c u l a r "normal mode frequency". Thus the coordinates r_ defined by Eq. [3.6] are indeed normal coordiantes f o r th i s problem. One notes from Eq. [3.9], however, that to^ = so that the normal mode frequencies, except f o r H = N, and f o r I = N/2 (which e x i s t s only i f N i s even), are doubly degenerate. Further, the normal coordinates r n are complex so that i t i s not po s s i b l e at a glance to v i s u a l i z e , from the transformation matrix A, the motion of the p a r t i c l e s . Nonetheless i t i s possible to determine the r e a l motion of the p a r t i c l e s by taking -33-linear combinations of two coordinates r corresponding to the same frequency. These two complications do not arise in a linear chain with free ends as w i l l now be seen in Section 3-3. 3-3 Linear Chain with Free Ends For a simple linear chain with free ends consisting of N particles interacting only with nearest neighbors through a harmonic force i t i s easier, for the purposes of this section, to deal with the [classical] equations of motion, rather than with the Hamiltonian. The reason for this is that i n this case the closure relations which would be needed for uncoupling the Hamiltonian, such as Eq. [3.7], are not simple. Further, the equations of motion are linear in the coordinates, whereas the Hamiltonian is quadratic. If the particles a l l have mass m and the springs a l l have force constant k, the equation of motion for the nth particle i s [3.10] ' m r = -k ( i ; - ^ V. ( r * " ^ - ^ or, letting u>2 = k/m, [3.11] rh ^ ( r^, - z ^ * r n _ , ) For free ends one lets r = ri and r „ M = r„. Wanted now is a trans-. „ o 1 N+1 N N formation r n =i £ a ^ x^ such that the obey x^ = -w^  x^; that i s they are normal coordinates. Putting these two expressions in Eq. [3.11], and using the fact that the x 's are independent, one obtains - 3 4 -[3.i2] . x \ a M + l f ^-',2 = 0 as the condition which is to be satisfied by the elements a ^ of the transformation matrix A. Following Weinstock (1971) , a solution of Eq. [3.12] i s given by [3.13] = n ^ sin 0, where <j>^  = cos 1[1 - co2/2u>2]. One may now use the boundary conditions for free ends which may be expressed as a^ +^ ^ = a^ ^ to obtain the allowable values of <|> . Thus putting the expression [3.13] into this boundary con-dition one obtains [3.14] • Sl« M 4>j ( 1 ^ from which i t follows either that cos o> = 1 so that 0)^ = 0 for a l l 1 (which i s the physically uninteresting N-fold degenerate case in which a l l modes are pure translation), or sin Ncji^  = 0 so that [3.i5] w; - ( i - r a-<)) for I.= 1, 2, ... N. Eq. [3.15] gives N distinct normal mode frequencies (0^, including the pure translation . Although, i n general i t i s not possible to give a closed form for the matrix A as was done in this and the previous sections, i t i s always possible, as mentioned in Section 3-1, to diagonalize the potential energy matrix U for any one-dimensional arrangement of particles and - 3 5 -springs. 3-4 Remarks To put a l l the nucleons of a nucleus into a common p o t e n t i a l w e l l i s a u s e f u l beginning f o r many nuclear problems. This standard p r a c t i c e i s the nuclear s h e l l model. The l i n e a r chain model being used i n t h i s t h e s i s o f f e r s a d i f f e r e n t handle on c e r t a i n nuclear problems. I t i s i n t e r e s t i n g , though not necessary for the present purposes, to speculate about the r e l a t i o n s h i p between the two models. This s e c t i o n , then, appends to Chapter 3 a b r i e f comment regarding the two models. In the l i n e a r chain the normal coordinates are those which have an o s c i l l a t o r y motion independent of one another. In the s h e l l model the p a r t i c l e s are put i n a common p o t e n t i a l w e l l independent of one another. This leaves room for conjecturing that the normal modes of the l i n e a r chain are the e n t i t i e s which correspond to the p a r t i c l e s i n the s h e l l model. In both cases i t i s through e x c i t a t i o n of these to higher states that the system absorbs energy. However, whereas i n the s h e l l model a ground state nucleus i s kept from having a l l i t s p a r t i c l e s i n the energy l e v e l of the,lowest energy p a r t i c l e by the P a u l i p r i n c i p l e , i n the l i n e a r chain a ground s t a t e nucleus i s kept from having a l l i t s normal modes at the energy l e v e l of the lowest energy normal mode by the f a c t that each one i s subject to a d i f f e r e n t e f f e c t i v e spring constant. In conclusion, then, i t has been seen i n t h i s chapter how normal coordinates may be treated i n general. I t should be r e i t e r a t e d that the normal coordinates by uncoupling the equations of motion or the Hamiltonian apply generally as an a i d to both c l a s s i c a l and quantum mechanical s o l u t i o n s . The next chapter while i l l u s t r a t i n g with a four-p a r t i c l e chain the techniques to be used i n t h i s work w i l l deal e x p l i c i t l y -36-with the correspondence between c l a s s i c a l and quantum mechanical systems. -37-CHAPTER 4 PROTOTYPE CALCULATIONS WITH FOUR PARTICLES In order to incorporate the discussion to this point into a concrete example, and in order to provide a visualizable i l l u s t r a t i o n for the techniques to be used henceforth, this chapter contains calcu-lations using a system of four particles. The four particles move in one dimension and are subject to harmonic forces which in Section 4-1 are arranged to correspond to a compound nucleus, and in section 4-2 to correspond to a single particle state in nuclear reactions. The overlap of the ground state wavefunctions of the "compound" system and the "single particle" system is calculated in Section 4-3. The energies of the two systems are compared in Section 4-4. Section 4-5 examines the residual interaction between the two systems, and i t s moments in the ground state of the single particle system. These are shown to be related to the moments of the strength function. Finally, an alternative method for obtaining the single particle interaction as an average interaction i s shown, in Section 4-6, to lead again to harmonic oscillator interactions. It w i l l turn out that, although the four particle system already contains much of the essential physics of interest, the small number of particles and low density of states results in one state of the compound system acquiring 97% of the strength of the single particle system. This precludes the possi b i l i t y of a meaningful cal-culation of the strength function. The methods learned, though, can be applied to larger, richer systems. -38-4-1 Compound System The compound system is characterized by having a l l the participating particles grouped together into one system. In the case of a linear chain with free ends having four particles and harmonic interactions between nearest neighbors i t can be depicted as in Figure 4-1. Fig. 4-1. - Compound system with 4 particles, k indicates the spring constant. Let r^ be the displacement to the right from equilibrium of the -i-th particle. If the particles a l l have mass m the [classical] Hamiltonian describing the system i s [4.1] f| , ^ ( r > r ; . r ; + ^ ) + ^ [ ( r , - ^ . ( r v -Using the results of Section 3-3 one transforms to a new set of coordi-nates x using a matrix transformation A. such that _r = A x. From Eq. [3.13] and Eq. [3.15] the elements of A are given by [ 4 . 2 ] <£. = s i * : j ()*-') tr/u - s i n (J-.)(K,-I) Tr/tv sin (k.-i) TT/tJ J k ' K where C, is to be chosen such that £ a 2, = 1. E x p l i c i t l y , then, one k j J k may write - 3 9 -" r l " "0.50 0.653 0.50 0.271" *2 0.50 0.271 -0.50 -0.654 x 2 [4.3] *3 = 0.50 -0.271 -0.50 0.654 *3 0.50 -0.653 0.50 -0.271 T T and i t i s r e a d i l y v e r i f i e d that A A = A A = 1^. Ins e r t i n g t h i s trans-formation, [4.3], into the Hamiltonian [4.1] gives [4.4] H - + *y *• *-K) + ^ (p.58(> + 2.0 ^ + •j.Wx,1 One notices that t h i s Hamiltonian does not couple the x's to each other and, thus, the x's are indeed normal coordinates f o r t h i s system. Further, l e t t i n g (jj£/k/m one sees from the Hamiltonian [4.4] that the [ c l a s s i c a l ] normal mode frequencies f o r t h i s system are co, = o w v - JoTsSV to - OHbS u [4.5] u>3 - J~J^ O J = /,Y'V »** which agree with the p r e d i c t i o n s which would be made using the general expression [3.15] f o r normal mode frequencies. Having w r i t t e n the c l a s s i c a l Hamiltonian i n terms of the normal coordinates i t i s necessary now to obtain from i t the quantum mechanical Hamiltonian. This t r a n s i t i o n from a c l a s s i c a l Hamiltonian i s obtained by r e q u i r i n g that the generalized coordinates q_ and the generalized momenta p s a t i s f y the commutation r e l a t i o n s -40-It can be shown that the transformation [4.3] from particle coordinates to normal coordinates preserve the commutation relations [4.6], and thus in the present problem one can take x as the generalized n coordinates and mi as the generalized momenta from which i t follows n that the quantum mechanical Hamiltonian corresponding to Eq. [4.4] is A realization of these requirements is obtained by taking [4.7] = - a i _ [4.8] H ~- H C M where [4.9] [4.io] H f = (-£ .£ * 5.0 K. It is well to point out that the coordinate R of the centre of mass of the system i s given by -41-[4.11] * ^ I^-Lll^L H so using transformation [4.3] one may write [4.12] x, - L ( r, + A + r, * - a R . Therefore the expression in Eq. [4.9] may be written [4.13] h and is seen to truly represent the motion of the centre of mass. Thus, as promised in Chapter 2, the problem has very effortlessly separated i t s e l f into the motion of the centre of mass and the internal motion. By considering only the states of the Hamiltonian H^ n t of Eq. [4.10] which does not contain x^ one is sure not to include any spurious centre of mass states. The Hamiltonian H^ n t in Eq. [4.10] describes three independent one-dimensional harmonic oscillators. The solution of the time-? independent Schrddinger equation with H^ n t is the product of the three wavefunctions associated with each of the harmonic oscillators in H. int These wavefunctions are well known (see, for e.g., Merzbacher 1961) so that the total wavefunction may be immediately written down as -42-[4.14] where the w are the classi c a l normal mode frequencies given in Eq. [4.5], n_^  is the number of quanta in the -tth normal mode, and H_ is the Hermite polynomial of order n. The energy eigenvalues corresponding, to this wavefunction are [4.15] £ a n n = K ^ ' O t ^ ^ ( v ' / v H ^ - N + ' / C ) k ^ and i t can be easily verified that the wavefunction i s properly normalized so that t4-16] ( ( ( K „ „ z«*;< - K< K< . Setting these compound system solutions aside for now, Section 4-2 considers a single particle system.' _43-4-2 Single Particle System The single particle system is characterized by having the participating particles grouped, actually, into two clumps. Each clump experiences the average f i e l d of the other as i f i t were a single particle - hence the name. If the two clumps interact via only a central force then, of course, the problem may be reduced to an equivalent one-body problem in terms of the reduced mass and the relative coordinates (Goldstein 1950). The division of the particles into two clumps may ascribe any number of the available particles to one of the clumps. In the present case of four particles each clump w i l l be chosen to contain half the particles since the physical problem eventually being aimed at is the interaction of 1 2C with 1 2C. These two clumps w i l l be made to interact with each other through a harmonic force coupling their centres of mass. To allow for the fact that the strength of the interaction between the two clumps could be different from that of the interaction between two particles, a spring of constant k' w i l l be associated with i t . The value of k 1 w i l l have to be discussed later. The single particle system can, thus, be depicted as in Figure 4-2. (jy-Wr-<E) , gMm—© i /jjfo 1 Fig. 4-2. - Single particle system with 4 particles. Bearing in mind that these are the same four particles belonging to the compound system of Section 4-1 but arranged differently here, one may -44-use the same notation and write the Hamiltonian describing this system as [4 In matrix notation this may be written [4.18] where [4.19] U = 1 -1 -1 1 0 0 0 0 0 0 1 -1 k' 1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 1 The normal coordinates, _, needed to uncouple H' are those in which _ is diagonal. Since 2 i s a real symmetric matrix i t is diagonalizable by an orthogonal transformation, _, such that r_ = JL _y_. Since for this arrangement of harmonic oscillators (see Figure 4-2) there exists no known transformation in closed form as for the compound system of Figure 4-1, the task requires the diagonalization of a 4x4 matrix for which there are standard methods. This algebraic procedure, however, is extremely tedious, and since i t is possible in the case of four particles to guess at the solution one is led to try the transformation - 4 5 -[4.20] —" j r H _ I 1 1 SI 0 o 0 ) _ T T It i s r e a d i l y v e r i f i e d that B B = B B = _1, and that i n s e r t i n g t h i s transformation into the Hamiltonian H' of Eq. [4.17] gives [4.2i] n'= s ^ V'v^ j * + + ^ , * 3> so that indeed the y's are normal coordinates for t h i s system. One sees also that the normal mode frequencies are to, - ° [4.22] w s c ^ w to As i n Section 4-1, the quantum mechanical Hamiltonian corresponding to H' i n Eq. [4.21] i s [4.23] H' - H \ H • H : h t where [4.24] ft' - t> _ _ _ C M 1 ^ 0 ^ - 4 6 -[4.25] Again, the transformation [ 4 . 2 0 ] shows that H ' ^ = H Q J J and that i t i s the proper description of the centre of mass of the system. Thus the states of interest are the eigenfunctions of H ' . which describes three independent harmonic oscillators, so that the complete wave function associated with H'. can be immediately written down as xnt [ 4 . 2 6 ] where the u)!^ are the normal frequencies of Eq. [ 4 . 2 2 ] and the rest of the notation i s as i n the compound system solutions [ 4 . 1 4 ] . The energy eigenvalues corresponding to this wavefunction are [ 4 ' 2 7 ] (v''*)**; + (v>>)*4 ^ v ' ^ - K , and i t can be easily verified that the wavefunction is properly normalized so that [ 4 . 2 8 ] ((( V ' */\ . . J V ci 7 = S B • L S , -47-Now that both the single particle solutions and the compound system solutions have been obtained, the next section w i l l be devoted to calculating the overlap of the two ground state wavefunctions. 4-3 Overlap of Ground State Wavefunctions In the context of this chapter one may restate the aim of this thesis as expressed in Section 1-7 by asking how good a model for the compound system of Figure 4-1 i s provided by the single particle system of Figure 4-2. The question needs to be asked for each state of each system. If for each single particle state there i s one compound state with which i t s overlap i s close to unity, then the model i s very good. If a l l the overlaps of the single particle state with compound states are close to zero, then the model is very bad. The behaviour with energy of the average overlap squared per average energy interval is the strength function (up to a constant) and i t characterizes how a single state of the single particle system behaves when the particles enter a compound system arrangement. As explained in the introductory chapter, this i s the quantity of interest in nuclear physics. Its evaluation requires the calculation of a great many overlap integrals. This section w i l l show how one of these, namely the overlap of the ground states, i s to be calculated. Wanted, then, i s the overlap [4.29] Coo - \ "fc.W d ^ where XQ(2£) the ground state of the compound system obtained from Eq. [4.14] by setting n 2 = n3 = n^ = 0, and (y_) is the ground state -48-of the single particle system obtained from Eq. [4.26] by setting n 2 = n3 = = 0. In order to evaluate the integral in Eq. [4.29] both functions in the integrand must be written in terms of the same variables. This is straightforward since, i t w i l l be recalled, _ = A x and r_ = B_ y_ so that [4.30] x -Letting C_ = B_ A, one has, e x p l i c i t l y , [4.31] C = 1 0 0 0 0 .92 0 -.38 0 .27 .70 .64 0 .27 -.70 .64 One sees that _C, which is the product of two orthogonal matrices, i s , as expected, also orthogonal, and that the Jacobian of the transformation from (x 2 , X3 , x^) to (y 2 , y 3 , y 4) is 1 and thus preserves the normalization of the wavefunctions. The integrand in the overlap consists then of the product of [4.32] Kir / 2* 2 * -49-with I T XT [4.33] \ * T T / The overlap of these two wavefunctions w i l l be a three-dimensional integral. In general, for N particles the overlap integral w i l l be (N-l)-dimensional (since in the internal wavefunctions being considered the pure translational mode has been excluded). One notices, however, that the product $ XQ contains the exponential of a general quadratic function of the x's. The coefficients of the x's in the quadratic contain the normal mode frequencies 102, W3, u^, 0J3, w^ , which are given in terms of u(H/k/m) in Eq. [4.5] and Eq. [4.22], Further, i f k' = ak, and one knows the number a, then aj£ (= /k' /m) can also be expressed in terms of w so that where M i s a real symmetric matrix containing a l l the numerical coef-ficients. E x p l i c i t l y , M i s given by where W1 is the diagonal matrix containing the ratios w^ /w along the diagonal, and W is the diagonal matrix containing the ratios u^ /co along the diagonal. (In doing numerical work a t r i a l value for the number a(= k'/k) must be inserted at this stage). Again, a real [4.34] [4.35] M = C T W' C +• W -50-symmetric matrix can always be diagonalized with an orthogonal trans-formation, and thus there exist a set of coordinates _ obtained from x by an orthogonal transformation x = _ _ such that the overlap integrand can be written [4.36] T The matrix (X M X) is diagonal. X i s formed as usual from the eigen-vectors of M. Finally, since Eq. [4.36] permits the integrand to be written in the form [4.37] ^ X 0 4 e e ~ ^ • • • e the three-dimensional overlap integral becomes a product of three one-dimensional integrals. This useful separation can be carried out for the overlap integral of any two linear harmonic oscillator states of N particles. Indeed, having obtained the matrix X from the eigenvectors of _ in Eq. [4.35] one has [4.38] ^ x 2. y = y where Y = C_ X. Inserting the transformations [4.38] respectively into any wavefunction x(x) and any wavefunction iKy), any overlap integral j...|\jj(y^ ^,(x^ can be written as a product (or, at worst, a sum ( - a - ' i - \ I of such products) of integrals of the form f (z^) e ^ Z ; -51-Integrals of this type can be found in tables of integrals. Thus, one of the most important features of the present one-dimensional harmonic oscillator model is that i t is a solvable problem for N bodies. That i s , the N-dimensional overlap integrals can be done analytically with no more computer assistance than the diagonalization of NxN matrices. Returning now to the four particle i l l u s t r a t i v e example of this chapter, one remarks that even diagonalization of a 4x4 matrix requires considerable effort i f done by hand. Further, this cannot be done without committing oneself to specifying the ratio k'/k. For small numbers of particles.however, these d i f f i c u l t i e s can be bypassed using an alternative method as follows. Consider the overlap [4.39] C0 0 \[\ Hi(c*) XJl) ^ with the wavefunctions as given in Eq. [4.32] and Eq. [4.33], By noticing from Eq. [4.22] that u)3 = u)^ , the overlap can be written i n the form [4.40] r °c ( -O.X* The integral over X 3 is a standard form and with the aid of tables may be done immediately. An expression for the integral over x 2 is given Bierens de Haan (1867, Table 28, #1). The answer, in terms of X1+, is incorporated into the integral over x^ which then becomes of the same form as the integral over X3 and is done immediately. This procedure, using the u 's and u|'s from Eq. [4.5] and Eq. [4.22] respectively, gives -52-[ 4 . 4 1 ] C 0 0 = 3 . 3 f c ^ , y / < ' ( r " V V + ^ ' At this point, to get a numerical answer, one has to f i x the value of k' with respect to k (or, equivalently, with respect to to) . What is being sought here i s the strength function which serves as an indicator of how good a model for the compound system is provided by the single particle system. Therefore one would l i k e to begin with that single particle system which i s the best possible model for the compound system. One way to obtain this - other ways w i l l be described in the following sections - i s to require that the choice of k' maximize the overlap C between the ground states of the two systems. Setting oo the derivative of C q _ i n Eq. [ 4 . 