S. De Baerdemacker

Linear reversible transformations in the Galois field GF(2) and linear reversible transformations in the field of real numbers show both resemblances and differences. The former constitute a finite group isomorphic to the general linear group GL(w, 2), the latter constitute an infinite, i.e. Lie, group isomorphic to the general linear group GL(w, R) (where w is the logic width of the computation, i.e. respectively the number of bits and the number of real numbers, processed by the computer). Generators of the former group consist of merely control gates; generators of the latter group consist of both control gates and scale gates.

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The spectral properties of a tractable collective model Hamiltonian are studied. The potential energy is truncated up to quartic terms in the quadrupole deformation variables, incorporating vibrational, γ-independent rotational, and axially deformed rotational structures. These physically significant limits are analysed in detail and confronted with well-established approximation schemes. Furthermore, transitional Hamiltonians in between the limits are presented and discussed. All results are obtained within a recently presented Cartan-Weyl based framework to calculate SU(1,1)×SO(5) embedded quadrupole collective observables.

In this article we report the results of detailed interacting boson model calculations with configuration mixing for the neutron-deficient Pb isotopes. Calculated energy levels and B(E2) values for ^188-196Pb are discussed and some care is suggested concerning the current classification on the basis of level systematics of the 4₁+ and 6₁+ states in ^190-194Pb. Furthermore, quadrupole deformations are extracted for ^186-196Pb and the mixing between the different families (0p-0h, 2p-2h, and 4p-4h) is discussed in detail. Finally, the experimental and the theoretical level systematics are compared.

The matrix elements of the quadrupole collective variables, emerging from collective nuclear models, are calculated in the natural Cartan–Weyl basis of O(5) which is a subgroup of a covering SU(1, 1) × O(5) structure. Making use of an intermediate set method, explicit expressions of the matrix elements are obtained in a pure algebraic way, fixing the γ-rotational structure of collective quadrupole models.

Very recently the atomic masses of neutron-rich Zr isotopes, from ⁹⁶Zr to ¹⁰⁴Zr, have been measured with high precision. Using a schematic Interacting Boson Model (IBM) Hamiltonian, the evolution from spherical to deformed shapes along the chain of Zr isotopes, describing at the same time the excitation energies as well as the two-neutron separation energies, can be rather well reproduced. The interplay between phase transitions and configuration mixing of intruder excitations in this mass region is succinctly addressed.