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-\u00a9 &0>

^ and ~~] -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 \u00ab\u2022 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.\u00a9rjx . [r,V\u00a9 [r,\u00a9rj \u00a9 [ r ^ rc9i] [r,anvV \u00bb[r,]* * fnV\u00aer* 0 @ ^ rc.92] [r,\u00a9aT - [n]\" \u00a9 [ra3\u00ae a \u00a9 [^ \u00a9[itf \u00a9 r,\u00a9lr*V \u00a9 . The reduction of representations such as [T, \u00a9 T 2 \u00a9 . ^ + . . . ] 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(<\u00a3\\ r Therefore [c.94] P\u00ab. r* * r(\u00a3)<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 ) \u00ae P(S) \u00ae \u2022\u2022\u2022 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 =\u2022 \\ 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, ( \u00a3 ) for even n [C.97b] [?Y '(~~