Nuclear Physics A436 (1985) 445457 @ North-Holland Publishing Company ANALYTIC EXPRESSIONS FOR ENERGY CENTROIDS AND WIDTHS IN THE MICROSCOPIC COLLECTIVE MODEL G. ROSENSTEEL Department of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Physics and Quantum Theory Group, Tulane University, New Orleans, Louisiana 70118, USA and J. P. DRAAYER Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA Received 24 September 1984 Abstract: Analytic formulae are given for the U(3) centroids of the collective Bohr-Mottelson potential in the microscopic collective model. In particular, formulae are reported for the centroids of the quadratic [Q. Q _ B’] and cubic [Q. (Q x Q) _ j? “cos3y] rotational scalars in the microscopic quadrupole operator. Favorable comparisons for ground-state intensities are achieved between shell-model diagonalizations and statistical predictions based upon the gaussian approximation to the energy density. These results suggest that statistical measures can be used reliably for truncation of the infinite-dimensional representation spaces of the microscopic symplectic collective theory. 1. Introduction The symplectic collective model is a completely microscopic generalization of the Bohr-Mottelson geometrical model l-l’). The symplectic theory extends Elliott’s SU(3) shell model to encompass core-excited basis wave functions as dictated by the quadrupole and monopole collective degrees of freedom. Thus, the model space of the symplectic theory is defined to be an irreducible unitary representation space for the real symplectic Lie algebra Sp(3, R). But, since the symplectic Lie algebra is noncompact, the symplectic model space is necessarily infinite-dimensional. Of course, the hamiltonian eigenvalue problem on an infinite-dimensional space poses, in general, an intractable technical obstacle. Only very special model hamiltonians with a large symmetry algebra, e.g. the harmonic oscillator Ho, may be solved analytically. Fortunately, because of the nuclear shell structure, the symplectic-model space can be truncated safely to span only the first few major oscillator shells. For example, in symplectic-model calculations in the ds shell for 20Ne and 24Mg, a truncation including 10ho oscillator excitations above the Otto ds shell was found to be adequate 4,5). Nevertheless, the dimension (145,530) of the 24Mg 1Oho truncated space is still formidable. For nuclei in the rareearth region, 445