4 1 ] with respect to co' equal to zero gives [ 4 . 4 2 ] = ° - * l n ^ or, equivalently, [ 4 . 4 3 ] ~~ 0. 7 ( 1 k With this value for k', the constant of the spring joining the two clumps in the single particle system, the overlap squared of the ground states i s [ 4 . 4 4 ] o. <n i - 5 3 -[4.44] C* = 0. 173 This high value indicates that the single particle system is a very good model indeed for the ground state of the compound system, and at the same time indicates that the strength function w i l l resemble a spike. Since the sum of a l l the overlaps squared must add up to 1.0, overlaps other than C q o w i l l be very small. The four particle system is in this sense too discrete to show the shape of the strength function. This is not the case when the number of particles is increased as w i l l be shown in the next chapter. Notice, however, that once having fixed the relationship between k' and k as in Eq. [4.43] one can obtain the numerical value of overlaps. In particular, as promised in Chapter 2, i t is not necessary to have a numerical value for k - the overlaps, and hence the strength function, are independent of k. This can also be seen using a dimensionality argument: since the overlaps are dimensionless numbers they can only contain the spring constants k and k' in ratios. There are, however, other ways besides the maximization of the ground state overlap to f i x the value of k' with respect to k. The next sections consider these alternatives. 4-4 Comparison of the Energies It was pointed out in Section 4-3 that one would l i k e the single particle spring constant k' to be such that the single particle system is the best model possible for the compound system. This was done by requiring that the overlap between the ground states of the two systems be maximized. One may, alternatively, use the ground state -54-energies as a criterion, and require, as in this section, that the ground state energies for the two systems match. From Eq. [4.15] and Eq. [4.5] the energy levels of the compound system are [4.45] & n v n v! = ^ w [0.76^ (nv + '/J + (V'v) ^ /• (V *0] From Eq. [4.27] and Eq. [4.22] the energy levels of the single particle system are [ M 6 ] C , v *" w + t^L'-^w^o H . * / , ^ , ^ Setting the ground state energies equal to each other, E 0 0 0 = E ' 0 0 0 ! gives [4.47] to ' = /./?? w and, consequently, L 4 - 4 8 ! It' = / . V J 7 This value for k' is different from the value for k' in Eq. [4.43] found by maximizing the overlap. One needs to ask how sensitive are the energy and the overlap to these changes in k'? Putting k' = 1.437k found in Eq. [4.48] by matching the energies into the expression [4.41] for the overlap gives -55-[4.49] OO so that [4.50] - 0.61 70 C, o.in k.) On the other hand putting k' = 0.769K, found in Eq. [4.43] by maximizing the overlap, into the expression ^ O Q Q in Eq. [4.46] for the ground state of the single particle system gives This reveals that both the energy and the overlap are remarkably insensitive to the choice of k'. In particular, the overlap changes by only 0.6% when k' i s changed by 100%. Thus, as i s desired, i t i s the clustering arrangement of the particles which dominate the physics -the choice of interaction constants are playing a minor role. leads to two different Hamiltonians. Their difference i s called the "residual interaction". The next section examines the role of the residual interaction in determining the shape of the strength function. 4-5 Moments of the Strength Function One of the main concerns of this thesis i s the question of the shape of the nuclear strength function. In the introductory chapter the strength function was defined in Eq. [1.4] as [4.51] E, = 2,0 % The way the four particles are clustered in the two systems -56-[4.52] Sc _ <Vj> where in the giant resonance interpretation of Lane, Thomas and Wigner [4.53] Y* =• (X"Y C The ( Y s ^ ) 2 are the single particle reduced widths of Eq. [1.13]. In cp this interpretation the energy dependence of the strength function i s contained in the energy dependence of the overlaps C 2^ > c_. The present approach using linear chains of harmonic oscillators permits, as seen in Section 4-3, the direct calculation of the C 2, as a function of X;cp the compound system energy E^, giving directly the shape of the strength function. Information on the shape of the strength function i s also contained in i t s moments with respect to the single particle energy levels E . It w i l l now be shown how these moments can be expressed cp in terms of the moments of the residual interaction in the single particle wavefunctions ^ Cp' Using the notation of the this chapter for single particle and compound system quantities, and letting V = H-H' be the residual interaction, one begins with the definition of the nth moment, M , of the strength function n [4.54] K S j ( E , - ^ ) " K ( £ , ) [4.55] -57-[4.57] [4.58] [4.59] [4.60] [4.61] [4.62] [4.63] D Therefore [4.64] [4.65] M V - ( * . F , r v . , ) [4.66] In the present four-particle linear chain problem the residual interaction V is the difference between the compound system Hamiltonian H of Eq. [4.1] and the single particle system Hamiltonian H' of Eq. [4.17], -58-namely [4.67] V - hi ( Illi - r* *-r-V where the r^ are the particle coordinates. The single particle ground state wavefunction over which the moments are to be calculated i s , from Eq. [4.26] and Eq. [4.22], where the y^ are the normal coordinates of this system. To calculate the integral in the f i r s t moment CO i t i s necessary to express the residual interaction V i n terms of the normal coordinates y_ by using the explicit form of the transformation L = i X given i n Eq. [4.20] whereupon the integral, since the exponential does not couple different y^'s, i s doable, yielding For now i t i s sufficient to notice that the value of to' (= 0.877 to) obtained in Eq. [4.42] by maximizing the overlap of the wavefunctions does not make Mj vanish, giving, i n fact, Mi = ( Y ^ ) 2 0.242 t u . This means that the strength function of the single particle system in the compound states i s not centered at the single particle energy. This result was anticipated in Section 4-4 where i t was shown, Eq. [4.51]» - 5 9 -that with the value of u)' which maximizes the overlap the ground state energies d i f f e r by 8%. This question of the centering of the strength function w i l l arise again in Section 4-6 where reference w i l l be made to the f i r s t moment M]^  of Eq. [4.70], requires squaring the residual interaction V of Eq. [4.67] and trans-forming i t from the particle coordinates r_ to the normal coordinates y_. However, the integral i s , for the reason expressed in the previous paragraph, s t i l l doable, and gives ment that the single particle system provide a good model for the compound system implies that the value of the strength function w i l l be appreciable only in a narrow range of energies around the single particle energy. This condition i s optimized by requiring M2 to be minimized with respect to ui ' which gives Calculation of the second moment [4.71] This provides yet a third alternative for determing u>' . The require-[4.73] w' * 1. 3.31 u> [4.74] comparable to the values obtained in Section 4-4 by matching the energies. - 6 0 -In the case of the second moment i t is to be noticed that the value oo' (= 0.877 oo) obtained in Eq. [4.42] by maximizing the over-lap does not make M2 i n f i n i t e , giving, in fact, M2 = (Y^p) 2 0.332 ^ i 2 o o 2 . Lane, Thomas, and Wigner (1955) have shown that any local complex potential w i l l lead to a strength function with an i n f i n i t e second moment. The present work has dealt only with local real potentials. Moreover, Vogt (1957) showed that the total cross sections are largely insensitive to a change in the form of the strength function provided the f u l l width at half maximum, W, is kept constant, and, further, that the second moment provides an upper limit on W, [4.75] 1/ J^l > W . This means that the width of the distribution of the C2, for the X ;cp four particle chain w i l l be less than 2.3 "Kco. This is indeed the case. Since the maximum of the C2, occurs at C 2 = 0.973, the sum rule on A;cp oo the squared overlaps does not permit an overlap to exist for which C2, = C 2 /2. Thus the width in this case is effectively zero. A ;cp oo The next section, 4-6, presents an alternative way of obtaining the interaction between the two clumps in the single particle system. The new method to be presented, closer to the s p i r i t of the optical model, w i l l turn out to lead to results much the same as those which were obtained in the previous sections by simpler methods. 4-6 The Model Interaction as an Average Over Exact Interactions The single particle system is here being conceived as a model for the compound system which can be called the "exact system". When the particles forming the compound system are torn apart into the -61-two clumps of the s i n g l e p a r t i c l e system, i n d i v i d u a l i n t e r a c t i o n s between p a r t i c l e s which now f i n d themselves i n d i f f e r e n t clumps are sundered. In t h e i r places one has a s i n g l e i n t e r a c t i o n between one clump and the other clump as a whole. I t i s p h y s i c a l l y s e n s i b l e that the i n t e r a c t i o n between the clumps - the model i n t e r a c t i o n - be taken as the average over the i n d i v i d u a l i n t e r a c t i o n s d i s s o l v e d i n the clumping process, i . e . as an average over exact i n t e r a c t i o n s . This i s , i n f a c t , the p r e s c r i p t i o n suggested by Michaud and Vogt (1972) f o r obtaining the model i n t e r a c t i o n s at each stage of the hierarchy of models described i n Chapter 1. I t i s equivalent to Hartree's s e l f - c o n s i s t e n t method, although Hartree's method leads, i n a d d i t i o n , to a renormalization of the i n d i v i d u a l i n t e r a c t i o n s w i t h i n each clump. While t h i s method of obtaining model i n t e r a c t i o n s has been prescribed for a long time (see, f o r e.g. Lane, Thomas, and Wigner 1955) i t has not been w e l l i l l u s t r a t e d i n the l i t e r a t u r e . I t i s not d i f f i c u l t to understand why t h i s should be so: r e a l i s t i c nuclear wavefunctions and p o t e n t i a l s are too complicated to make the c a l c u l a t i o n f e a s i b l e . Here, again, the l i n e a r harmonic o s c i l l a t o r systems come to e x h i b i t t h e i r main v i r t u e : c a l c u l a b i l i t y . In the present problem, then, the exact system i s represented by V(*lx) V(s^ V0Jt,) Ghtm—0~^WP—Qr-om—© where the i n t e r p a r t i c l e distance s ^ = r ^ - r ^ , and the i n t e r p a r t i c l e i n t e r a c t i o n ^(s^..) = h k s 2 ^ . The model system i s represented by -62-where the distance between the centres of mass of the two clumps i s [4.76] S l t | V = ^ _ r^r; _ and V(sj2 3k) - V(s) is the interaction to be determined. In the previous sections of this chapter V(s) was simply assigned a harmonic oscillator behaviour with spring constant k'. In this section V(s) is obtained as the average of the interactions i t replaces - in this case V(s23). This averaging is to take place over the internal motion of each of the two clumps, that i s [4.77] ~- ^ V ( S r t ) ut($,0 Uj($ i t) JSn J h i -co where the internal wavefunction of a clump, u 0 ( s ^ j ) > is t n e ground state wavefunction of particles i and j interacting through V ( s ^ ) . That i s , [ 4 . 7 8 ] u . ( s ^ . e , r ( -The task now is to calculate V(s) using Eq. (4.77]. It w i l l not necessarily turn out to be a harmonic oscillator potential as has been assumed heretofore. It is to be noted that in the integral [4.77] the coordinate -63-s ^ i s not independent of S12 A N U &3k> but they may a l l conveniently be expressed in terms of the normal coordinates y_ using the trans-formation [4.20] which gives [4.79] 1 3 ^ **> '* A " S With these coordinates the integration is straightforward and gives [4.80] VCS^ * i - k s " ^ One, f i r s t of a l l , notices immediately that the resultant interaction i s indeed a harmonic os c i l l a t o r interaction. This gives support to the assumption of harmonic oscillator interactions made i n the previous sections. Secondly, one notices that the model spring constant k' = k, a value intermediate between k' = 0.769 k obtained in Eq. [4.43] by maximizing the ground state overlap and k' = 1.437 k obtained in Eq. [4.48] matching the energies. Since i t was seen i n Section 4-4 that the overlap is hardly affected by this change in k', the overlap can s t i l l be considered to be maximized with the choice k' = k. Thirdly, the interaction [4.80] has an added constant energy fico/4/2 which is precisely the value needed to insure that the strength function i s centered around the single particle energy. This is apparent by considering the f i r s t moment obtained in Eq. [4.70]. Setting a)' = w, one sees that the quantity ^ 01/4/2" in the single particle Hamiltonian, H', w i l l reduce the residual interaction, V = H-H', by just -64-th e right amount to make Mj, vanish. The vanishing of the f i r s t moment of the strength function has been shown by Lane, Thomas and Wigner (1955) to be a general property of model interactions obtained by the averaging procedure of this section. The present example has the ab i l i t y to show this property in action. The present chapter has shown by i l l u s t r a t i o n how the linear harmonic oscillator chains are adapted, because of their analytic simplicity, to test the calculational tools of nuclear physics. How-ever, not much more than what has been done in this chapter with the four particle system can be done without resorting to computer assistance for handling the matrices. The rest of this work w i l l consist of computer assisted calculations which begin in the next chapter with an examination of the effect of increasing the number of particles, N. -65-CHAPTER 5 MODEL INTERACTION FOR SYSTEMS WITH N PARTICLES The previous chapter showed the method by which overlaps are to be c a l c u l a t e d , and i l l u s t r a t e d i t by c a l c u l a t i n g the overlap of the ground states of two systems of four p a r t i c l e s . The best model wavefunction Is that s i n g l e p a r t i c l e wavefunction which has the la r g e s t overlap with the compound system wavefunction. I t was found i n Chapter 4 that, with four p a r t i c l e s , the best s i n g l e p a r t i c l e wavefunction i s a very good model f o r the compound system wavefunction - i n f a c t , too good: the strength of the sin g l e p a r t i c l e states resides 97% i n a s i n g l e compound s t a t e , and not enough strength i s l e f t i n other compound states to permit a determination of the shape of the strength function. The present chapter inquires whether increasing the number of p a r t i c l e s has any e f f e c t i n d i l u t i n g the strength of the si n g l e p a r t i c l e system. Let the t o t a l number of p a r t i c l e s be N (N = 2, 4, 6, 24). For each value of N a compound system i s formed and sol u t i o n s obtained i n the manner indicated i n Section 4-1. Next, as i n Section 4-2, the s i n g l e p a r t i c l e system i s formed by d i v i d i n g the p a r t i c l e s into two clumps with N/2 p a r t i c l e s each. The two clumps are allowed to i n t e r a c t through a harmonic force with spring constant k' i n terms of which the solutions are found. Following the procedure of Section 4-3 the overlap of the two ground state wavefunctions i s found. This requires that both the normal coordinates x of the compound system and the normal coordinates y_ of the sin g l e p a r t i c l e system be expressed i n terms of the i n t e g r a t i o n coordinates z^ , described i n Section 4-3, which permit the (N-l)-dimension-a l overlap i n t e g r a l to be w r i t t e n as a product of (N-l) one-dimensional i n t e g r a l s . The overlap depends on the r a t i o of spring constants k'/k, - 6 6 -and i t i s maximized numerically with respect to this ratio. The results are shown in Figure 5-1 which displays, for each value of N, the maximum ground state overlap and the corresponding value of the ratio k'/k. It i s seen from Figure 5-1 that although the maximum overlap decreases gradually as the number of particles, N, i s increased, even for N=24 i t has only decreased to C 2 q = 0.829 (corresponding to k' = 0.195 k). However, although the strength of the single particle ground state seems to s t i l l be rather concentrated on one compound state (there being only 17% of strength l e f t to be distributed among other states), with higher N the other states appear much closer together. In particular, for N=24 the density of states i s higher and the excited states are closer to the ground state than for N=4. This i s apparent from the expression for the total energy of a linear harmonic oscillator chain. [5.i] E = I h + " '0 where the normal mode frequencies for a compound system are given i n Eq. [ 3 . 9 ] as [5.2] (J, = ( * si'fl TjA The higher density of states suggests that i t is possible that the distribution of the remaining 17% of the strength give an indication of the shape of the overlaps C2t__ as a function of energy. Further, i f one considers an excited state of the single particle system, i t s higher energy w i l l locate i t in the midst of a larger density of compound states (the density of states increases as the energy goes up). At these higher energies there is no reason to • - . \ \ — % _ ( — • - - _ « C o o • ? 