On the transition from shell structure to collective behavior. A simplified shell-model study

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To study the feasibility of carrying out shell-model calculations in nuclei with active protons and neutrons in different major shells, the following simple idealized model has been studied: (i) Proton and neutron configurations are chosen to be (f 5 2p 3 2p 1 2)np and (g 7 2d 5 2d 3 2s 1 2)nn, so that results for the separate proton and neutron basis states to be used in any approximation scheme can be compared with the results for exact shell-model calculations, (ii) The proton and neutron single-particle energies for these active shells are separately taken to be degenerate. (iii) The two-body interaction is approximated by the simple surface delta interaction (SDI). To effect the severe truncation of the full shell-model space needed to make such a shell-model study possible the separate proton and neutron parts of the shell-model basis are built from a superposition of the favored pair states of the SDI (with J ≠ 0, as well as J = 0). In the neutron configuration (g 7 2d 5 2d 3 2s 1 2)nn = 4, for example only three of the 94 shell-model states with Jn = 2 are retained in the truncation scheme. In this highly truncated basis both the energies and the strong B(Ek) values for the transitions from these states to similar favored states with other J-values are within a few percent (or better) of the results of exact shell-model calculations. A truncation of the shell-model space based on such superpositions of favored pair states leads to a manageable shell-model basis (dimensions ≦ 200). (a) The number of states in the separate proton and neutron parts of the basis are small enough (8-13 for the proton space, 15-30 for the neutron space). They are also the key states in the following sense (b) They include the low-lying energy eigenstates of the separate p-p and n-n parts of the interaction (c) They contain most of the collective coherence of the separate proton and neutron configurations. (d) The matrix elements of the n-p part of the interaction between the favored states are in general very large compared with the matrix elements between a favored and an excluded state. The latter effect is studied from several aspects, in particular in terms of sum rules for the matrix elements of the surface multipole operators from which the n-p part of the SDI is built. For most of the low-lying favored states the sum over all favored states gives more than 90 % of the total sum rule for the squares of matrix elements of the surface multipole operators. The results of shell-model calculations in this truncation scheme, with np = 4 or 6, and nn = 4, show many of the features of a quadrupole vibrational spectrum. The presence and exact nature of a 0+ member of the 0+, 2+, 4+ "two-phonon triplet" is dependent on the inclusion of the key favored states with seniorities of 6. © 1972.

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Nuclear Physics, Section A

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