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


A one-dimensional chain of particles interacting through harmonic forces is 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 ¹²C+¹²C total reaction cross section. A nested hierarchy of harmonic oscillator systems is constructed to parallel the hierarchy of three stages through which the ¹²C+¹²C 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 in which six groups of 4 particles interact with each other via another average potential, and a "compound" system in which all 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 is defined, 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 its shape is compared to the widely used Lorentzian shape. Group theory is 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 ¹²C+¹²C experiments. Attention is drawn to the consequences of using a Lorentzian strength function for cross section extrapolations.

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