10 ,% IH It, IS 3.0 Fig. 5-1. - The maximum overlap, C 2 , between the ground states of the single particle and compound systems, and the corresponding single particle spring constant ratio, k'/lc, as functions of the number of particles, N. -68-expect that there exists a compound state which w i l l monopolize most of the single particle strength as in the case of the ground state. This should permit the calculation of enough closely spaced overlaps C2___ t o give a f a i r indication of the shape of their distribution. To correspond most closely to the number of particles involved i n the 1 2C+ 1 2C reaction of interest, the remaining chapters w i l l r e s t r i c t themselves to systems with N=24. The next chapter w i l l report the calculations of overlaps for N=24 which this chapter suggested as worth-while: overlaps of the ground state of the single particle system with excited states of the compound system; and overlaps of an excited state of the single particle system with excited states of the compound system. -69-CHAPTER 6 OVERLAP INTEGRALS FOR A SYSTEM WITH 24 PARTICLES As indicated i n Chapter 1, in the giant resonance model of Lane, Thomas, and Wigner (1955), the strength function is proportional to C'X'cp /D where ^ . C p -*-s t n e average overlap obtained by averaging over a few neighboring values of X, and D is the average spacing between the energy levels E... The quantity of interest i s thus [6.1] X> A E . / # AB where # indicates the number of states X included in the sum, i.e. i t i s the number of states over which the average i s done; and AE is the averaging interval which must be large enough to average out local fluc-tuations, and yet not so large that i t obscures the shape of the strength function. This chapter determines J c 2 /AE as a function of compound ^ X; cp system energy E^ for two different values of cp. In Section 6-1 ij> i s taken as the ground state of the single particle system. In Section 6-2 ^ Cp i s taken as the excited state of the single particle system having 2 quanta of excitation i n the 12th normal mode. 6-1 Single Particle System in i t s Ground State It i s well to remember that the integral over a l l space of the product of an even function with an odd function vanishes. A harmonic oscillator wavefunction is the product of Hermite polynomials with an exponential. The exponential i s an even function of the coordinates. A -70-Hermite polymial i s an even or odd function of Liu- coordinates depending on whether i t i s of even or odd order r e s p e c t i v e l y . The order of the Hermite polynomial i s equal to the number of qiutnta i n the normal mode with which the Hermite polynomial i s associated. Thus the ground state wavefunction w i l l be an even function of the coordinates and w i l l have zero overlap with any odd wavefunction. In p a r t i c u l a r , the ground state of the s i n g l e p a r t i c l e system w i l l have zero overlap with a l l compound system states containing a t o t a l of 1, 3, 5 ... quanta of e x c i t a t i o n . overlap of the s i n g l e p a r t i c l e ground state with the compound ground state. This was a c t u a l l y done i n Chapter 5 giving C 2 = 0.829. Next come the compound states with 1 quantum, which are odd functions, and, thus, a l l t h e i r overlaps with the s i n g l e p a r t i c l e ground state vanish. T h i r d l y , one considers compound states with 2 quanta. modes there are 23 states having 2 quanta i n the same normal mode, and there are 23 x 22 = 253 states having 1 quantum i n each of two d i f f e r e n t normal modes. These two types of states with 2 quanta must be kept separate f o r the purposes of c a l c u l a t i n g the overlap i n t e g r a l . The reason f o r t h i s r e s t s i n the transformation to the i n t e g r a t i o n coordinates £ which are described i n Chapter 4. If 2 quanta are both i n the Zth normal mode, the Hermite polynomial involved i s To proceed systematically, then, one begins by c a l c u l a t i n g the Since a l i n e a r system of 24 p a r t i c l e s has 23 i n t e r n a l normal 21 but i f one quantum i s i n the Zth mode and another i s i n the j t h mode, then -71-second case the contribution to the integral comes from X ^ Th u s a different computer program i s needed in each case to handle the elements of the matrix X together with the proper constants in the Hermite poly-nomials. Carrying out these computations i t turns out that out of a possible 276 overlaps only 78 do not vanish (the reasons for this w i l l be explored in Appendix C). The results are gathered in Figures 6-1, 6-2 and 6-5. Next in complexity are the compound states with 3 quanta. These, as explained above, are odd functions of the coordinates and a l l their overlaps with the single particle ground state w i l l vanish. Adding one more quantum to the compound system gives the 4 quanta states. In this case there are five types of states (equal to the number of partitions of 4) depending on whether the 4 quanta are a l l in the same mode or distributed among two, three, or four modes. Since there are 23 modes i t i s easy to show that there are 14,950 compound states having 4 quanta. However, only 1,365 of these w i l l have a non-vanishing overlap with the ground state of the single particle system (see Appendix C). Carrying out each of the five types of integrals separately gives the results gathered in Figures 6-3, 6-4, and 6-5. The compound system states having 5 quanta w i l l , again, since they are odd functions of the coordinates, have no overlap with the single particle ground state. However, adding up at this point the -72-Fig. 6-1. - The number, m, of 2-quanta compound states having non-zero overlap with the single particle ground state as a function of the compound state energy, E^. The energy E^ is in units of -Rco above the compound ground state energy. Fig. 6-2. - The average overlap squared, ^C 2 ), between 2-quanta compound states and the single particle ground sBIte as a function of compound state energy, E^. The squared overlaps have been multiplied by 10h. The energy units are the same as in Figure 6-1. -73-100 Ex Fig. 6-3. - The number, m, of 4-quanta compound states having non-zero overlap with the single particle ground state as a function of the compound state energy, E^. Conventions are the same as in Figure 6-1. 3 h Fig. 6-4. - The average overlap squared, <(c2. ^ , between 4-quanta compound states and the single particle ground sfUte as a function of compound state energy, E . Conventions are the same as in Figure 6-2. l.lt 4+a.fe or I f •03 Fig. 6-5. - The strength, Jc 2/AE, of the single particle ground state in the compound states with 0, 2, and 4 quanta as a function of compound state energy, E^. The energy E^ is i n units of -fiu) above the compound ground state. AE i s the energy interval over which the overlaps C 2 are summed. -75-distribution of strength of the single particle ground state among com-pound states shows that 82.9% went to the 0 quanta compound state 13.3% went to the 2 quanta compound states 2.9% went to the A quanta compound states which accounts already for 99% of the strength. Therefore no more effort w i l l be spent in pursuing higher excited states of the compound system -their effect on the strength function w i l l be small. Looking at the results in Figures 6-1 through 6-5, i t i s apparent that there are two distinct factors contributing to the shape of the strength function: the density of states and the average overlap. Figure 6-5 seems to indicate that although the gross structure of the strength function i s , as considered by Lane, Thomas, and Wigner (1955), a reflection of the behaviour of the overlaps C 2 , the density of states X;cp' associated with different number of quanta might be responsible for intro-ducing some systematic variations into the shape. The present case (where the single particle system is i n i t s ground state) does not allow stronger conclusions for two reasons: a disproportionate amount of the strength resides in a single compound state, and a l l the available com-pound states are of higher energy than the single particle state being considered which makes vi s i b l e only the high energy side of the strength function. To overcome these limitations Section 6-2 considers the spread of an excited single particle state into compound states. 6-2 Single Particle System in an Excited State The strength of the single particle ground state in the compound states resides 83% in one compound state; the remaining strength being distributed among higher energy compound states. Since a more uniform -76-distribution of the strength would serve to better show off the shape of the strength function one ought to begin with a higher energy single particle state which because of i t s placement among a higher density of compound states can be expected to distribute i t s strength more equitably. A preliminary calculation of the overlaps of the 1-quantum single particle states with compound states shows that 50% of the strength is s t i l l concentrated on one compound state. However, the trend i s promising, and one is led to examine next the strength of a 2-quanta single particle state. To choose this state so that i t does not occur near either end of the distribution of 2-quanta compound states, one places the 2-quanta in the 12th normal mode of the single particle system. This state w i l l be called the ni 2(2) single particle state, and this section examines the distribution of i t s strength among compound system y states. Again, one can, at the outset eliminate from consideration the compound states with an odd number of quanta since these are odd functions of the coordinates and w i l l have zero overlap with the ni 2(2) single particle state which i s an even function of the coordinates. To proceed systematically, then, one begins by taking the overlap of the n12(2) single particle state with the compound ground state which gives C 2 = 0.0000218. Next one considers the compound states with 2 quanta. As mentioned in Section 6-1, there are 276 of these and there are two types of overlap integrals depending on whether both quanta are in the same mode or i n different modes. However, only 91 of these have non zero overlap with the nj 2(2) single particle state. The largest overlap gives C = 21% which shows that indeed, as expected, the strength i s not con--77-centrated mostly on one state. The rest of the 2-quanta compound states account for an additional 61.5% of the strength. It w i l l be convenient to c a l l the strength of an n-quanta single particle state in the m-quanta compound states S^m n^. Thus the strength of the ni2(2) single particle state in the 2-quanta compound states i s S^°\ The strength S^°^ is shown in Figure 6-6. The 4-quanta compound states number 14,950 and involve five different types of overlap integrals as w i l l be recalled from Section 6-1. In the present case only 1,807 states have non-zero overlap with the n12(2) single particle state, and they contribute 15.6% to the strength. (2) The strength S ' is also shown in Figure 6-6. The 6-quanta compound states can be shown to number 376,740, and to involve eleven types of overlap integrals (equal to the partitions of the number 6). However, i t is perhaps not necessay to do these since 98% of the strength has already been exhausted, and thus the contribution »le. ,(4) (4) of S to the shape of the strength function w i l l be almost negligible. Nevertheless i t i s possible to obtain an indication of the shape of S' (2) as follows. Plotting, as in Figure 6-7, S , the strength of the ni 2(2) single particle state in the 4-quanta compound states (from (2) 1Figure 6-6), and S , the strength of the single particle ground state in the 2-quanta compound states (from Figure 6-5), one sees that they are very similar to each other. Although the number of states involved i n each case i s quite different - 1807 and 78, respectively - their contri-butions to the total strength are not dissimilar - 15.6% and 13.3% respectively. Likewise, the strength of the nj 2(2) single particle state in the 2-quanta compound states i s 82.5% of the total, while the strength of the single particle ground state in the compound ground state i s 82.9% -78-t. ^ .t<>1 .of AE .01 .001 . copy Fig. 6-6. - The strengths, £c2/AE, of the n 1 2(2) single particle state in the compound states with 2, 4, and 6 quanta as a function of compound energy, . The three types of strength are called S ( ° ) , S ( 2 ) , and S(4) respectively. s(4) has not been calculated but obtained by approximating i t with the strength of the single particle ground state in the 4-quanta compound states. E^  and AE are as described in Figure 6-5. The energy of the ni 2(2) single particle state is 2.435 lioj above the ground state. .ot . 0 3 .01 h r i . J •CO • oi -a. .Fig. 6-7. - Comparison of the strength SK2} of the ni 2(2) single particle state in 4-quanta compound states with the strength s( 2) of the single particle ground state in 2-quanta compound states. S( 2) i s obtained from Figure 6-5. s( 2) is obtained from Figure 6-6 but has been shifted back along the energy axis by 2.435 -fiio - the energy of the n 1 2(2) single particle state above the ground state. -80-of the total. Thus, one can consider the strength function to be approximately independent of the state of the single particle system. In (4) particular, one can get a good indication of the shape of S , the strength of the ni 2(2) single particle state in ^ the 6-quanta compound (4) » states, by looking at S , the strength of the single particle ground state in the 4-quanta compound states. Thus, Figure 6-6 also contains (4) S obtained in this way. One of the most intriguing features of Figure 6-6 is i t s terraced shape which indicates that the contributions to the strength come from different populations of states. In a given energy interval one of the strengths S ^ , S ^ , or S ^ dominates the others. To see whether this competition between two or more populations has been seen in nuclear data, the literature on neutron cross sections was scanned for any evidence of deviations from the expected Porter-Thomas distribution (see Chapter 8) of the level width fluctuations. Rohr and Friedland (1967) reported just such a discrepancy in the neutron scattering on 5 1V. Some enumeration of shell model states in the compound nucleus was carried out to see i f alpha-like quartet excitations, which compete favorably i n energy with single particle excitations (Arima, G i l l e t , and Ginocchio, 1970), would yield a second population of states accounting for the observed width fluctuations. A further search of the literature, carried out at the same time, revealed, however, that S t i e g l i t z , Hockenbury, and Block (1971) had repeated the experiment and shown that proper accounting of p-wave resonances removes any discrepancy with the Porter-Thomas distribution. This question, which is only incidental to the main theme of this thesis, was not pursued further, but remains a subject for future work. -81-Figure 6-6 provides, in histogram form, a distribution of the average of the C2/D. As described in Chapter 1, this must be compared to a Lorentzian shape. Having done the overlaps, the C 2 are now known. Thus, the next step in the procedure is to determine the function D(E), the average spacing between states. This forms the subject of Chapter 7. -82-CHAPTER 7 THE DENSITY OF STATES In Section 6-2 the strength of the ni 2(2) single particle state in compound states was computed by evaluating the overlaps # and summing them in an energy interval AE to obtain £ c 2 - c /AE as a function of compound system energy E^. In the limit where the summing interval contains only one non-zero overlap the quantity ^C^^/AE becomes C 2 /D, where D is the average spacing between compound states x-\ having A j C p A non-zero overlap with the single particle state ^p- A curve f i t t e d to the quantities C2.cp/D (as a function of E^) w i l l average out the fluctuations i n C^^/B, and, i f i t i s good f i t , w i l l reproduce the shape of the histogram of Figure 6-6 which provides such an average directly. Before this strength function can be obtained by f i t t i n g a curve to the quantities C^.^/D (which w i l l be done in the next chapter) each one of the overlaps C 2 > cp(E^), which was evaluated in the previous chapter, must be divided by the corresponding average spacing D(E^). This chapter concerns i t s e l f with obtaining D(E^). As a f i r s t approximation one could take [7.i] D eg = E x + > ~ E x-However, an average taken in this way would be a very local average, and the resulting spacing D would transmit i t s local fluctuation to the strength C? /D. At the other.extreme one could take the total energy A ;cp interval containing non-zero overlaps and divide by the total number of overlaps to obtain -83-[7.2] which, while certainly providing an overall average, would give a con-stant spacing insensitive to i t s dependence on E^. Of course, one may also take an intermediate position and average over an energy interval large enough to contain a few but not a l l of the overlaps. This has, i n fact, already been illustrated in Figures 6-1 and 6-3 which indicate that not only both of the above objections are s t i l l applicable, but, in addition, one would be introducing a certain amount of unwanted structure into the strength function due to the discontinuous nature of the histo-grams. the distribution of states with a continuous function. The shape of the histograms 6-1 and 6-3 themselves suggest that the functional dependence of the density of states could be Gaussian. In fact, Wong and French (1972) have shown that two-body interactions do lead to a Gaussian partial level density, and Wong and Wong (1973) have shown that this agrees with shell model calculations. Thus for the present case the Gaussian form w i l l be adopted for the level density. P 3 i s a parameter which permits the distribution to be skewed in order to accommodate the effect of a small sample size. Using a least squares procedure to f i t the density D - 1(E) of Eq. [7.3] to the distribution of the 91 levels of the 2-quanta compound A l l the above shortcomings may be circumvented by approximating [7.3] -84-system having non-zero overlap with the ni 2(2) single particle state gives [7.4] P! = 2.80 P2 = L U p 3 = 0.023 pit = 63.8 for the parameters to be used in Eq. [7.3]. This, then, is a l l the i n -formation which is needed to tabulate the quantities C2/D as a function energy which permits, in the next chapter, a Lorentzian curve to be fitt e d to them. -85-CHAPTER 8 THE SHAPE OF THE STRENGTH FUNCTION The Gaussian shape adopted in Chapter 7 for the density of states D - 1(E) permits the strength C? /D to be obtained as a function of X; cp energy. In the giant resonance interpretation of Lane, Thomas, and Wigner the average of this function has, as discussed in Section 1-2, a Lorentzian shape. In order to put in evidence to what extent <fc2 //D is indeed N X;cp' Lorentzian, this chapter w i l l f i t a Lorentzian curve to the strength of the ni2(2) single particle state in the 2-quanta compound states. These states account for 82.5% of the strength and are represented in Figure 6-6 as S^°^. Figure 6-6 serves, as well, to show that only S^°^ (and not g(o) + s(2) g(o) + s(2) + g(4) ) w i l l b e s u i t a b l y r ep resented by a Lorentzian. The Lorentzian curve f i t t e d to S^°^ w i l l be an average over the fluctuations of the quantities C2/D. In order that this average be indeed a representative average the nature of the fluctuations of C2/D about their (local) mean must be known beforehand. Porter and Thomas (1956) have argued that the fluctuations In the reduced widths Y? (which are Ac proportional to C 2. ~ s e e Ecl« [1'13]) obey a chi-squared distribution with one degree of freedom. That i s , they are described by a probability density function [8.1] f(x} = j ~ x e Win-where x = Yxc/v'xV * I n n u c l e a r Physics the distribution [8.1] i s called a Porter-Thomas distribution. -86-To determine whether i n fac t the f l u c t u a t i o n s of the C2/D obey a Porter-Thomas d i s t r i b u t i o n reference i s made to the r e s u l t s c o l l e c t e d i n Figure 6-6. Each of the C2/D i s divided by the l o c a l average i n each of the ten boxes i n the histogram S^°^ of Figure 6-6. C a l l i n g these quantities x, one pl o t s i n Figure 8-1 t h e i r frequency d i s t r i b u t i o n t o -gether with a Porter-Thomas d i s t r i b u t i o n . Comparison of the two d i s t r i -butions i n Figure 8-1 indicates that the Porter-Thomas d i s t r i b u t i o n does represent the f l u c t u a t i o n s o f the C2/D. Now that the d i s t r i b u t i o n of the f l u c t u a t i o n s of the C2/D are known, one can proceed to f i t a curve through t h e i r mean. A standard method f or f i t t i n g a curve i s the maximum l i k e l i h o o d method. The l i k e l i h o o d f o r obtaining a set of points i s defined as the product of the p r o b a b i l i t y d e n s i t i e s evaluated at the given points. The curve which best estimates the given points i s that f o r which the l i k e l i h o o d (or, equivalently, the logarithm of the l i k e l i h o o d ) i s maximized. I t turns out that i n connection with a Porter-Thomas p r o b a b i l i t y density, as i n Eq. [8.1], the best estimate of a f i t obtained by the maximum l i k e l i h o o d method coincides with a good v i s u a l f i t , i . e . i f there i s no sc a t t e r the points a l l l i e on the curve. This i s not t r i v i a l but i t i s a character-i s t i c of the chi-squared d i s t r i b u t i o n s of which the Porter-Thomas i s a s p e c i a l case. The proof i s as follows. The d i s t r i b u t i o n [8.1] used by Porter and Thomas belongs to the class of chi-squared d i s t r i b u t i o n s [8.2] F (*) olx — (xj x e J x where n i s the "number of degrees of freedom", and x i s the r a t i o of a -87-v » X o oi -a £ D C <H I-S O 40 6 4 •5 30 s: 1 0 i m Fig. 8-1. - The distribution of C2/D about 10 local averages. The quantity x i s defined as the ratio of C2/D to i t s local average. The C2/D are in this case the strengths of the n i 2 ( 2 ) single particle state in the 2-quanta compound states. The dashed lines re-present a Porter-Thomas distribution. -88-a quantity to i t s average (say, x = y/y)• Thus Now, given for y a specific value y°, what is the best estimate of y? The likelihood L(y) that a particular choice of y leads to the given value y° i s defined by [8.4] L (V) = f'(^) Obtaining the best estimate of y by maximizing the logarithm of L(y) with respect to y gives [8.5] y y° which is independent of the number of degrees of freedom n. This says that a curve describing the behavior of y w i l l pass through the [single] point y° as one would expect from a good visual f i t . Having established the appropriateness of the maximum l i k e l i -hood method for the present case, one now wants the parameters,T and E Q of the Lorentzian curve [8.6] y ( E ) -that describes the average behaviour of the strengths C2/D of the ni£(2) single particle state in the 2-quanta compound states. The quantities C2/D were found in Chapter' 7 (there were 91 of them), and here they are -89-given the symbol y° (E.. ) . The normalization constant N i n Eq. [8.6] i s A designed to allow f o r the f a c t that S^°^ contains only 82.5% of the strength and to minimize the e f f e c t of v a r i a t i o n s i n the average l e v e l spacing D. I t i s chosen such that [8.7] Z y = I y ° ( ^ Now, the l i k e l i h o o d f o r obtaining the observed value y°(E^) given the predicted or mean value y(E^) i s , according to the Porter-Thomas d i s t r i -bution [8.1], given by [8.8] ~- l^y°(£x) y(*>) « f ( y ° ( ^ / v ( ^ ) ] " / i . If there are 91 points, [numerical] maximization of [8.9] or = Zl ^ Lx with respect to.T and E q gives [8.10] T ~- 0. 