Elsevier

Nuclear Physics A

Volume 119, Issue 1, 21 October 1968, Pages 221-232
Nuclear Physics A

Pairing and deformation in a solvable model and the validity of different approximation methods

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Abstract

We consider an exactly solvable nuclear model in order to test the validity of various well-known approximation methods. The model Hamiltonian is built on two shells and contains two essentially different interactions (pairing and monopole coupling); it exhibits typical features of realistic Hamiltonians of heavier nuclei and leads in addition to an understanding of specific nuclear properties in the transitional regions.

The energies and the expansion coefficients of the lowest states are determined as functions of the characteristics parameters of the model, i.e. the particle number and the strengths of the interactions. The calculations are based on the generalized quasi-spin method. The Hamiltonian H is expressed by a set of operators which define the Lie algebra of O5; the matrix elements of H for the relevant representations of this algebra are obtained by means of an algebraic computer programs which, in addition, takes care of the remaining diagonalization.

The numerical results thus obtained are represented graphically in several diagrams. The characteristics dependences of the relevant quantities on the parameters show the invariance of the “pairing solution” (sharp seniority) with respect to the addition of the second interaction within a well-defined interval of relative strength (pairing region) and prove the existence of a transitional region with strong variations of nuclear properties (characteristic admixtures of higher seniorities) to be followed by a “deformed” region with extremely low-lying levels. In addition, we find typical “vibrational levels” within the pairing region and their continuous transition to the “rotational states” of the deformed domain. The validity of the Frank-Condon principle is illustrated by the characteristic behaviour of excited states in the transitional region.

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