Abstract
Background: The multiconfigurational dynamical symmetry (MUSY) is the common intersection of the shell, collective, and cluster models for the multi-major-shell problem. It is able to describe the spectra of different configurations in different energy windows in a unified way. It is based on some heuristic arguments, related to the connection of the wave functions, and energy eigenvalues. The detailed mathematical background has been worked out so far only for the simplest case of the two binary cluster configurations.Purpose: I intend to construct the exact algebraic framework for the general case of the MUSY, i.e., for any number of configurations and any number of clusters or nucleons. As an illustrative example, the spectrum of different configurations of the $^{44}\mathrm{Ti}$ nucleus is described by a simple Hamiltonian.Methods: Classification schemes defined by different algebra chains need to be combined; in particular, those of the major shell scheme and of the particle index scheme.Results: A class of Hamiltonians, which is invariant under the transformations from one configuration to the other, is determined. In case of the $^{44}\mathrm{Ti}$ low-lying shell-model spectrum, as well as $^{40}\mathrm{Ca}+^{4}\mathrm{He}$ and $^{28}\mathrm{Si}+^{16}\mathrm{O}$ cluster states are obtained in a unified way.Conclusions: The MUSY is based on two pillars: (i) a unified multiplet structure for shell, collective, and cluster model states and (ii) a Hamiltonian which is invariant with respect to the transformations from one configuration to the other.
Highlights
The multiconfigurational dynamical symmetry (MUSY) [1,2] is a unifying symmetry that connects the fundamental structure models of atomic nuclei: the shell, collective, and cluster models [3]. It is a composite symmetry in the sense that each configuration has a usual [U(3)] dynamical symmetry, and in addition a further symmetry transforms these configurations into each other
The MUSY defines the allowed binary cluster configurations of the shape isomers; i.e., it provides us with those reaction channels, which can populate these extremely deformed states, or in which they can decay
The MUSY was first introduced as a connecting symmetry of different cluster configurations [1]. It was called multichannel dynamical symmetry, referring to the reaction channels which define the cluster configurations. The reasoning for this possible composite symmetry was based on the connection of the wave functions, and as a consequence on the relations of the energy eigenvalues
Summary
The multiconfigurational dynamical symmetry (MUSY) is the common intersection of the shell, collective, and cluster models for the multi-major-shell problem. It is able to describe the spectra of different configurations in different energy windows in a unified way. It is based on some heuristic arguments, related to the connection of the wave functions, and energy eigenvalues. Results: A class of Hamiltonians, which is invariant under the transformations from one configuration to the other, is determined. In case of the 44Ti low-lying shell-model spectrum, as well as 40Ca + 4He and 28Si + 16O cluster states are obtained in a unified way. Conclusions: The MUSY is based on two pillars: (i) a unified multiplet structure for shell, collective, and cluster model states and (ii) a Hamiltonian which is invariant with respect to the transformations from one configuration to the other
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