09/(f E 0 This, then, i s the r e s u l t of the maximum l i k e l i h o o d method. In Figure 8-2 the Lorentzian strength function with the parameters [8.10] Is com-pared to the histogram f o r the average strengths previously shown i n Figure 6-6. Several features are apparent from Figure 8-2. Although the Fig. 8-2. - The strengths of Figure 6-6 compared to a Lorentzian strength function with parameters, T = 0.0918 -nco and E q = 2.493 "KOJ obtained through a maximum likelihood procedure. -91-Lorentzian f i t is adequate near the central (E s; 2.5 -fioo) region of the o strength, in regions farther from the centre than about 1 "fiio on either side the f i t is noticeably deficient. However, the next lowest ( n i 2 ( l ) ) and next highest (ni 2(3)) single particle states occur at 1.2 -hoi and 3.7 iico, respectively, so the Lorentzian does describe the strength i n the region where one would expect i t (see Section 1-2) to be applicable. Interestingly, the deviations of the strength function from the Lorentzian differ on the high and low energy sides: in the higher energy region of the curve the Lorentzian underestimates the strength, while on the lower energy region the Lorentzian overestimates the strength. The reason for this i s that although a single particle state w i l l preferentially populate a certain type of compound state (S^°^ carries 82.5% of the strength), the strength i s in principle spread among a l l possible compound states (subject, of course, to the conservation laws), and there are many more states at higher energies than at lower energies. Further, since i n actual nuclei, as in harmonic oscillator systems, the total density of states increases with increasing energy, one should be prepared, when extrapolating cross sections with Lorentzian strength functions down to lower energies, to overestimate the cross section. This w i l l be par-ticularly significant i n the region below the lowest single particle resonance. The f i t t i n g i n this chapter of the Lorentzian curve to the strength function completes, together with the discussions of Chapters 2 to 7, the machinery needed for the discussion of strength functions proposed in the introduction. The last four chapters have dealt with i t s application to the spread of a state of the single particle system into states of the compound system. Its application to -92-the spreading of states in a hierarchy of systems w i l l now be considered in Chapter 9. -93-CHAPTER 9 A HIERARCHY OF SYSTEMS The promise made in the introduction - to study the behaviour of the strength function i n the hierarchy of harmonic oscillator systems which are the counterparts of the three stages of the 1 2C+ 1 2C reaction -has yet to be f u l f i l l e d . However, the method to be used is that which has been developed after the introduction through the chapters which preceded the present. This chapter, then, w i l l not re-elaborate on the method, but w i l l indicate how i t is to be extended to encompass the hierarchy, and w i l l succinctly quote the results obtained therefrom. The hierarchy i t s e l f and the choice of spring constants i s presented i n Section 9-1. Section 9-2 contains the evaluation of the strength functions. As re-marked in Section 4-3, because of the properties of harmonic oscillator wavefunctions, no numerical integration i s required, but computer assistance was employed in diagonalizing matrices, gathering results, and f i t t i n g curves. At the speed of the IBM 370 a total of 1.2 hours of central processing unit time were required for these tasks. A single overlap can take up to about 40 seconds, but subsequent overlaps of• the same type need only a few additional seconds each. 9-1 The Hierarchy The three stages through which Michaud and Vogt (1972) have de-picted the 1 2C+ 1 2C reaction were illustrated in Figure 1-5. The three harmonic oscillator systems which correspond to the three stages were i l -lustrated i n Figure 1-6, and called the single particle system, the doorway system, and the compound system, respectively. It should be re-marked that the single particle system of the present chapter i s not the - 9 4 -same as the single particle system used in Chapters 5 to 8 . The single particle system used then was simply a division of the compound system into two sections of 1 2 particles each. The single particle system used in this chapter reflects the clustering properties of 1 2C described i n the introduction. With this observation one sees that the systems used in this chapter form a natural hierarchy in the sense that, as one pro-ceeds from the single particle system through the doorway system to the compound system, at each step an average interaction is replaced'by a more exact interaction - one closer to the compound system. The idea, then, i s to study how a single particle state is spread among doorway states, and how one of these doorway states is in turn spread among com-pound states. The quantity which reflects this spreading i s the strength function, and i t w i l l have to be determined, as was done in Chapters 6 , 7 , and 8 , for each step of the hierarchy. One begins by solving each system one at a time. Thus, f i r s t , using t r i a l values for k', the spring constant of the interaction between quartets, and for k", the spring constant of the interaction between dodecaplets, the classical Hamiltonians are written down. Next, they are diagonalized and written i n terms of the normal coordinates which provides at the same time the normal mode frequencies. These frequencies depend on the choice of k' or k". Having obtained the classical Hamiltonians in terms of the normal coordinates, the quantum mechanical Hamiltonians follow immediately together with their solutions which consist of products of 2 3 harmonic oscillator wavefunctions similar to those in Eq. [ 4 . 1 4 ] and Eq. [ 4 . 2 6 ] . The next step is to determine k' and k" by maximizing the overlap C q o between the ground states. This is done numerically by varying the t r i a l values of k' and k", and doing the integration using the integration coordinates z described in Section -95-4-3. The result for the maximum overlap between the ground state of the compound system with the ground state of the doorway system i s [9.1] C 2_ c = 0.6978 corresponding to a value k' = 0.4797 k. Similarly, the maximum overlap between the ground state of the doorway system and the ground state of the single particle i s [9.2] C 2 . = 0.9474 s-d corresponding to a value k" = 0.2890 k. These values of k' and k" permit the normal mode frequencies to be written down as is done i n Table 9-1. The Hamiltonians, which w i l l not be used further in this chapter, are, to avoid repetition, written down ex p l i c i t l y i n Appendix C where their symmetry properties are investigated. A comparison of the ground state overlaps[9.1] and [9.2] already reveals something about the hierarchy. They make i t clear that as far as the ground states are concerned the single particle system i s 95% l i k e the doorway system, while the doorway system i s only 70% l i k e the compound system. It w i l l be recalled from Chapter 1 that i n the 1Z-C+ C reaction the intermediate structure in the cross section has widths an order of magnitude smaller than the single particle resonances. This is interpreted to mean that the doorway structure lasts longer than the single particle structure because, in terms of the residual interactions (see Eq. 11.24] and Eq. [1.25]), the doorway is more like the compound system than l i k e the single particle system. This i s in contradistinction -96-TABLE 9-1 THE NORMAL MODE FREQUENCIES OF THE THREE SYSTEMS IN THE HARMONIC OSCILLATOR HIERARCHY [All frequencies are in units of to (=/k/m) ] Normal Mode Compound Doorway - Single Particle Number System System System 1 0.0 0.0 0.0 2 0.131 0.179 0.219 3 0.261 0.346 0.346 4. 0.390 0.490 0. 346 5 0.518 0.600 0.600 6 0.643 0.669 0.600 7 0. 765 0.765 0.765 8 0.885 0.765 0.765 9 1.000 0.765 0.765 10 1.111 0. 765 0.765 11 1.218 0. 765 0. 765 12 1.319 0.765 0. 765 13 1.414 1.414 1.414 14 1.504 1.414 1.414 15 1.587 1.414 1.414 16 1.663 1.414 1.414 17 1.732 1.414 1.414 18 1.794 1.414 1.414 19 1.848 1.848 1.848 20 1.894 1.848 . 1.848 21 1.932 1.848 1.848 22 1.962 1.848 1.848 23 1.983 1.848 1. 848 24 1.996 1. 848 1.848 -97-to what has just been found for the ground states of the harmonic os-ci l l a t o r hierarchy. Will this behaviour of the ground states carry over into the strength function for excited states? Section 9-2 examines, thus, the strength function beginning with an excited state of the single particle system. 9-2 Spreading Widths for Excited States Wanted in this section are two spreading widths: f i r s t the spread of an excited single particle state into doorway states w i l l be calculated, and then the spread of one of these doorway states into compound states w i l l be calculated. In each case the spreading width w i l l be given, as in Chapter 8, by the width of the Lorentzian f i t to the strength function. Putting 2 quanta i n the second normal mode of the single particle system creates the n 2(2) state. When this state is spread among 2-quanta doorway states i t i s found that the n 2(2) doorway state acquires 58% of the strength of the n 2(2) single particle state. As was seen i n Section 6-1 this disproportionate concentration of the strength does not show off the strength function to best advantage. Therefore one must begin with a higher excited state. Consider the n 2(4) state of the single particle system which contains 4-quanta i n the second normal mode. As w i l l be seen in Appendix C, because of the degeneracies evident in Table 9-1, not a l l the 14,950 4-quanta doorway states w i l l have a non-zero overlap with the n 2(4) single particle state. The results in histogram form can be seen in Figure 9-1. To f i t this strength function with a Lorentzian one follows the F i g . 9-1. - The strength, S , of the n 2(4) s i n g l e p a r t i c l e state i n 4-quanta doorway s t a t e s , and the strength, S, , of the ^ (4 ) doorway state i n 4-quanta compound states. In each case the best Lorentzian f i t has Been superimposed on the histograms obtained from the calculated overlaps. -99-procedure of Chapter 8: the density of states i s described by a Gaussian functi o n , each overlap i s divided by the value of the average spacing at the corresponding energy; and the r e s u l t i n g C2/D i s f i t t e d with a Lorentzian using the maximum l i k e l i h o o d method. T h e b e s t f i t i s obtained with the Lorentzian parameters [9.3] ;T = 0.301-nu> [9.4] E Q = 1.073 -Kto In Figure 9-1 the r e s u l t i n g curve has been superimposed on the histogram. The next step i s to choose f o r i n v e s t i g a t i o n one of the doorway states i n t o which the n 2(4) s i n g l e p a r t i c l e s t a t e deposited i t s strength. The ^ ( 4 ) doorway state occuring at an energy of 1.958 'ho) above the ground state l i e s part way between the centre and the t a i l of the strength f u n c t i o n j u s t c a l c u l a t e d . I t received 0.01% of the strength which i s neither d i s p r o p o r t i o n a t e l y high nor low, and thus as a t y p i c a l state i s a good candidate f o r the present purposes. Therefore the task now i s to repeat a l l the above c a l c u l a t i o n s beginning with the ^ ( 4 ) doorway s t a t e to study i t s spread i n t o compound s t a t e s . The r e s u l t i n g histogrammed strength function together with the best Lorentzian f i t to i t are also shown i n Figure 9-1. The parameters of the Lorentzian i n t h i s case are [9.5] , r = 0.974 -Rio [9.6] E =• 2.509 fiu -100-Comparison of Eq. [9.3] with Eq. [9.5] shows that the width of the strength function for the ^ (4) doorway state into compound states i s three times larger than the width of the n2(4) single particle state i n doorway states. This confirms what was anticipated by the ground state overlaps in Section 9-1: the doorway is a better model for the single particle system than for the compound system. This i s also almost ap-parent by looking at the normal mode frequencies in Table 9-1. The quartet clustering in the doorway and single particle systems introduce degeneracies that are not present in the compound system. It would thus seem that the harmonic oscillator hierarchy can be more closely aligned with i t s nuclear counterpart by adjusting the spring constants within the compound system such as to mimic the strong internal binding of alpha particles. The ensuing compound system might then look lik e Figure 9-2 where the spring constant k between particles in the same quartet i s much larger than the spring constant k' between particles in different quartets. Fig. 9-2. - A possible modification of the compound system whereby quartet clustering i s reproduced by taking k>>k'. -101-CHAPTER 10 SUMMARY AND DISCUSSION The context within which this work has been performed was de-fined in the introductory chapter. The key question raised was "How does the state of a quantum mechanical system evolve when i t is disturbed?" In Section 1-1 this question was put on a quantitative basis in terms of the overlap integral between two quantum mechanical states. In Section 1-2 i t was shown that the giant resonance interpretation of Lane, Thomas, and Wigner for the nuclear optical model centres around the average behaviour of certain overlap integrals. This behaviour with energy serves to define the nuclear strength function, which i n this case i s expected to have a Lorentzian shape. Why is this shape Lorentzian? Section 1-3 described some early attempts to answer this question. The models used, though elegant, provided l i t t l e connection with the physics. Section 1-6 described some heavy ion reactions whose cross sections have recently revealed some resonance structure intermediate between the widely spaced single particle resonances and the narrowly spaced compound nucleus resonances. The role of this intermediate structure in the calculation of ste l l a r reaction rates was discussed i n Section 1-5. Section 1-6 argued how the alpha particle doorway states conceived by Michaud and Vogt could be responsible for these intermediate resonances. Section 1-7 proposed a hierarchy of one-dimensional harmonic oscillator systems i n terms of which one might not only revive the question of the Lorentzian shape of the strength function, but also investigate quantitatively how quartet doorways give rise to intermediate structure. Chapter 2 discussed several features of the model. One would have liked to begin by choosing for the harmonic oscillator parameters -102-those values which give the best approximation to a r e a l i s t i c nuclear potential. It turned out, however, that the study of the strength function i s independent of the numerical values of these parameters. Harmonic oscillator potentials have been used extensively i n nuclear physics, most importantly in the shell model where special care has to be taken to eliminate spurious states due to motion of the centre of mass of the system. Will these kinds of states affect the present calculation of the strength function? It was shown that since the Hamiltonian of a linear chain can be written as a sum of a term describing the centre of mass motion and a term describing the internal motion, the eigenfunctions of this second term w i l l be the desired internal wavefunctions. Thus, in the present problem, in contrast to the shell model, the elimination of spurious states was straightforward. Both classical and quantum mechanical systems of coupled harmonic oscillators are most easily solved when described i n terms of a set of coordinates called normal coordinates which effectively uncouple the oscillators. The normal coordinates are obtained from the particle coordinates by a linear transformation and Section 3-1 showed that this transformation exists for any possible arrangement of the connecting springs i n a linear chain. Sections 3-2 and 3-3 illustrated this by giving i n closed form the transformation to normal coordinates for two special cases. With a small number of particles i t i s possible to do many calculations by hand. Chapter 4 treated chains with four particles in order to develop the techniques which were later generalized to systems of N particles. Sections 4-1 and 4-2 obtained the wavefunctions for two different arrangements of the four particles. The procedure illustrated -103-how the quantum mechanical solutions are obtained beginning with the classical Hamiltonian, and also how the centre of mass motion gets separ-ated out. Section 4-3 showed how overlap integrals are to be done using a particularly appropriate set of integration coordinates. Oscillator parameters, other than those connecting individual particles, were de-termined by maximizing the overlap of the ground states, by matching the ground state energies, and by minimizing the second moment of the strength function. It was shown how the moments of the strength function can be expressed i n terms of the moments of the residual interaction. Although this result is widely known, the proof i s not spelled out i n the l i t e r -ature. Finally, i n Section 4-6 i t was shown how a model interaction can be obtained as an average over exact interactions. In the case of har-monic oscillator exact interactions, the model interaction obtained i n this way i s also harmonic, and contains in addition a constant term which has^ the effect of bringing the ground state energies into closer alignment. Chapter 5 was used to generalize the treatment of Sections 4-1, 4-3, and 4-3 to N particles. It was shown that as the number of particles increased from 2 to 24 the maximum squared overlap between the ground states of the "single particle" arrangement and the "compound" arrangement decreased gradually from 1.0 to 0.83. It was concluded that the more particles involved, the more the strength w i l l be distributed, and hence the more conspicuous w i l l be the shape of the strength function. In the remaining chapters the linear chains consisted of 24 particles. Chapter 6 continued the work of chapter 5 by calculating with the appropriate overlap integrals, the strength of the single particle ground state in compound states, and the strength of the ni 2(2) single particle state in compound states. It was found that the strength for spreading into states with the same number of quanta of excitation -104-dominated the strength for states differing by 2 quanta which, in turn, were larger than the strength for states differing by 4 quanta from the i n i t i a l state. See Figure 6-6. It is clear that i n a given energy interval certain types of states are preferentially populated over certain others. The preferred>ones are those whose excitation i s most l i k e the i n i t i a l state; that i s , most l i k e the entrance channel. Using the con-jecture of Chapter 3 one may put in evidence the nuclear counterpart to this undemocratic behaviour. It is to be reiterated that i t was not certain normal modes that were found to be preferentially populated, but certain types of excitations (for example, 2 quanta distributed i n a l l possible ways among a l l the normal modes). Correspondingly, in the nuclear problem, one would not expect the preferential excitation of certain particles, but, perhaps, of certain types of excitations (for example, 8-particle - 4-hole), which are most similar to the entrance channel. Preferential excitation of certain types of states has in fact been observed in heavy ion scattering experiments by Middleton et a l (1971), Voit et a l (1973), and B a l l i n i et a l (1974). This constitutes some of the most compelling evidence for the interpretation of inter-mediate structure i n 21tMg in terms of alpha-cluster configurations. The strength function depends not only on the overlap integrals but also on the density of states. Chapter 7 showed that the Gaussian shape used by Wong and French for the nuclear part i a l level density i s also appropriate for the present problem. This enabled a tabulation to be made of the quantities C2/D - the squared overlap divided by the cor-responding average spacing. In Chapter 8 a Lorentzian curve was f i t t e d to the tabulated C2/D. It was shown that the fluctuations of the C2/D obey a Porter-Thomas -105-distribution, and that the maximum likelihood method of curve f i t t i n g i s appropriate for a l l chi-squared distributions. The Porter-Thomas distribution is a special case of these. Using the maximum likelihood method a Lorentzian curve was f i t t e d to the strength function of the ni2(2) single particle state in the 2-quanta compound states. The results were plotted in Figure 8-2 and are