Study of the shift of two-phonon poles in deformed nuclei by the method of boson expansions

Abstract

The situation with regard to two-phonon collective states in even-even deformed nuclei is obscure. According to the Bohr-Mottelson model deformed nuclei must have two-phonon collective states; this was recently pointed out once more in [i]. According to the interacting boson model the low-lying states must include states with large two-boson components [2]. In the framework of the quasiparticle-phonon nuclear model [3] the study of the twophonon states showed [4] that allowance for the Pauli principle, i.e., allowance for the exact commutation relations between the phonons, leads to a shift of the two-phonon poles to higher energies. In [5], the corresponding secular equations were solved. Allowance for the Pauli principle leads to a shift of the centroids of the energy of the collective two-phonon states by 1-2 MeV. At excitation energies of 3-4 MeV the strength of the collective two-phonon states must be distributed over many nuclear levels. It was concluded on this basis in [5] that in even-even deformed nuclei there are no collective twophonon states. In the description of two-phonon collective states in deformed nuclei there is a contradiction between the quasiparticle-phonon model and the Bohr-Mottelson and interacting boson models. As is shown in [6], there is a contradiction between the quasiparticle-phonon model and the interacting boson model in the description of a number of other nonrotational states in deformed nuclei. We note that according to [7] allowance for the Pauli principle in even-even spherical nuclei does not lead to a large shift in the energies of the two-phonon states, in agreement with the experimental data on the twophonon collective states in spherical nuclei. An analysis of experimental data led to the conclusion in [8] that there are no twophonon collective states in deformed nuclei. New experimental data on 1~SEr in [9,10] showed that the levels previously treated as two-phonon levels have a dominant one-phonon component. The absence of two-phonon quadrupole states with energy less than 2 MeV in 168Er is explained in [i,Ii] by the large anharmonicity of the $ and ~ vibrations due to the shape of the nucleus in the form of a triaxial ellipsoid. The absence of two-phonon octupole states in a number of Ra, Th, and U isotopes [12] is explained in many studies by the existence of a stable octupole deformation. We note that the new experimental data on 168Er [9,13] agree well with calculations in the quasiparticle-phonon model and with the calculations in [14] of the structure of the single-phonon states and do not agree with the calculations in the interacting boson model. In view of the importance of clarifying the situation with regard to two-phonon collective states of even-even deformed nuclei, it is necessary to study the behavior of the two-phonon states by a different mathematical formalism. Since allowance for the Pauli principle in the two-phonon components of the wave functions plays a decisive role, it is expedient to make the calculations with phonons constructed from the operators of "true" bosons. It is necessary to establish whether there is a large shift of the twophonon poles in the problem when formulated in this way. The present paper is devoted to