Group theory of the interacting Boson model of the nucleus

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Abstract
Recently Arima and Iachello proposed an interacting boson model of the nucleus involving six bosons, five in a d and one in an s state. The most general interaction in this model can then be expressed in terms of Casimir operators of the following chains of subgroups of the fundamental group U(6): U(6) ⊆U(5) ⊆O(5) ⊆O(3) ⊆O(2), U(6) ⊆O(6) ⊆O(5) ⊆O(3) ⊆O(2), U(6) ⊆SU(3) ⊆O(3) ⊆O(2). To determine the matrix elements of this interaction in, for example, a basis characterized by the irreducible representations of the first chain of groups, then we only need to evaluate the matrix elements of the Casimir operators of O(6) and SU(3) in this basis as the others are already diagonal in it. Using results of a previous publication for the basis associated with U(5) ⊆O(5) ⊆O(3), we obtain the matrix elements of the Casimir operators of O(6) and SU(3). Furthermore, we obtain explicitly the transformation brackets between states characterized by irreducible representations of the first two chains of groups. Numerical programs are being developed for these matrix elements from the relevant reduced 3j symbols for the O(5) ⊆O(3) chain of groups that were programmed previously.