discussed below. The hierarchy of systems (single particle, quartet doorway, and compound) proposed in the introduction was studied in Chapter 9 which re-quired the extension and application of the method developed in Chapters 2 to 8. It was found that the Lorentzian shape f i t t e d to the strength function of a single particle state in doorway states had a width of ;T = 0.30 (in units of "Rtd) , whereas the one f i t t e d to the strength function of a doorway state in the compound states had a width of ; = 0.97. These results, though at odds with the expectations from the alpha-particle model for the 1 2C+ I 2C reaction, are not, retrospectively, d i f f i c u l t to understand; they simply expose the fact that, with the harmonic oscillators, the single particle system i s closer to the doorway system than the doorway system i s to the compound system. Indeed the single particle system differs from the doorway system by the placing of one spring, while the doorway system differs from the compound system by the placing of five springs (see Figure 1-6). Thus the spreading of the doorway into compound states was wider because the residual interaction was larger, i.e. the doorway Hamiltonian l e f t out more terms from the compound Hamiltonian than the single particle Hamiltonian l e f t out from the doorway Hamiltonian. Further, the results had been anticipated by the overlap of the ground states: single particle with doorway - 95%; doorway with compound - 70%. -106-The widths obtained in Chapter 9 must be contrasted with the experimental broad single particle widths and the narrower doorway widths of the 1 2C + 1 2C reaction. However, one remarks that while the known clustering properties of the 1 2C nucleus have been expressly designed into the corresponding harmonic oscillator single particle system, no cognizance was taken of possible clustering effects in the structure of 2l+Mg in the arrangement of the compound harmonic oscillator system. This is what made the doorway model so different (in the sense of having a large spreading width) from the compound model. Though, in 2lfMg the ground state has very l i t t l e quartet clustering (Basu 1972), Arima, G i l l e t , and Ginocchio (1970) have shown that, energetically, quartet excitations can be expected among low lying shell model states. Had the compound harmonic oscillator system been made to include quartet clusters with strong internal binding as in Figure 9-2, the spread of the doorway state might have been much narrower. One of the implications of the present work i s , then, the suggestion that the narrowness of the intermediate structure in the 1 2C + 1 2C reaction i s due to the existence of quartet clustering in the compound nucleus 2 4Mg at the excitation energies being probed.. In Chapter 8 a single particle state was spread among compound states. In Chapter 9 the single particle state was spread among doorway states, and one of these was spread among compound states. In order to examine the applicability of the commonly used Lorentzian shape for the energy dependence of <C2>/D, a Lorentzian curve was f i t t e d in each of the three cases to the calculated <C2>/D. Considering for the moment only the strength of the i n i t i a l state i n states of the same number of quanta, i t can be seen from Figures 8-2 and 9-1 that the distribution of <C2>/D f a l l s off in energy more rapidly -107-than a Lorentzian. This was not entirely unexpected. A Lorentzian curve has an i n f i n i t e second moment, but any calculated distribution w i l l have f i n i t e moments. ^In the central regions, that is near the resonant energy, the distribution i s not unlike a Lorentzian while a Gaussian curve, on the other hand, would f a l l off much too rapidly. More importantly, however^ adding to the strength function the contribution due to states of different quanta, i t was seen that the Lorentzian curve overestimated the strength in the low energy side and underestimated i t in the high energy side. The latter effect was directly ascribable to the level spacing D. Although the average overlap <C2> for the states of different quanta i s much smaller than for the states of same number of quanta as the i n i t i a l state, many more states can be formed with a larger number of quanta, hence increasing the level density. Therefore, although the assignment of Lane, Thomas and Wigner (1955) of the giant resonances to the behavior of the <C2> i s corroborated i n this model, the decreasing average level spacing D w i l l lead to a skewing of the Lorentzian shape of the resonances. The other physical consequence that was implied in these results is that an optical model extrapolation to lower energies from a measured resonance w i l l tend to overestimate the absorption cross section, provided that there i s no new physics below the lowest resonance. It is not possible, however, to declare the optical model cross section as an upper limit on the cross section since the harmonic oscillator model cannot, nor was i t designed to, treat effects such as "absorption under the barrier" (Michaud, Scherk, and Vogt, 1970) which might influence the cross section at low energies. Such effects w i l l require the examination of individual nuclei. In short, the harmonic oscillator because of i t s remarkable -108-solvability is a useful testing ground for many of the tools of theoretical physics (see also, for example, Moshinsky, 1969). The linear chains used in this work have served to make manifest the preferential population of certain types of states, the non-Lorentzian shape of the strength function, the second moment as a measure of the width of the strength function, and the importance for heavy ion reactions of cluster states in the compound nucleus. A l l these features may survive the harmonic oscillator approximation and they should certainly colour the way in which the data i s examined. It would be interesting in the future to include spin and isospin in the model and to investigate the emergence of tightly bound ot-like clusters while examining the special role, i f any, of 4-body forces. Actually, for springs of zero length the harmonic oscillator systems are solvable exactly in spaces of any dimension. A study of the formalism needed to incorporate f i n i t e length springs in more than one dimension could provide an insight into suitable mechanisms for nuclear matter calculations with repulsive core potentials. Another area where this model could be useful i s a study of the connection between free particle interactions and effective interactions i n nuclei: the system could be solved both exactly and in a shell-model-like approximation. In conclusion i t can be said that the f u l l usefulness of the harmonic oscillator model has by no means been exhausted and that one can look forward to seeing i t in f r u i t f u l service for many years. -109-APPENDIX A NUCLEAR POTENTIALS AND HARMONIC OSCILLATOR POTENTIALS It was not necessary in this work to employ particular numerical values for the parameters of the harmonic oscillator potential used to represent the nucleon-nucleon interaction. Nevertheless, this appendix discusses how the parameters of the harmonic oscillator would be chosen to best approximate a nuclear potential. In Section A-1 this i s done by using one of the nuclear potentials from the literature. A better approximation i s obtained i n Section A-2 by constructing an N-N potential that i s better suited to a harmonic oscillator approximation than those in common use. A-1 Approximation of a Potential by a Harmonic Oscillator Potential In order to approximate an arbitrary potential V(r) by a harmonic oscillator potential [A.l] vH.O. n one can expand V(r) in a Taylor series about r obtaining where r i s chosen such that o [A.3] = 0 and require that -110-[A.4] » for a l l n > 2. The range of r for which condition [A.4] holds w i l l define the range of applicability of a harmonic oscillator potential approximation to the potential V(r). This procedure w i l l now be illustrated by applying i t to an analytically very manageable phenomenological three Gaussian potential (3GRS) developed by Tamagaki (1968) as a substitute for more r e a l i s t i c potentials . The three range Gaussian potential used by Tamagaki (1968) for the *S state i s [A.5] V (r) = + £30 <> XL where r is in units of 1 0 - 1 5 m and V is in units of MeV. This potential can be found to have a minimum at r = r Q = 1.03 fm. The range of applicability of a harmonic oscillator approximation to Eq. [A.5] is then given by condition [A.4] as [A.6] In this region V can be approximated by 3 G [A.7] Ho v Vo - ^ * (f-\.ol) where V q = -53 MeV and k = 653 MeV/fm2. However, the ground state energy of the potential V R 0 in Eq. [A.7] is E Q = + 63.542 MeV. When -111-this value i s compared to the 1S system of two nucleons, which Is unbound by about 0.5 MeV, one obtains a clear indication that the harmonic oscillator potential [A.7] is too steep: the value of the spring constant k i s too large. This i s because the harmonic oscillator is being forced to approximate the relatively hard core of the 3GRS potential. This i s not surprising. The N-N potentials which are usually constructed are not designed to be approximated by a harmonic oscillator potential and thus they do not resemble a parabolic well. The next section w i l l aim at constructing a new N-N potential e x p l i c i t l y designed to lend i t s e l f easily to a harmonic oscillator approximation. A-2 Construction of a Nucleon-Nucleon Potential Wanted in this section i s a nucleon-nucleon potential which reproduces N-N scattering data f a i r l y well and which most easily lends i t s e l f to a harmonic oscillator approximation. The most straight-forward of such potentials i s a harmonic os c i l l a t o r well matched on to a one pion exchange t a i l . The matching radius has to be chosen to make f u l l use of the known val i d i t y of the one pion exchange potential (OPEP) for large distances. Breit (1962) reviewed this question and concluded that for r > 2.9 fm the OPEP is dominant. Later work (Tamagaki 1967) has indicated that the OPEP can probably be trusted down to 2.0 fm, and, in this section, this value w i l l be used as the matching radius. The centre of the harmonic oscillator well w i l l be chosen to coincide with the minimum of the 3GRS potential as found i n Section A-1; namely, r =1.0 fm. Thus, one seeks a potential of the form [A.8a] V --112-[A.8b] \/= ' V 6 5: r > 2-0 fm where u = m c/fi and f 2 / 4 i T = 0.08. The potential in Eq. [A. 8a] i s the ir harmonic oscillator (HO) potential, and Eq. [A.8b] is the OPEP. The task i s to determine the two parameters V q and k. Two methods w i l l be used -a) Matching the value and slope of the HO potential to the OPEP at r = 2.0 fm gives V =3.14 MeV and k = 2.35 MeV/fm2. This o value of V q makes the potential of Eq. [A.8] extremely shallow compared to typical N-N depths of ~ 50 MeV. The ground state energy of this potential, however, is + 3.82 MeV, which i s of the same order of mag-nitude as the vir t u a l state of the so-called "singlet deuteron". It Is necessary to ask now whether this potential can be improved by relinquishing the requirement of a smooth slope at r - 2.0 fm. b) Matching the values of expressions [A.8a] and [A.8b] at r = 2.0 fm gives the relationship [A.9] k -- X ( V . - I. <H) M e V / f Now by matching the scattering length from the potential [A.8] to the s s experimental value (either a = -17 fm or a = -23.7 fm; see Moyes r nn np 1972), one should be able to determine the value of V , the remaining parameter. In practice one chooses a t r i a l value for V q and calculates the scattering length numerically. A choice of V q = 30 MeV gives a = -21 fm. Making V q smaller by about 1 MeV w i l l increase the scattering length to the singlet neutron-neutron value. Making V q larger by about 1 MeV w i l l decrease the scattering length to the singlet neutron-proton -113-value. Thus, V q = 30 MeV is a good order-of-magnitude estimate. The Value of the spring constant k is then determined from Eq. [A.9], which gives k = 56 MeV/fm2. A harmonic oscillator well with these parameters has a ground state energy of +4 MeV, which, again, is of the same order of magnitude as the virtual state of the singlet deuteron. This appendix has served to display the f e a s i b i l i t y of a harmonic oscillator approximation to r e a l i s t i c nuclear potentials. It is not necessary to pursue further the question regarding the best choice of parameters. Their numerical values are only incidental to the developments in this thesis, which addresses the question of the spread of a state of one system into states of another system. The treatment which was followed uses harmonic oscillator forces with arbitrary parameters. The results are independent of their numerical values. It i s appropriate to point out here that non-nearest-neighbor forces in a system consisting of many particles may also be treated successfully with harmonic osc i l l a t o r forces. One simply mimics the weakening of the nuclear force with increasing distance by using successively weaker springs (smaller k) to connect successively more distant neighbors. Although the problem of linear chains considered in this thesis would s t i l l be solvable using non-nearest-neighbor forces, these would constitute an added complication. A preliminary calculation indicated that the spring constants associated with next-to-nearest-neighbor forces are only 1% of those associated with nearest neighbors. Since the physical effects that are studied here do not depend on the inclusion or non-inclusion of these added forces (whose effects, in any case, are small), use of these i s avoided, and the calculations employ nearest neighbor forces only. -114-APPENDIX B DIFFICULTIES WITH THE EXCLUSION PRINCIPLE The linear chains of nucleons which are used in this thesis are made up of fermions. It i s the purpose of this appendix to study the place of the Pauli Exclusion Principle in the formulation of the problem. For the purposes of i l l u s t r a t i o n , the discussion w i l l be presented in reference to a three-particle chain - the simplest non-t r i v i a l example. See Figure B-l. °< fi Y Fig. B-l. - A simple linear chain of particles. The labels a, 3, Y denote positions on the chain; they do not label the particles. If the three particles 1, 2 and 3 are identical fermions, the wave-function ¥ describing the system must be totally antisymmetric under the exchange of any two particles, and i t may be written [B.i] 9 = VO.i.z)-s^/,3FO + - Wuti + Win) ' f(3.*.') where 1, 2, 3 in the argument of are to be understood as representing a l l the coordinates of each of these particles respectively, the functions ^ ( i , j , k ) are the exchange degenerate eigenfuctions for the same Hamiltonian and same energy as and A is an antisymmetrizing operator. The wavefunction V is then the unique antisymmetric combination of the i K i , j , k ) . -115-F i r s t of a l l , however, i t i s necessary to establish that a l l the particles are identical. This means that any one of the particles 1, 2, or 3 may appear with equal probability in any one of the positions a, $, or y °n the linear chain. The identity of the particles must also be reflected i n the Hamiltonian describing the system which must remain unchanged whenever any two particles are substituted for each other. Indeed, i t i s possible (Messiah 1961) to define particles as identical whenever the Hamiltonian has this property. An appropriate Hamiltonian describing the system of Figure B-l with harmonic oscillator interactions of spring constant k between particles in adjacent position i s [B.2] H • 2 f > v . vtr where p^ and m are, respectively, the momentum and mass of the -cth particle. The interaction V „ between a particle at site a and a particle at site 3 is i + k where r^ is the displacement from equilibr ium of the -Cth particle, and Q a(i) i s a projection operator which acting on a wavefunction gives zero unless the -cth particle is in the position a. Using a convention on the ordering of the arguments of iKj,k,£) such that (j,k,£) indicates that ^th particle at a, the feth particle at 3, and the Ith particle at y» one may write concisely -116-Clearly now, the Hamiltonian of Eq. [B.2] i s symmetric under exchange of any two particles, and produces a harmonic interaction between adjacent particles and no Interaction between non-adjacent particles. It acts on the total antisymmetric wavefunction to give [B.5] H * * E * and on the exchange degenerate wavefunctions to give [B.6] • H +i$,K*} E • It i s now possible using the explicit effect [B.4] of the operators Q a(i) o n the wavefunctions iKj,k,£) to simplify the SchrSdinger equation [B.6] as follows. [B.7] X so that the SchrOdinger equation [B.6] becomes -117-[B.8] .TT i f * A ^ J where j (and k, and I) = 1, 2, 3. Finally, since ^ =£iKl,2,3), i t is necessary to solve only That i s , one need only consider particle 1 fixed in the f i r s t position, particle 2 fixed in the second position', etc. , and obtain Y, at the end, by antisymmetrizing I/J(1,2,3) with respect to the particle coordinates. This last step, extremely simple in principle, and manageable with only three particles, becomes prohibitive with twenty four particles. The reason is the following. If equation [B.9] did not couple the coordinates of the different particles, then one would be able to write ^(1,2,3) as a product function and ¥ would be given by the 3 x 3 Slater determinant formed with the <J>'s (or an N x N determinant for N particles). However, Eq. [B.9] does couple the coordinates of different particles, and requires, for its solution, transformation to a new set of (uncoupled) coordinates x» -118-[B.ll] which are called the "normal coordinates". (A single underline denotes a column matrix, a double underline a square matrix) Then In the case of 24 particles a single antisymmetrized wavefunction ¥ would consist of 24! (= Avogadro's number) terms, each term consisting of a product of 24 functions, and each of these functions depending on a l l 24 particle coordinates with respect to which the antisymmetrization is to be carried out. The fact that these wavefunctions are used in overlap integrals does not significantly reduce the complexity of the problem as can be appreciated even by considering chains of two particles. Let Y and $ be two antisymmetrized wavefunctions each describing two particles. Writing [B.12] [B.13] [B.14] enables their overlap integral to be written -119-[B.15] where the last step is obtained by relabeling the variables of integration. Thus, out of 4 possible integrals only 2 have to be evaluated. For the case of three particles the same argument shows that out of a possible 36 terms in the overlap integral 6 have to be evaluated. In general, for N particles the antisymmetrized wavefunctions have N! terms, and the overlap integral of two of them has (N!) 2 terms out of which only N! are distinct. This is the d i f f i c u l t y that led to the restriction to unsymmetrized wavefunctions in the body of this work. The simple harmonic oscillator i s often used to i l l u s t r a t e the elegance of the number representation and the role of creation and annihilation operators together with their commutation relations. However, the operators which create or annihilate states of the harmonic oscillator obey boson commutation relations, and the states of a set of uncoupled harmonic oscillators can be identified with the states of an assembly of bosons. When written in terms of normal coordinates the Hamiltonian of a linear chain is indeed uncoupled, but the antisymme-trization has to be carried out with respect to the particle coordinates -not the normal coordinates. Even in the number represenation, then, i t would be necessary to transform from the normal labels to the particle labels and to carry out the antisymmetrization as above with no gain in economy of labour. For this reason the coordinate representation has been used throughout this work. -120-APPENDIX C PREDICTION OF NULL OVERLAP INTEGRALS In determining a strength function by the method used in this work, a great many overlap integrals have to be calculated. One wonders, naturally, i f a considerable amount of labour may be saved by examination of the structure of the integrals, or of the structure of the Hamiltonians themselves. In this appendix the spreading of a single particle state into doorway states w i l l be considered by examining in Section C-1 the overlap integral between the single particle state and a doorway state, and by using group theory in Sections C-2 to C-5 to exploit the symmetry of the Hamiltonians into yielding predictions about the overlaps of their wavefunctions. In Section C-2 the orthogonality relation between two irreducible representations of a group is used to show that two wavefunc-tions w i l l have zero overlap unless they are both basis for the same representation of the group. The common invariance group G of the two Hamiltonians is found in Section C-3. In Section C-4 the representation of the group G, found in Section C-3, is decomposed into the irreducible representations of the group. Finally, i n Section C-5 the eigenfunctions are classified according to the representations of G and use i s made of the theorem of Section C-2 to obtain predictions about the overlap integrals. This appendix is not a contribution to the methods of group theory, but consists of an application of standard techniques to the overlap problem. C-1 Structure of the Overlap Integral Consider a single particle state with m2 quanta in the second normal mode and no quanta in a l l other modes. Its wavefunction is -121-[ c i ] ^ Cy) Nl H « K e, f [• I 7 ^] . Consider an arbitrary doorway state [C.2] -Y (x) - A/ UJKK) //„ Ut*t) •'• H„ W p l - Z x f ^ l . The overlap integral of interest in the determination of the strength function is CC.3] 1 ^ = j ^ (Y) ^ ( x j ^ where dT is an element of volume in the space of the y's or x's. It is appropriate to note that both the coordinates y_ and the coordinates x are related to the particle coordinates by an orthogonal transformation; thus they span the same space and are related to each other by an ortho-gonal transformation C^, [C4] I = £ X It i s , further, advantageous to note that the matrix C_ has the block diagonal form [C.5] which permits the overlap [C.3] to be written as - 1 2 2 -[C.6] f In Eq. [C.6] i t is understood that y = y(x) according to Eq. [C . 4 ] , and since the transformation _C i s orthogonal, each of the sub-matrices in the block diagonal form w i l l also be orthogonal, and i n each case the magni-tude of the Jacobian w i l l be +1. Now, f i n a l l y , observing in Table 9 - 1 that [c.7a] <o7 w 8 ^ -- - -- u > I V -- u>; = u>; *\ « ; „ * u>; --[c.7b] «v w,, w t f H i u ) n w i f , u>; -- u>; r - o > ; t * - - « ; f [C.7c] u ) „ u s v - O i v , w,s-_ ^  u ^ , ^ , ^ , w ^ . . « ' „ -- < , one may write * i ••• H « l I ( ( V a O e " ^ ^ <V<** -123-Hence, using the orthogonality r e l a t i o n s of the Hermite polynomials one finds For the s p e c i a l case where 4 quanta are a v a i l a b l e to the doorway system, these can be d i s t r i b u t e d among the 23 v i b r a t i o n a l modes to give 14,950 possible doorway s t a t e s . But according to Eq. [C.9] a doorway state having any quanta i n any of the modes 7 through 24 w i l l have zero overlap with the s i n g l e p a r t i c l e state n^O. Thus f or the purposes of c a l c u l a t i n g these overlaps, the 4 quanta i n the doorway system need only be d i s t r i b u t e d among modes 2 through 6 to give only 70 p o s s i b l e non-zero overlaps. The conclusion of t h i s s e c t i o n may be summarized by remarking that the orthogonality properties of the transformations and of the Hermite polynomials, together with the f o r t u i t o u s e q u a l i t y of many of the eigenfrequencies of the two systems, permit the p r e d i c t i o n of a great many zero overlaps. The remaining sections of t h i s appendix w i l l examine how many of these zero overlap i n t e g r a l s are non-fortuitous, and may be ascribed, instead, to the common invariance properties of the Hamiltonians belonging to the overlapping wavefunctions. In the example j u s t discussed, for which only 0.47% of the 14,950 overlaps are non-zero, i t w i l l turn out that group theory removes only 50% of the overlaps to be c a l c u l a t e d . Nonetheless, the an a l y s i s which follows could be a more us e f u l t o o l i n other cases. C-2 Group T h e o r e t i c a l P r e d i c t i o n of N u l l Overlap Integrals Using the notation of t h i s thesis t h i s s e c t i o n w i l l serve as a -124-reminder that i f two wavefunctions Y Cr) a n ^ $ ( r) a r e basis functions n m for two inequivalent irreducible representations of a group G, their overlap integral I is equal to zero. The overlap integral I i s defined by r nm [C.10] Now l e t [ C . l l ] r' ~ Gr . r where G_ is an element of the group G of matrix transformations on the r_. Then [C.12] 1 ^ , r') I f ( f r - V ) Arj |T| where the Jacobian J is the determinant of G - 1, so that ]J|=1 for ortho-gonal transformations. Now relabel the coordinate of integration by simply dropping the prime to obtain Here x (G - ; ir) i s a new function of r. Call i t P_ X (r)» i«e, n ~ — — G n — If G is an invariance group of the Hamiltonian of Xn(£.)> then this new function of r_ may be expressed as a linear combination of -125-functions Y ?( r) belonging to the same eigenvalue of energy as x (G 1 r ) n - n = [C.15] P f r X,(C) = Z X n , ( r ) The matrix D0(G) is a representation of the element G of G. This re-presentation has been labeled by the superscript a. A similar treatment of ^ (G_-1r) leads to the representation D (G) , and permits the overlap IC13] to be written as [C.16] 1^ = £ ( X n . ( t ) ^ , C O . Since Eq. [C.16] holds for every element G of the group G , one may sum over a l l the elements G and divide by the number of elements |G | in the group G to obtain But the well-known (see for example Wigner (1959) p. 83) orthogonality relation between matrices of two irreducible representations states that [C.18] I J > " ( » w (^m*. = ° ^ cut 4 . Therefore, unless x ( r) and CO are both basis functions for the same n — m irreducible representation of G, their overlap, I , w i l l be zero. nm C-3 The invariance Groups of the Single Particle and the Doorway Systems In the previous section, Section C-2, i t was shown that two wavefunctions may have non-zero overlap only i f they each form part of a -126-basis for the same irreducible representation of the invariance group which i s common to their two Hamiltonians. In this section one asks just what are the invariance groups of the two Hamiltonians at hand, namely, the single particle Hamiltonian and the doorway system Hamiltonian. An invariance group of a Hamiltonian is a group of coordinate transformations every one of whose elements commutes with the Hamiltonian. Such a group is now sought for the Hamiltonian of the single particle system. Consider the following single particle system On^yt^/rti-Q o/K,vO-*,M>-fc-0 OiwO^O-^O I F " -Trtrrr k" Fig. C-1. - The single particle system m where the coupling between individual particles i s by means of a spring of constant k, the coupling between "a particles" i s by means of a spring of constant k', and the coupling between " 1 2C particles" i s by means of a spring of constant k''. Let r^ be the displacement to the right from equilibrium of the -tth particle. Each of the 24 particles has mass m. The Hamiltonian of this system i s then [C.19] where [C.20] VI 6 -6 ft -e> -6 -6 C -ft a s -ft -6 -6 B •a B a -ft -6 -4 • f t -e> -4 a % a -ft c. -d -6 -a 3 a -a e -e •6 8 6 ft 4 " " 1 4 -127-where [C.21] 1 -1 1 1 1 I ^ i x i . - i X. -i 3 - i .L.»-_L c = X X. X. 2. -I Z, -I "«! 1 i i i i i i i i 1, % . Z 3. By looking at Figure C-l i t is apparent that the system i s invariant under certain coordinate transformations, [C.22] r r ' = s and, i n fact, i t can be verified that the Hamiltonian [C.19] i s invariant i f i s any of the following matrices ( i = 1,2,3,4,5,6,7,8,9) or their products. [C.23] Jit i i aw m 01 •11< 11 J V ' l 1 -128-where [C.24] 1 = These matrices ( i = 1,2, ... 9) are the generators of a 512 element (non-abelian) group G'. This group i s a subgroup of the symmetric group on 24 objects, i.e. the group of a l l possible permutations on 24 symbols. G' i s an invariance group of the Hamiltonian [C.19]. Now consider the following doorway system O I N J ^ W D t»0~avOnO cyxr*0**OftQ I T l » 1 1 J*rv- I I OVO^O^O OyrC^Orrt-o 1 1 1 <&r*— 1 Fig. C-2. - The doorway system where the coupling between individual particles i s by means of a spring of constant k, and the coupling between "a particles" i s by means of a spring of constant k 1. Then in the same notation used i n the previous paragraph for the single particle system, the Hamiltonian for the doorway system may be written [C.25] H, 2. r T 1 r A, " = ~ 2* Id ~ where [C.26] o -6 -6 C -6 — - -6 t -6 -ft C -6 -6 -129-Again, by looking at Figure C-2 one sees that the system is invariant under certain coordinate transformations tc.22] r — * r' ^  S £ which leave the Hamiltonian [C.25J invariant, where S^, in this case, can be any of the previously defined ( i = 1,2, ... 7) or any of their products. These matrices g^ ( i = 1,2, ... 7) are the generators of a 128 element (non-abelian) group G which is a subgroup of the group G'. G i s an invariance group of the Hamiltonian [C.25]. Since i t is also an invariance group of the single particle Hamiltonian [C.19], G i s the common invariance group of both Hamiltonians under consideration. C-4 The Representations of the Common Invariance Group G In Section C-3 the invariance group G, common to both the single particle Hamiltonian, and the doorway system Hamiltonian, was found. The elements of G were matrices having 24 x 24 entries, and these, in fact, constitute a 24-dimensional representation of G, D(G). This representa-tion D(G) w i l l , i n general, be reducible, which, i n this case, means that i t can be written as a direct sum of irreducible representations of the group G. Wanted, then, in this section, are the irreducible representa-tions of G and the decomposition of D(G) i n terms of them. It is convenient to begin by noting the abstract properties of the group G. The group G has 7 generators g ^ i = 1,2, ... 7, which have the following relations -130-[C.27a] -z\ = E ( t h e i d e n t i t ^ f o r 1 = 1 > 2 > 3 > 4 » 5 > 6 > 7 [C.27b] g ± g j - 8 ^ f ° r j = 1> 2' 3' 4' 5' 6 [C.27c] g ?S 1 = 8 68 7 [C.27d] g ?S 2 = § 5g ? [C.27e] g 7 § 3 " Sfy The irreducible representations of G can be found most easily by finding, f i r s t , the irreducible representations of a subgroup H of G, and, then, using the theory of induced representations to find the representations of G. Let H be the subgroup of G with generators g^ ( i = 1,2,3,4,5,6). H w i l l then be an abelian group of order 64, i.e. i t i s of index 2 in G (which has 128 elements), and one may write [C.28] <r - H *• <^ H Since a conjugate class i s the set of H' obtained by H' = H^ H H~, an abelian group w i l l have as many conjugate classes as elements; in this case, 64 conjugate classes. For each conjugate class of H there exists an irreducible representation of H. If d^ is the dimension of the -Lth representation, and N is the number of irreducible representations, then in v [C.29J ^ c Zl ^ order of the group = 64 which can only be satisfied i f d^ - 1 for a l l i . In other words, each one of the 64 irreducible representations of H is one dimensional. Note -131-that since for each H e H, H 2 - E, a one dimensional representation d(H) w i l l have each d(H) = ± 1, and the 64 representations are each obtained from one of the 2 6 ways that + 1 and -1 can be assigned to the 6 generators of H. Now that the representations of H have been found, i t i s necessary to consider the group G (of which H i s a subgroup), and to re-c a l l the following definition: an element G of G belongs to the orbit L t d j ^ ) ] i f d j (Hi) ^ dj (G H i G - 1) . One remarks that, in the present case, a l l elements H of H belong to L[d..(H^)] because H is abelian, and thus H H. YTl = H., so that t r i v i a l l y d, (H.) ^ d.(H H. H _ 1). If, i n addition, d..(H^ ) ^  d j (S7 ^ g7 _ 1)» then %j and a l l the elements of G belong to L[dj(H ±)], so that L[dj(H ±)] = G. If, however, d^(H±) X dj (g7 g 7 - 1 ) , then <gl and a l l gyH elements of G do not belong to L t d j O ^ ) ] , so that LtdjO^)] = H. The orbit of H in G, as defined in the previous paragraph, plays a role in inducing a representation D(G) from a representation d(H) according to the following theorem: a) If L[d^(H^)] = G, then one obtains 2 irreducible representa-tions of G: [C.30] 0 M H i ) ~- o/(Hc) D (^,HC) , [C.31] cO }(Hi} - <*(*C) - - l b (^H;) p b) If L j d j ^ ) ] = H, then one obtains only 1 irreducible re-presentation of G: -132-[C.32] o *** D<V)--0 The proof of this theorem can be found i n many textbooks dealing with induced representations, for example Jansen (1967). The symbol *** represents expressions involving d.. (H^) and d.. (gj 1L g7 - 1) which w i l l not be needed in subsequent calculations. By enumeration of the 64 representations for the generators of H, i t i s easy to see that there are 8 cases for which [C.33] and thus, according to part (a) of the above theorem, these lead to 16 one-dimensional irreducible representations for G. The 56 remaining representations of fi pair up so that IC34] and thus, according to part (b) of the theorem, one obtains 28 two-dimensional irreducible representations for G. A useful check i s provided by expression IC.29]: [C.29] I N : order of the group which in this case -133-[C.35] « /** + 2$ ' 2* -- tft verifies that a l l the irreducible representations of G have been identified. The groundwork has now been l a i d for asking the question, what irreducible representations of G are contained i n the 24-dimensional representation D(G)? In other words, wanted is an explicit expression for [c.36] ~ Z. h CO i where is the number of times that the irreducible representation D^(G) is contained i n D(G), and the sum i s a direct sum. Taking the trace of both sides of Eq. [C.36] and using the orthogonality relation between irreducible representations of a group, one obtains which w i l l be used repeatedly in what follows. The application of Eq. [C.37] is simplified by noting that [C.38] T r " ° so that [C.39J u- - - L - L T r T r [ D C ( H ) ] r« us H The required traces are list e d in Table C-I. -134-TABLE C-I TRACES OF THE (REDUCIBLE) REPRESENTATION D(H), AND OF SOME OF THE IRREDUCIBLE REPRESENTATIONS d ±(H) Notation: The 6 generators of H are labeled H^  with j = 1,2,3,4,5,6. Vl ~ Hk£ Element k TrCDCiy) Tr(d 2 a y ) Tr(d 2(H k)) . . . E 2 4 it i 1 1 - i 1 2 i -1 3 2 0 i 1 4 i 1 5 i 1 6 > i 1 12 - l -1 13 - i 1 14 - i 1 15 - i 1 16 • - i 1 23 i -1 24 16 i -1 25 i -1 26 i -1 34 i 1 35 i 1 -135-TABLE C-I (Continued...) Element k TrODOy) TrCd 2(H k)) . . • 36 1 1 45 1 6 1 1 46 1 1 56 / 1 1 123 s -1 -1 124 -1 -1 125 -1 -1 126 -1 -1 134 -1 1 135 -1 1 136 -1 1 145 -1 1 146 12 -1 1 156 -1 1 234 1 -1 235 1 -1 236 1 -1 245 1 -1 246 1 -1 256 ' 1 -1 345 1 1 346 1 1 356 1 1 456 / 1 1 -136-TABLE C-I (Continued...) Element k. TrCDGy) TrCdi <\)) Tr(d 2(H k)) . . . 1234 \ -1 -1 1235 -1 -1 1236 -1 -1 1245 -1 1246 -1 -1 1256 -1 -1 1345 -1 1 1346 8 -1 1 1356 -1 1 1456 -1 1 2345 1 -1 2346 1 -1 2356 1 -1 2456 1 -1 3456 > / 1 1 12345 -1 -1 12346 -1 -1 12356 l -1 -1 12456 -1 -1 13456 -1 1 23456 1 -1 123456 0 -1 -1 -13 7-The identity representation d^CH) induces 2 one-dimensional representations of G: DjCG): DjCG) = 1 (the identity representation) and, •Di:t(G): D I l ( H ) = 1 a n d D l l ( g 7 H ) = - 1 -Let u be the number of times D-^G) is contained in D(G). Then, H Similarly, [C.41] W Thus, so far, one may write [C.42] D (&) ~ 6 b 1 ( 2 ^ © © others. It is useful to note now that any representation of G when restricted to H is a representation of H, and to ask which irreducible representations of H are contained in the 24 dimensional representations D(H). Let be the number of times that the irreducible representation d±(H) is contained in D(fi). Then, [C.43] ' v. , J _ y T r [ > < » ] f r [ ^ ( M ] - 1 3 8 -Now, let djCH) be the identity representation. Let d^H) for i = 1 , 2 , 3 , 4 , 5 , 6 , be the representation of H for which the -Lth generator g_^  is represented by - 1 and a l l other generators are represented by + 1 . Then, i t is easy to apply expression I C . 4 3 ] using the traces tabulated in Table C - I to find that [ C 4 4 ] V T = IX [ C . 4 5 ] Y , - , v; - V , v r - • = A. . Therefore I C 4 6 ] D ( H ) ~ P - o l^ i n© Ad . fH ) 0 A C/JH) © A d a f H ) © a-o/^rt) Note that in Eq. [ C . 4 6 ] the representations on the Right Hand Side ex-haust the dimensionality ( 2 4 ) of D(H), and, thus, only these irreducible representations and no others are contained i n D(H). Now i t i s necessary to ask, which representations of G w i l l , when restricted to H, result in the representations of Eq. I C . 4 6 ] ? Using the symbol 4> to mean "restricted to", i t i s already clear that [ C . 4 7 ] & ( ? ) I H = D (H) D x (£) I H ' * '' cl, (fi) j) (£) I H "- (H) . I C . 4 8 ] IC49] -139-Further, by making use of the group properties [C.27] one observes that [C.50] so that IC51] % Sfc ^ 7 Thus the theorem on inducing representations quoted earlier in this chapter indicates the existence of a representation T>\ (G) such that [C.52J 0 J P(^H^') 0 0 H, a re-ducible representation of H. In fact, using Eq. [C.51], one may write [C.53] 3>. C5-) I H = OI,(HJ © d f c(£) . Similarly, and with analogous notation, [C.54] IC55] -140-It i s necessary here to observe carefully that since di(H) and dg(H) induce only T>i (G) , only Di (G) w i l l , when restricted to H, yield d} (H) and d5(H). This observation permits one to write, f i n a l l y , [C.56] D(Z) ~6lt(Z\® t\W © MX?) « a M £ ) s X7>*M as the only decomposition of D(G) in terms of the irreducible representa-tions of G which is consistent with Eq. [C.46]. Expression [C.56] i s therefore the decomposition of D(G) sought for at the beginning of this section. C-5 Classification of the Eigenfunctions As was shown in Section C-2, the prediction of a null overlap integral between two wavefunctions depends on whether or not the two wavefunctions are each part of a basis for the same representation of a group G. One of the wavefunctions being considered i s the single particle wavefunction if1 (y) which is an eigenstate of the Hamiltonian H g of Eq. [C.19]. The other is the doorway system wavefunction X 00 which i s an eigenstate of the Hamiltonian H^of Eq. [C.25]. The group G is the common invariance group of the two Hamiltonians. It is necessary, then, in this section, to classify the states (y_) by specifying for each set of quantum numbers m the representation of G for which ^ (y_) serves as a basis function. A similar c l a s s i f i c a t i o n must also be made for the states X (x). Since the wavefunctions are known functions of the coordinates, n — i t i s appropriate to begin by considering the transformation properties of the coordinates. -141-C-5(a) Transformation Properties of the Coordinates The representation D(G) has the particle coordinates r_ as i t s basis functions; i.e. the representation D(G) consists of matrices of size 24 x 24 which act on the 24-dimensional vector r_. The wavefunction X (x) i s , however, written in terms of the normal coordinates n — ' [C.57] X - A where A is a real, orthogonal, 24 x 24 matrix. The effect of the representation D(G) on the coordinates x can be found by performing the similarity transformation [C.58] ,x where the representation D (G) is a representation equivalent to D(G) and having the coordinates x as basis functions. Carrying out the trans-formation [C.58] for the 7 generators g^ of the group G one obtains [C.59] where i = 1,2,3,4,5,6, -142-[C.60] 6 < 4 and M_^ 2» ^ 3 > M5> ^7 a r e » each, non-zero 6 x 6 matrices, (1) One sees immediately, by inspection, that xi i s a basis for D^CG), X £ is a basis for D^(G), X3 i s a basis for D^(G), xi^ i s a basis for D^CG), X5 i s a basis for D.j.(G), xg i s a basis for D^j(G). (2) Again, by inspection, one sees that X13, x ^ , X 1 5 , x^g, X 1 7 and xj 8 a r e each a basis for d.j.(H). d.j.(H) induces D^CG) and D^CG). However, after the previous paragraph, (1), and because of the decompo-sition [C.56] of D(G) in terms of irreducible representations of G, there are only 3 Dj(G) and 3 D^CG) which have not yet been assigned basis functions. Thus, there exist 6 linear combinations of xj3, x j i , , . . . , xj8 such that each of 3 of them forms a basis for D^CG), and each of the other 3 forms a basis for D^CG). Nevertheless, i f one wishes to adhere to the normal coordinates as given, i t suffices to state that fx13» xl«t» •••» x18^ i s a basis for 3 D^G) © 3 D ^ G ) . -143-(3) It remains to assign bases to the six 2-dimensional representations 2 (G), 2 D 2(G), and 2 0 3(G) out of X7 , xs, ...» x i 2 and x^g, X 2Q> •••» x2i+. Consider { X 7 , . . . , X } 2 } . It must form a basis for a 6-dimensional representation of G which is equivalent to a direct sum of 3 of the 6 irreducible representations above. This 6-dimensional x representation i s in fact given in IC.59] by D (G)^ > where i , j = 7,8, ...» 12. Recalling that two equivalent representations w i l l have the same trace, i t i s possible to identify which 3 irreducible representa-tions, when summed directly, are equivalent to the 6-dimensional representation by using the following compilation of traces. 1 = 1 2 3 4 5 6 Tr[D 1(g ±)] = 0 2 2 2 2 0 = [1] Tr[D 2( g j L)] = 2 0 2 2 0 2 =[2] T r [ D 3 ( g i ) ] = 2 2 0 0 2 2 = [3] so that [1 i 9 1 « 9 2] = 2 4 6 6 4 2 [1 « 9 1 fi 9 3] = 2 6 4 4 6 2 [1 « 9 2 « 9 2] = 4 2 6 6 2 4 [1 « j 2 e »3] = 4 4 4 4 4 4 [1 « a 3 e J 3] = 4 6 2 2 6 4 [2 t » 2 i i 3] = 6 2 4 4 2 6 [2 « J 3 6 J 3] - 6 4 2 2 4 6 -14 4-These must be compared with, what is obtained from direct calculation on the matrices of Eq. IC.59] which yield [C.61] i -1 - o r t, = 1.5,..., 4 This permits an immediate and unique assignment of {X7, X12} a s the basis for a representation equivalent to ~D\ (G) © D2(G) © D3 (G) . (4) By elimination, (x^g, X24} is a basis for a 6-dimen-sional representation equivalent to Di(G) @ D2(G) @ 03(G). The above 4-step reasoning can be repeated for the normal co-ordinates y_ of the single particle system in order to find their y ~ transformation properties. Let D (G) be a representation equivalent to D(G) but having the coordinates y_ as basis functions. By straightforward calculation i t s matrices are found to have a structure similar to the matrices D (G) given in Eq. IC.59] with the exception that [C62] /v| = (1)' By inspection one sees that yi is a basis for D^(G) y2 is a basis for D (G) -145-{y3>yiJ i s a basis for D ] . ( G ) © V (G) {y5»y6^ i s a basis for B^G) © D J JC G ) ( 2 ) ' By inspection one sees that y^3, yit*, • • • > yi8 a r e each a basis for d^.(H). d^(H) induces D ^ C G ) and D ^ ^ ( G ) of which there are now only 3 of each to be assigned bases. Thus, {yi3, ynt, •••> yi8^ i-s a basis for 3 D ^ G ) © 3 V^G) . ( 3 ) ' Consider y-j, y j 2 ' They transform among themselves under the group G according to D ( G ) ^ , where i , j = 7 , 8 , 1 2 . The trace of this representation can be found by direct calculation to be Comparing this with the table of traces in paragraph ( 3 ) above, one arrives, similarly, at a unique assignment of (y7, yi2^ a s t n e basis for a representation equivalent to ( G ) © D 2 ( G ) © D 3 ( G ) . (4)' By elimination, (yig, y2^} Is a basis for a 6-dimensional representation equivalent to ( G ) @ D 2 ( G ) © D 3 ( G ) . C-5(b) Transformation Properties of the Exponential The wavefunction of a system with harmonic interactions, li k e the doorway system under consideration, consists of a product of Hermite polynomials times the exponential factor [ C . 6 3 ] for IC64] e -14 6-Wanted are the transformation properties of the wavefunctions. Having just obtained i n Section C-5(a) the transformation properties of the co-ordinates, one now proceeds to examine the transformation properties of the exponential part of the wavefunction. Since several modes of the doorway system share the same frequency, i t is possible to write the exponential in Eq. [C.64] as [C.65] € i W; where a l l the modes labeled by kj have the same frequency co^  . It i s important to remark now that for each value of j the x. form a basis for a representation of G as described in Section C-5(a). With this remark in mind, i t is useful to r e c a l l the following theorem: If {Yi ... F } is a set of basis functions for a n representation of a group G, then F 2 is invariant under a l l elements of G. ^ Proof: [C.66] [c.68] = z: D« (e ) . w F, ^ IC.69] * II ^  f. fl.fi.D. -14 7-Hence the factor V x 2 in Eq. JC.65] i s invariant under a l l elements of the group G, and hence the entire exponential i s invariant under any element of G. Thus the transformation properties of the wavefunctions are determined exclusively by the transformation properties of the Hermite polynomials they contain. C-5(c) Transformation Properties of the Fundamental Wavefunctions Having seen that the exponential in the wavefunction i s i n -variant under a l l elements of the group G, i t follows t r i v i a l l y that the ground state which contains only the zeroth Hermite polynomial, [C70] H, - I is also invariant under a l l elements of the group. The wavefunctions containing one quantum of excitation in one mode are called the funda-mental wavefunctions. Since the fundamental wavefunctions contain only the f i r s t Hermite polynomial [C.71] H, (0 = 2? they w i l l transform in the same way that the coordinates transform. That i s , a state with one quantum in the -ith. mode has the same transformation properties as the -cth normal coordinate. Further, having found the transformation properties of the normal coordinates in Section C-5(a), i t is straightforward to classify a l l the fundamental wavefunctions according to energy and to representation of G. This i s done in Table C-II where a subscript n., = 1 on a wavefunction indicates that there is one quantum in -148-the <tth mode, and no quanta in any of the other modes. TABLE O i l CLASSIFICATION OF FUNDAMENTAL WAVEFUNCTIONS ACCORDING TO ENERGY AND TO REPRESENTATION OF G Single Particle System Doorway System Hi) H1 Vi-t V V V V n s i * . I i a, - ^ 3 V V ' - M l V i (JQ V V X i i ) Energy This table may be used directly to determine when an overlap integral between a single particle state and a doorway state w i l l be zero by -149-making use of the theorem of Section C-2. For example [C.72] ( (?) X ( M o/T - O because ^(y_) i s a basis function for D^CG), but x(x) is a basis function _m2=l _ ng=l for Di(G) © D 2(G) © D3 (G). C-5(d) Transformation Properties of the Overtone Levels An overtone state i s defined as having more than one quantum of excitation in one frequency, and no quanta in other frequencies. Note that more than one mode may share the same frequency. An overtone wave-function w i l l consist of a product of Hermite polynomials times an exponential factor, and since the exponential factor was shown in Section C-5(b) to be invariant under the action of the group, the symmetry species of the wavefunction is that of i t s Hermite polynomial(s). Suppose that there are d normal coordinates r^, r 2 , . .., r^ with equal frequencies to^  = C J 2 = . . = 0 ) ^ = 0). Suppose that n quanta are to be distributed into modes with frequency w. This w i l l give rise to an f-fold degenerate overtone level with f = f(d,n). A wavefunction for this level w i l l be [c.73] 0 oc // ( ^ r . ) H 0 ... H c^r r„) where ni + n 2 + ... + n^ = n. Now, since the Hermite polynomial of order i , H^(5), has as the highest power of £, the wavefunction [C.73] may be written -150-[C.74] (b °< ( r."' r rtl ... rAd + smaller powers of the r's). Hence, i t is apparent that the degree f of degeneracy of the level i s equal to the number of ways one may write [c.75] r,"' . .. r/J with ni + n 2 + ... + n^ = n. Now i t i s convenient to r e c a l l the following definition If.r(G) i s a^representation of G transforming T\, r 2 , among themselves, then [r] n(G) is a representation of G transforming the products ( r l 1 r 2 2 ... r^d) , (r? 1 r 2 2 ... r ^ d ) , ... among themselves. Here ni+n2+...+n^ = nj+n2+...n^ = n. [T] n(G) is called the symmetric Kth power representation. Now, the f wavefunctions $ w i l l form a basis for a representation of G, A(G). The f products of the type ( r j ^ r ^ 2 ... r ^ d ) w i l l also form a basis for a representation of G, namely, [r]n(G).' It follows, therefore, that these 2 representations must have the same dimension. Further, since the transformation A(G) is linear, i.e. [C.76] P & <j>, = Z ^ j i t w i l l only mix together equal powers of the coordinates. However, this transformation of the powers of the coordinates among themselves is ac--151-complished, as defined above, by the symmetric nth power representation. Hence, re.??] - [r]n (£) In other words, the wavefunction of the state with n quanta distributed among modes which have the same frequency w i l l be a basis function for the [reducible] nth power representation [ r] n(G). C-5(e) Transformation Properties of Combination Levels A combination state i s one in which two or more distinct normal frequencies are excited with at least one quantum each. Let <j> ^ ^ ^  be the wavefunction of such a state. Then, i f for example, 2 modes are excited, where <f>^  i s the state having .ni quanta i n the mode with normal coordinate r i , and <{>j ' i s the state having n 2quanta i n the mode with normal co-ordinate r 2 . Then the action of an element G of the group G on the wave-function i>f<^^ of Eq. [C.78] may be written - i _ A - w i k L-MU <P; *; [C.78] 0 ; cx [C.79] [C.80] IC.81] -15 2-where A'(G) and A"(G) are the representations of G for which <j>^  and <f>j 1 respectively, are basis functions. Recall, however, that the direct product A of two matrices A* and A", written A = A' fi A", is given by t c - 8 2 1 A.. % Therefore, [C.83] In other words, the combination level ^ ^ j j transforms lik e the direct product of the representations transforming and 4>j '. C-5(f) Transformation Properties of the Most General Wavefunction The foregoing may be applied to determine the transformation properties of, for example, the most general doorway state. The doorway system has only 8 distinct frequencies. Its wavefunction can, thus, be written [C84] Y = 7 n n i ° 3 \ V n, n„ n„ Therefore, bearing in mind the transformation properties of the funda-mental doorway wavefunctions as li s t e d in the Table C-II in Section C-5(c), X n w i l l transform under the representation [C.85] A (&) -153-A similar expression applies, of course, to the most general single particle state i> . The two wavefunctions x and ^ may have a non zero m n m overlap only i f the transformations for which they are basis functions contain, when reduced, at least one common irreducible representation of the group G . Thus for the analysis of any given overlap i t w i l l be necessary to reduce a representation of the form [C.85] for the doorway system, and a similar representation for the single particle system. It i s to be noticed that, although A Q (G ) * n Eq. IC.85] is written in terms of i r -reducible representations of G , i t is not written in reduced form; i.e. i t i s not written as a direct sum of irreducible representations. To carry out this reduction one makes repeated use of the expression [C.37], namely, [ C 8 6 ] ~- JL Z TrO*<S>] -Tr\>(<^l IG-I a where u, i s the number of times that the representation A contains the i a irreducible representation D^. For this purpose one needs the traces of the representation which can be obtained with the help of the ex-pressions given i n Eq. [C.87] to Eq. [C.92] below. A general expression for the trace [ x ] N ( G ) of the symmetric nth power representation [ T ] N ( G ) i s given by Lyubarski (1960), from which [c.87] [xYtQ-) - \ T ^ C O «• i [C.88] [ X ] ' ^ ) ~- i + \ X 1 O) -15 4-[C.89] [XT J *M * i *('^ " 1 * aM The symmetrized nth product representation of a d i r e c t sum of 2 representations i s given i n reduced form by Watson (1972) as follows. [c.90] rr.©rjx . [r,V© [r,©rj © [ r ^ rc9i] [r,anvV »[r,]* * fnV®r* 0 @ ^ rc.92] [r,©aT - [n]" © [ra3® a © [^ ©[itf © r,©lr*V © . The reduction of representations such as [T, © T 2 © . ^ + . . . ] n i s e a s i l y obtained from the immediately preceding three expressions by i t e r a t i o n . Further, since some of the i r r e d u c i b l e representations of G are one-dimensional, i t i s convenient to gather here some r e s u l t s f o r the case of one-dimensional representations. Let r(G) be a one-dimensional r e -presentation, i . e . i t has one basi s function r ; i . e . [c.93] p r - V(<£\ r Therefore [c.94] P«. r* * r(£)<2> >>> r" -155-Thus i t follows from the d e f i n i t i o n of the symmetric nth power representation given i n Section C-5(d) that [C.95] [VY(l) = ^ 0 ) ® P(S) ® ••• n times, i . e . for a one-dimensional representation the symmetric nth power r e -presentation i s the same as the d i r e c t product representation. Now l e t \ii be the number of times that the i d e n t i t y representation, (G) i s con-tained i n the d i r e c t product representation T(G) S r(G). Then, from Eq. [C.86],. [C.96] yux = J - X P ( 0 =• \ s i n c e , ( G ) = 1 for a l l G e G and r(G) contains r(G) once. However, since.T(G) fi r(G) i s one dimensional,Tj(G) exhausts i t s dimensionality. Hence,T(G) fl,r(G) =.Ti(G), and i n general, [C.97a] [VY (1) - P, ( £ ) for even n [C.97b] [?Y '(<H - T (t) f o r odd n. C-5(g) P r e d i c t i o n of N u l l Overlaps - An Example The foregoing machinery permits the p r e d i c t i o n of whether or not the overlap i n t e g r a l I of Eq. [C.3] between any s i n g l e p a r t i c l e wavefunction ^ m and any doorway system wavefunction X n must equal zero. As an example of the procedure to be followed, the s p e c i a l case treated i n Section C-1 from a d i f f e r e n t point of view, w i l l be reanalyzed here for comparison. -156-Consider a single particle state with 4 quanta in the second normal mode. Since the corresponding fundamental wavefunction transforms like the 1-dimensional representation D^CG), the 4 quanta wavefunction \j> w i l l transform lik e the identity representation D^CG). Consider a l l doorway system states with 4 quanta. As given i n Eq. [C.85], their wave-functions transform under the representation [c.98] = ixr® w " 3 ® W*- ® W*" ® W4 ® with n 2 + n 3 + n^ + ns + ng + xij + n}3 + nj 9 = 4. Thus, following the theorem of Section C-2, the overlap integral I between the two wave-nm functions w i l l be zero i f A (G) does not contain D T(G). a ' I Let u be the number of times the representation A^CG) contains the representation D^(G). Then, from Eq. [C.86], i t follows that [C99] ^ , _L Z T r l A j * ) ] In the next 5 paragraphs Eq. [C.99] is applied i n turn to each of the 5 possible partitions of 4 quanta with the aid of the expressions [C.87] to [C.92], Note that the possible values of the index i , indicating the normal frequency having n^ quanta, are 2,3,4,5,6,7,13, and 19. (1) The case n^ = 4. It is easily shown that each of the re-presentations [ D j j l V [DJ] 4, ID2 © D 2 © D 3 ] \ and [3D « 3 T > n ^ k contain Dj.. Therefore, any doorway state with 4 quanta in any one frequency may have a non-zero overlap with m2=4 single particle state. -157-(2) The case n = 3, n = 1, with i ^  j . Again, by direct calculation using Eq. [C.99] one can show that the representations [ D ^ ] 3 fi [ D J J ] 1 , [Dj] 3 S I D J . ] 1 , I D J ] 3 fi [3DI © M ^ ] 1 , [ D ^ . ] 3 fi [3DZ © 3D I ] ;] 1, DDj © 3D I ] :] 3 fi [D ^ 1 , [3^ © 30.^] 3 fi [ D ^ 1 , and [Dl © D 2 © D 3 ] 3 fi [Di © D 2 © D3] 1 are, in this case, the only representa-tions containing D^.. Thus, the following numbers of overlaps between the m2=4 single particle state, and the doorway state with 3 quanta in 10^ and 1 quantum in to. are predicted to be zero. i = 2 2 2 2 3 3 3 3 3 4 4 4 4 5 5 5 5 5 6 6 6 6 3 = 3 5 7 19 2 4 6 7 19 3 5 7 19 2 4 6 7 19 3 5 7 19 # = 1 1 6 6 1 1 1 6 6 1 1 6 6 1 1 1 6 6 1 1 6 6 i = 7 7 7 7 7 7 13 13 19 19 19 19 19 19 j = 2 3 4 5 6 13 7 19 2 3 4 5 6 13 # = 56 56 56 56 56 336 336 336 56 56 56 56 56 336 This adds up to 1976 overlaps of this type predicted to be zero. (3) The case n^ = 2, n^ = 2, with i ^  j . In this case, i t can be shown that each of the representations [B^]2, [ D . ^ ] 2 , [3D^ © 3D ] 2 , and [Di © D 2 © D 3 ] 2 contain D . Therefore, i f a state of the doorway system has 2 quanta in one frequency and 2 quanta in another frequency, none of i t overlaps with the m2=4 single particle state have to be zero. (4) The case = 2, n^ = 1, = 1, with i ^ j / k . In this case, i t can be shown that the representations -158-[Dj] 2 8 [Di © D 2 © D3] 1 8 [Di © D 2 © D 3 ] J , [ D j j l 2 ® [Dl © D 2 © D3] 1 8 [Di © D 2 © D3] 1, [3D © 3 D n J 2 fi [ D l © D2 © ^ l 1 8 I D1 © D2 © D3] 1, [ D j ] 2 8 [D I X] S I D J J ] , [ D n ] 2 8 I D I X ] 8 [ D ^ ] , [3DX © 3 0 ^ ] 2 8 [D^] 8 [ D n J , [ D x ] 2 8 [D r i] fi [3Dj © 3 D n ] f I D I I ; ] 2 fl ID^] a [3DX © 3 0 ^ , [D].]2 8 [Dj.] 8 [3D][ © 3D I ] ;], [ D J J ] 2 a [Dj] fl [D].] , [ D ^ 2 8 [Dj] 8 [3DX © 3 D n ] , [3Dj © 3D I ] ;] 2 8 [Dj] 8 ID J], [Di © D 2 © D3] 2 8 [Dj] 8 [D^ , [D], © D 2 © D3] 2 8 [D ] 8 [D i ; [], [Di © D 2 © D3] 2 8 [Dj] 8 I3D I © 3D I ] ;], [Di © D 2 © D3] 2 8 [DJ.J.1 8 [D J J], and [D x © D 2 © D 8 [D^] 8 [3D^ © 3D ] are the only representations of this type which contain D^. Therefore, the following numbers of overlaps between the m2=4 single particle state, and the door-way state with 2 quanta in and 1 quantum i n co. and i n u are predicted 1 J K to be zero. i » 2,3,4,5,6 2,3,4,5,6,13 7,19 i - 2,4,6 2,3,4,5,6,13 7,19 k = 3,5 7,19 2,3,4,5,6,13 # - 18 1860 2772 This makes 4650 overlaps of this type predicted to be zero. (5) The case = n^ = n^ = n^ = 1, with i^j^k^-c. The re-presentations [Dj] ® [Dj] 9 ID I ] ;] 8 [D I ] ;], [D];] 8 [Dj] » [D^] 8 [3Dj. ® 3 D n ] , IDJ] a [DJ J ] 8 [ D I X ] a [3DI © 3 D n ] , [ D ^ 8 [D^J 8 J D ^ J 8 [3Dj © 3D I ] ;], [Di © D 2 © D 3J 8 [Dj © D 2 © D3] 8 [D ] » [Dj], [Di © D 2 © D 3], [Di © D 2 © D 3] a [Dj] £ I D J J ] , and [Di © D 2 © D 3] 8 ID! © D 2 © D 3] 8 [ K J J ] 8 iDjj] can be shown to be the only representations containing DT. -159-Therefore the following numbers of overlaps between the m2=4 single particle state, and the doorway state with 1 quantum in each of i>i±, oiy to, , and u» are predicted to be zero. i = 3,5 7,19 7,19 j - 2 2,3,4,5,6 13 h = 4 2,3,4,5,6 2,3,4,5,6 I = 6 2,3,4,5,6 2,3,4,5,6 # = 2 120 720 These are 842 overlaps of this type predicted to be zero. In conclusion, a straightforward group theoretical analysis of the overlaps between the single particle state m2=4 and a l l the possible doorway states with 4 quanta has led to the prediction of a total of 7468 null overlaps. This i s to be contrasted with the 14,880 nu l l overlaps predicted i n Section C-1 by means of an examination of the structure of the integrals themselves. Thus, a group theoretical analysis i s only of limited usefulness i n the present study of strength functions. The reason for some of the limitation of this analysis was pointed out i n paragraph (2) of Section C-5(a). In effect the wavefunctions which in Table C-II are classified according to representations of the group G are not basis for irreducible representations of G. Although i t would have been possible to search for a transformation of the wavefunctions among themselves such that the new wavefunctions being basis for irreducible representations of G could have resulted i n a larger number of zero predictions, i t would have been d i f f i c u l t to connect in a detailed manner the n u l l overlaps of the new wavefunctions with the overlaps of the wave-functions actually used. -160-BIBLIOGRAPHY Almqvist, E. , Bromley, D. A., and Kuehner, J. A., Phys. Rev. Lett. 4_, 515 (1960) 19 19 Report of the f i r s t three intermediate resonances in the -^ C-H- C reaction. Almqvist, E., Kuehner, J. A., McPherson, D., and Vogt, E. W., Phys. Rev. 136, B84 (1964) The f i r s t application of Ericson fluctuations (due to overlapping compound nucleus levels) is made using 1 2 C ( 1 2 C , a) 2 0Ne. Compound level widths and spacings are discussed. Arima, A., G i l l e t , V., and Ginocchio, J., Phys. Rev. Lett. 25, 1043 (1970) By calculating the energies of quartet structures in even-even N=Z nuclei from 1 2C to 5 2Fe i t i s shown that these can compete favorably in energy with single nucleon excitations. Arima, A., Broglia, R. A., Ichimura, M. , and Schafer, K., Nucl. Phys. A215, 109 (1973) Four-nucleon direct transfer reactions such as 1 6 0 ( 1 6 0 , 1 2C) 2 0Ne are discussed by considering 1 6 0 = 1 2 C + a and 2 0Ne = 1 60 + a. Arnett, W. D. , Astrophys. Space Sci. 5_, 180 (1969) A treatment of explosive 1 2 C burning in supernovae. At the onset, however, hydrostatic carbon burning may occur near Tg « 0.3 (see p. 190). , and Truran, J. W., Astrophys. J. 157, 339 (1969) Using new estimates for the reaction rate the estimated temperature range for carbon burning i s increased to T = (0.8 - 1.2) x 10 9 K. B a l l i n i , R., Cindro, N., Fouan, J. P., Kalbach, C , Lepareux, M., and Saunier, N., Nucl. Phys. A234, 33 (1974) It is found that 2 0Ne + a has intermediate resonances at the same energies as 1 2C + 1 2C. The 2 3Na + p reaction does not. This supports the a-cluster doorway interpretation. Basu, M. K., Phys. Rev. C6, 476 (1972) A study of a clustering in the ground states of light nuclei. Clustering i s almost complete in the ground state of 1 2C, but i s much less so for 21*Mg and heavier nuclei. Baudet, W. D., and Salpeter, E. E., Astrophys. J. 155, 203 (1969) A discussion of the conditions (masses and temperatures) necessary for hydrostatic carbon burning in stars. Bethe, H. A., Intermediate Quantum Mechanics, Benjamin, New York (1964) The Hartree-Fock self-consistent method i s described in Chapter 6. -161-Biarens de Haan, D., Nouvelles tables d'intggrales defines, Amsterdam (1867) A useful compendium of definite integrals. Table 28, #1, permits the evaluation of integrals of the exponential function of a general binomial expression without the use of a principle axis transformation. Bloch, C , Nucl. Phys. 3, 137 (1957) The strength function and i t s moments are discussed in terms of a simple model that gives a Lorentzian strength function shape. , Nucl. Phys. _4, 503 (1957) A unified formulation of the theory of nuclear reactions based on writing the scattering matrix as the Green function for the Schrbdinger equation. Special cases lead to the Wigner resonance model, to the optical model, and to direct interactions for inelastic scattering. __, in Nuclear Physics (Les Houches Summer School 1968) ed. by C. de Witt and V. G i l l e t , Gordon and Breach, New York (1969) A review of s t a t i s t i c a l nuclear theory covering level densities, s t a t i s t i c s of level spacings and partial widths, average cross sections, and cross section fluctuations. Block, B., and Feshbach, H., Ann. Phys. _23, 47 (1963) The f i r s t use of the concept "doorway state" to interpret the systematic deviations of the neutron strength function around the optical model prediction. Breit, G., Rev. Mod. Phys. 34, 766 (1962) Reviews some aspects of N-N scattering including the v a l i d i t y of the OPEP. Concludes (p. 800) that the OPEP is the main interaction for r > 2.9 fm. Bressel, C. N., Kerman, A. K., and Rouben, B., Nucl. Phys. A124, 624 (1969) They show that a Hamada-Johnston potential with a repulsive square well core of height ~ 670 MeV yields the best f i t to date for the phase shifts. De Benedetti, S., Nuclear Interactions, Wiley, New York (1964) Excellent introduction to a l l aspects of nuclear interactions "containing a description of the basic facts and a clear presentation of the theory". E l l i o t * J. P. and Skyrme, T. H. R., Proc. Roy. Soc. Lond. A232, 561 (1955) They sort out the centre of mass motion from the internal motion of the system in a shell model with a harmonic osc i l l a t o r potential. Spurious states are those that do not refer to the internal motion. Endt, P. M. and Van der Leun, C , Nucl. Phys. A214, 1 (1973) A review of nuclear energy levels for A = 21-44. In particular, there is a complete treatment of 21+Mg, i t s energy levels, decays, and resonances where 21+Mg is the compound nucleus. Extensive up-to-date references. -162-Farrel, J. A., Bilpuch, E. G. , and Newson, H. W. , Ann. Phys. 37_, 367 (1966) One of a series of papers from Duke U. on s- and p-wave neutron spectroscopy for A = 40-96. "Together the Wigner and Porter-Thomas distributions explain the data very well". Feshbach, H., in Reaction Dynamics ed. by E. W. Montroll et a l . , Gordon and Breach, New York (1973) 1968 Summer School in Mexico. Nuclear reactions are dealt with in 45 pages covering scattering theory, transition amplitude, resonance theory, optical model, energy averaging, and intermediate structure. , in Proceedings of the International Conference on Nuclear Physics, Munich, 1973, ed. by J. de Boer and H. J. Manz, North-Holland, Amsterdam (1973) A brief survey of intermediate structure in compound nuclear reactions including nucleon and heavy ion projectiles. _ _ _ _ _ _ Porter, C.E., and Weisskopf, V. F. , Phys. Rev. 96, 448 (1954) The introduction of the optical model as a tool for the description of average neutron cross sections. _______ Kerman, A. K. , and Lemmer, R. H. , Ann. Phys. 41, 230 (1967) A thorough and detailed introduction to the subject of intermediate structure and doorway states. Garg, J. B., ed. S t a t i s t i c a l Properties of Nuclei (Albany Conference 1971), Plenum Press, New York (1972) Contributions by Wigner, Feshbach, Lane, Mahaux, etc. on spacings, correlations, strength functions, level densities, intermediate structure, etc. Gibson, W. M., Nuclear Reactions, Penguin, England (1971) In page 152 can be found a clear account of the principles of thermonuclear reactions including Coulomb barrier penetration and reaction rates. Goldstein, H., Classical Mechanics, Addison-Wesley, Reading (1950) This well-known text gives a clear and elegant presentation of many topics in classical mechanics. Chapter 10 deals with small oscillations and the principle axis transformation to normal coordinates. Hanson, D.L. , Stokstad, R. G. , Erb, K. A., Olmer, C , Sachs, M.W. , and Bromley, D. A., Phys. Rev. C9, 1760 (1974) A review of the existence and non-existence of intermediate structure in heavy ion reactions. The level density of the compound nucleus is important but the details of nuclear structure must also be invoked in some (unknown) way. Harrington, D. R., Phys. Rev. 147, 685 (1966) The ground state of *2C is calculated quite successfully using s-wave interactions among 3 r i g i d alpha particles. -16 3-Henley, E., and Thirring, W., Elementary Quantum Field Theory, McGraw-H i l l , New York (1962) Quantum Field Theory is treated non-relativistically. The f i r s t chapters deal with a vibrating line of atoms, the harmonic oscill a t o r , and coupled oscillators. Periodic boundary conditions are used. Hermanson, J. C., and Robiscoe, R. T., Am. J. Phys. 41, 414 (1973) Short note on non-exponential decay. Hodgson, P. E., Annual Review of Nuclear Science, 17_, 1 (1967) A brief review of the optical model for the nucleon-nucleus interaction up to 1967. Hohn, F. E., Elementary Matrix Algebra, Macmillan, New York (1964) A very elementary but complete treatment of linear algebra. Very readable. A l l concepts are clearly defined. Nice exposition of orthogonal transformations, eigenvalue problems, etc. Igo, G., Hansen, L. F., and Gooding, T. J., Phys. Rev. 131, 337 (1963) As obtained from quasielastic (a, 2a) scattering at 910 MeV, i t is claimed that "alpha clusterization in carbon i s nearly 100% complete". Ikeda, K., Marumori, T., Tamagaki, R., Tanaka, H., Hiura, J., Horiuchi, H., Susuki, Y., Nemoto, F., Bando, H., Abe, Y., Takigawa, N., Kamimura, M., Marumori, T., Takada, K., Akaishi, Y., and Nagata, S., Prog. Theo. Phys. Supp. No. 52 (1972) The entire issue entitled Alpha-Like Four Body Correlations and  Molecular Aspects in Nuclei contains 7 articles devoted to topics such as Nuclear Forces and Clusterization, 8Be, etc. Jackson, A. D., Lande, A., and Sauer, P. U., Phys. Lett. 35B, 365 (1971) Calculations on the trinucleon bound state using Reid's soft core potential and a harmonic oscillator basis. Jansen, L., and Boon, M. , Theory of Finite Groups. Applications in Physics, North-Holland, Amsterdam (1967) Useful theorems of the theory of induced representations are presented. Page 161 treats the special case of a subgroup of index 2. Kennedy, H. P., and Schrils, R., eds., Intermediate Structure in Nuclear  Reactions, U. of Kentucky Press, Lexington (1968) June 1966 lecture series on some of the early ideas on the topic. Contributions by R. Lemmer, L. S. Rodberg, J. E. Young, J. J. Griffin,, A. Lande. Krutchkoff, R. G., Probability and S t a t i s t i c a l Inference, Gordon and Breach, New York (1970) * A very useful and readable text discussing many distribution functions and the relations between them. -164-Lane, A. M., Rev. Mod. Phys. 32, 519 (1960) A careful discussion of the concept of reduced width and a computation of the experimental reduced widths for particle and photon channels in light nuclei. , Thomas, R. G., Wigner, E. P., Phys. Rev. 98, 693 (1955) A giant resonance model is introduced to account for the success of the optical model in nucleon-nucleus scattering. The nuclear strength function and the moments of the residual interaction are discussed and calculated. , and Thomas, R. G., Rev. Mod. Phys. 30, 257 (1958) A comprehensive review of the R-Matrix theory of nuclear reactions. It i s also shown (pp. 305, 306) that the average cross section can be expressed in terms of the strength fuction whether or not the compound nucleus levels overlap. Latham, W. P., Jr., and Kobe, D. H., Am. J. Phys. 41, 1258 (1973) A simple model is solved exactly, and also using the Hartree-Fock and the Maximum Overlap methods. The two approximation methods are compared. Which method is better depends on the problems in which one is interested. Lieb, E. H., and Mattis, D. C., eds., Mathematical Physics in One  Dimension, Academic Press, New York (1966) A collection of reprints with introductory text i l l u s t r a t i n g the vast usefulness of one-dimensional prototypes in a wide range of physical problems. Lindgren, R. A., Trentelman, J. P., Anantaraman, N., Gove, H. E., and Jundt, F. C., Phys. Lett. B49, 263 (1974) Alpha particle transfer reactions can populate (sd) 3 (fp) states in 3 2S and 3 6A N with strength comparable to (sd)1* states. Lyubarski, G. Ya., The Application of Group Theory in Physics, Pergamon Press, London (1960) Expressions for symmetrized multiple products of representations are found in p. 75. The symmetry species of excited states of an oscillatory system are considered in p. 278. Macfarlane, M. H. , and French, J. B. , Rev. Mod. Phys. 32,, 567 (1960) A discussion of reduced widths and spectroscopic factors. Information from stripping reactions i s used to study the structure of light nuclei. Mahaux, C., Annual Review of Nuclear Science, 23, 193 (1973) A brief study of the status of intermediate structure in nuclear reactions. Mathews, J., and Walker, R. L., Mathematical Methods of Physics, Benjamin, New York, 1964 A large number of useful techniques are explained. Ch. 6 deals with coordinate transformations and diagonalization of matrices. Ch. 14 discusses distribution functions, moments, etc. -165-Mazarakis, M. G. , and Stephens, W. E., Astrophys. J. 171, L97 (1972) The 1 2C+ 1 2C reaction is measured down to 2.54 MeV in the cm. Intermediate structure is seen as low as ~ 4 MeV; at lower energies large error bars preclude any conclusions re intermediate structure. Mello, P. A., and Flores, J., in Proceedings of the International Conference on Nuclear Physics, Munich, 1973, ed. by J. de Boer and H. J. Mang, North-Holland, Amsterdam (1973) Vol. 1, p. 234 • The level density is a global property of nuclear spectra, while the level spacing distributions are local properties. Level spacings for a particular have a Wigner distribution. If a l l J i r values are allowed the distribution is Poisson. Mendez-Moreno, R. M., Moreno, M., and Seligman, T. H., Nucl. Phys. A221, 381 (1974) Using the best available phenomenological a-a potentials the spectra and form factors of 1 2C and *60 are calculated in the a-particle model. Merzbacher, E., Quantum Mechanics, Wiley, New York (1961) A textbook in elementary quantum mechanics. In Chapter 5 the Schr6"dinger equation i s solved for a harmonic osc i l l a t o r potential. The orthogonal eigenfuctions of the harmonic osc i l l a t o r are given. Messiah, A., Quantum Mechanics, North-Holland, Amsterdam (1961) A well-known, pedagogically written textbook. Appendix B of Vol. 1 is a useful treatment of special functions (Coulomb, spherical Bessel, etc) and associated formulae. Chap. XVIII discusses the Hartree-Fock method. Michaud, G., Astrophys. J. 175, 751 (1972) A discription of the status of the 1 2C+ 1 2C, 1 2C+ 1 60, and 1 60+ 1 60 stell a r reaction rates. New rates are calculated with a soft core optical potential giving a factor of 10 increase over previous calculations. _, Scherk, L., and Vogt, E., Phys. Rev. CI, 864 (1970) An "equivalent square well" i s constructed for diffuse edge nuclear potentials such that nuclear reaction calculations depending on wave properties are simplified. , and Vogt, E. W., Phys. Rev. C5, 350 (1972) The 1 2C+ 1 2C reaction is analysed phenomenologically using an optical potential to describe the gross energy dependence of the cross section. A framework is given in which to understand the inter-mediate structure of the cross section in terms of alpha-particle doorway states. Middleton, R. , Garret, J. D. , and Fortune, H. T., Phys. Rev. Lett. 27_, 950 (1971) The 1 2 C ( 1 2 C , a) 2 0Ne reaction i s shown, at one angle, to preferentially populate quartet states in 2 0Ne compared to other states of the same spin and parity. The quartet states consist of two sd-shell a particles outside a 1 2C core. -16 6-Moshinsky, M., The Harmonic Oscillator in Modern Physics: From Atoms  to Quarks, Gordon and Breach, New York (1969) An exploration of the application of harmonic oscillator states to a variety of many-body problems. Mostowski, J., and Wddkiewicz, K., Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron, & Phys. 21, 1027 (1973) A discussion of the decay law of unstable states. It is shown that at large times there are deviations from the exponential decay law. Noyes, H. P., Annual Review of Nuclear Science, 22, 465 (1972) A review of the N-N effective range parameters, a ^ ^ a® n because the scattering length is too sensitive to higher order failures of charge independence. Porter, C. E., ed., S t a t i s t i c a l Theory of Spectra: Fluctuations, Academic Press, New York (1965) A collection of important reprints together with an introduction by Porter which includes an application of st a t i s t i c s to prime numbers. Contains work by Wigner, Porter, Mehta, Dyson, Ericson. , and Thomas, R. G., Phys. Rev. 104, 483 (1956) The usefulness of a particular chi-squared distribution in describing the fluctuations of reduced widths is pointed out. The number of "degrees of freedom" is a lower bound on the number of open channels. Preston, M. A., Physics of the Nucleus, Addison-Wesley, Reading (1962) Very clear and didactic source book. Extensive treatment of N-N interactions. Sect. 18-3 deals with the strength function. Sect. 17-2 deals with the s t a t i s t i c a l distribution of reaction widths. Reeves, H., Stellar Evolution and Nucleosynthesis, Gordon and Breach, New York (1968) A brief, very readable book, suitable as an introduction to the subject. Reid, R. V., Jr., Ann. Phys. 50, 411 (1968) Excellent review of N-N potentials. Lots of references. Also constructs his own potentials to f i t latest phase shifts. Often quoted. His cores, however, are s t i l l too hard for a harmonic oscillator approximation. Robiscoe, R. T. , and Hermanson, J. C , Am. J. Phys. 40, 1443 (1972) Deviations from exponential decay due to the dependence of the decay rate on the amplitude at earlier times is discussed. Non-exponential decay implies non-Lorentzian line shapes. Rohr, G., and Friedland, E., Nucl. Phys. A104. 1 (1967) The distribution of neutron widths in b*V from 5 to 16Q keV is found not to obey a Porter-Thomas distribution. -167-Scherk, L., and Vogt, E. W., Can. J. Phys. 46, 1119 (1968) A concise description of how single-particle effects are to be handled with an application to alpha decay. Discussion of widths, reduced widths, spectroscopic factors. Schiff, L. I., Phys. Rev. B133, 802 (1964) The electromagnetic form factors of 3H and 3He are calculated in terms of the properties of the ground state wavefunction. The choice of wavefunctions is discussed in terms of the isospin formalism and the Pauli principle. Gaussian wavefunctions are given. , Quantum Mechanics, McGraw-Hill, New York (1968) A well-known textbook. Chapter 10 presents a particularly clear treatment of identical particles. Shapiro, I. S., Sov. Phys.-Usp. 4_, 674 (1962) Review of the optical model for nucleon and heavier particle scattering. The unexpectedness of i t s success for composite particles i s pointed out. Siddappa, K. , Sriramachandra Murty, M., and Rama Rao, J., Ann. Phys. 83, 355 (1974) Neutron strength functions of nuclei in the deformed region. The introduction gives a nice account of the meaning and significance of the concept of neutron strength function. Stie g l i t z , R. G., Hockenbury, R. W., and Block, R. C. Nucl. Phys. A163, 592 (1971) Taking into account the contribution of p-waves to the neutron cross section on 5 1V eliminates the disagreement with the Porter-Thomas distribution found by Rohr and Friedland. Stokstad, R. G., Hanson, D., Olmer, C., Erb, K., Sachs, M. W., and Bromely, D. A., in Proceedings of the International Conference on Nuclear Physics, Munich, 1973, ed. by J. de Boer and H. J. Mang, North-Holland, Amsterdam (1973), Vol. 1, p. 548 The discovery of heavy ion reactions leading to compound nuclei of level densities similar to 2tfMg and not possessing intermediate structure was a puzzle to the understanding of intermediate structure. Tamagaki, R., Rev. Mod. Phys. 39, 629 (1967) Combines meson-theoratical predictions at large distances (> 1.5 fm) with pheonomenological description at small distances to obtain a soft core N-N potential. , Prog. Theo. Phys. 39, 91 (1968) Contains parameters for various types of potentials in various states including a very useful (for calculations) phenomenological three-range Gaussian potential for the N-N interaction. -16 8-Tourreil, R. de, and Sprung, D. W. L., in The Two-Body Force in Nuclei, ed. by S. M. Austin and G. M. Crawley, Plenum, New York, 1972 They report the successes of a super soft core potential in f i t t i n g two-body data and nuclear matter properties. Vogt, E. W., Rev. Mod. Phys. 34, 732 (1962) A very didactic presentation of the theory of low energy nuclear reactions. , in Proceedings of the International Conference on Properties of Nuclear States, Montreal, 1969, ed. by M. Harvey, R. Y. Cusson, J. S. Geiger and J. M. Pearson, Presses de 1'Universite de Montreal (1969), p. 5 An early description of the alpha particle doorway states for the intermediate structure in the 1 2C+ 1 2C reaction was given in the context of a review of nuclear models. , , Nuclear Astrophysics unpublished lecture notes (1971) An introduction to the basic nuclear physics of astrophysics including level widths, barrier penetration, strength function, nuclear models, reaction rates, etc. , and Lascoux, J., Phys. Rev. 107, 1028 (1957) The role of the moments and the widths of the strength function and of the overlap coefficients i s discussed. A relationship is derived between the second moment and the width of the strength function. , McPherson, D., Kuehner, J., and Almquist, E., Phys. Rev. 136, B99 (1964) Analysis of the 1 2 C ( 1 2 C , a) 2 0Ne reaction provides a thorough test for the s t a t i s t i c a l compound nucleus theory. Level spacings and widths for the compound nucleus 2l+Mg are discussed. Voit, H., Ischenko, G., and S i l l e r , F., Phys. Rev. Lett. _30, 564 (1973) Experimental evidence for alpha-particle doorway states in the 1 2C+ 1 2C reaction is presented lending support to the framework of Michaud and Vogt. Voit, H., Duck, P., Galster, W., Haindl, E., Hartmann, G., Helb, H. D., S i l l e r , F., and Ishenko, G., Phys. Rev. CIO, 1331 (1974) A review of correlated intermediate resonances shows they occur only for 1 2C+ 1 2C, 1 2C+ 1 60, and 1+He+20Ne. Two empirical conditions for their observation are i) low level density in the compound nucleus and small number of open channels, and i i ) both nuclei must be a nuclei. Watson, J. K . G., J. Mol. Spectrosc. 41, 229 (1972) Expressions for [Vi 0 r 2 8 ••• ]° are given in terms of the symmetric multiple power representations of T^, r 2 ••*. Weinstock, R., Am. J. Phys. 39, 484 (1971) Exact solutions to linear chains of harmonic oscillators are found with a l l possible boundary conditions. Chains are both uniform and with single mass defect. -16 9-Weisskopf, V., Physics Today 14, 18 (July 1961) A qualitative discussion of nuclear reactions containing implicitly the concept of doorway state later cemented by Block and Feshbach. Wigner, E., Ann. Math. 62, 548 (1955) , Ann. Math. 65, 203 (1956) Motivated by the giant resonance model of Lane, Thomas, and Wigner, strength functions for eigenvectors of certain kinds of random matrices are calculated. ( , Group Theory, Academic Press, New York (1959) A well-known textbook containing many important theorems of group theory - particularly those useful in quantum mechanics. Wilson, E. B., Decius, J. C., and Cross, P. C. , Molecular Vibrations, McGraw-Hill, Toronto (1955) A very readable brief treatise on vibrations dealing with c l a s s i c a l vibrations, quantum mechanical vibrations, symmetries, selection rules. Wong, S. S., and French, J. B., Nucl. Phys. A198, 188 (1972) The level density i s Gaussian i f there are only 2-body interactions and has a Wigner semi-circular shape i f there are many-body (> 7) interactions. Wong, S. S. M., and Wong, S. K. M., in Proceedings of the International  Conference on Nuclear Physics, Munich, 1973, ed. by J. de Boer and H. J. Mang, North-Holland, Amsterdam (1973) Vol. 1, p. 240 Comparison of a Gaussian energy level distribution with that obtained from shell model calculations give favorable results. Zeitnitz, B. , Maschuw, R. , Suhr, P., and Ebenhcih, W., Phys. Rev. Lett. 28. 1656 (1972) Latest (1972) description of experiment and analysis giving *S neutron-neutron effective range parameters. 

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