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Physica Scripta

On the Ability of the Interacting Boson Model to Describe Nuclear Deformation Effects

A Bohr, Ben R. Mottelson

Published Jun 1, 1982• 13 citations
Publisher

Abstract

The characterization of many-fermion wave functions as condensates of bosons built out of pairs of particles is further analyzed. It is emphasized that the properties of the many particle systems are extremely sensitive to small components in the structure of the boson. In particular a truncation of the boson to the dominant angular momentum components may lead to results that are qualitatively different from those obtained from the full wave function.

DOI
10.1088/0031-8949/25/6b/003

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Physica Scripta•Nov 1, 1983•2 citations
The characterization of many-fermion wave functions as condensates of bosons built out of pairs of particles is further analyzed. It is emphasized that the properties of the many particle systems are extremely sensitive to small components in the structure of the boson. In particular a truncation of the boson to the dominant angular momentum components may lead to results that are qualitatively different from those obtained from the full wave function.
One considers the superfluid (SF) state of a Bose liquid with a strong repulsion between bosons, in which at T=0, along with a weak single-particle Bose-Einstein condensate (BEC), there exists an intensive pair coherent condensate (PCC), analogous to the Cooper condensate in a Fermi liquid with an attraction between the fermions. Such a PCC emerges in a system of bosons due to an oscillating sign-changing momentum dependence of the Fourier component of the pair interaction potential, which is characteristic of a certain family of repulsion potentials. In such cases, the Fourier component is negative in some domain of nonzero momentum transfer, which corresponds to an effective attraction. The collective effects of renormalization (``screening'') of the initial interaction lead to a suppression of the repulsion and an enhancement of the effective attraction. It is the ratio of the BEC density to the full density of the liquid n0/n≪1n_0/n\ll 1n0​/n≪1 that is used as a small parameter---unlike in the Bogolyubov theory for a quasi-ideal Bose gas, in which the small parameter is the ratio of the number of supracondensate excitations to the number of particles in an intensive BEC, (n−n0)/n0≪1(n-n_0)/n_0\ll 1(n−n0​)/n0​≪1. A closed system of nonlinear integral equations for the normal and anomalous self-energy parts is obtained, in the framework of a renormalized perturbation theory built on combined hydrodynamic and field variables. In the framework of the hard-spheres model, a spectrum of quasiparticles is obtained, which is in good accordance with the experimental spectrum of elementary excitations in superfluid 4^44He. The question of applicability of the Landau criterion in the absence of quantum vortices is discussed.
The interacting boson model introduced in Section 33 has been described from the point of view of the collective model of the nucleus. The 5 components of a d -boson have been related to the 5 degrees of freedom of a shape, with constant volume, which may have quadrupole deformations. The s -boson has been added to simplify the construction of eigenstates which belong to irreducible representations of the U (6) group, the fully symmetric ones being characterized by an integer N . We saw how simple Hamiltonians including single boson energies and boson boson interactions give very good description of various kinds of collective motion.
Deformation is fundamental to understanding nuclear structure. We compare two ways to efficiently realize deformation for many-fermion wave functions, the leading SU(3) irreducible representation and the angular-momentum-projected Hartree-Fock state. In the absence of single-particle spin-orbit splitting the two are nearly identical. With realistic forces, however, the difference between the two is nontrivial, with the angular-momentum-projected Hartree-Fock state better approximating an ``exact'' wave function calculated in the fully interacting shell model. The difference is driven almost entirely by the single-particle spin-orbit splitting.
Variational calculations within the pairing plus quadrupole model are carried out for a set of particles moving in a single j -shell and for values of the coupling constants giving rise to the main features of strongly deformed nuclei. Two calculations are carried out and the corresponding results compared. In the first, particles are allowed to align in the mean field (particle aligned model). In the second, pairs of particles coupled to angular momentum λ = 0 and λ = 2 align themselves in the average field (pair aligned model). Although many features of the results are qualitatively similar, even for well deformed systems, there are important quantitative differences. In particular, the single-particle occupancies predicted by the pair aligned model display a diffusivity around the Fermi surface (effective pairing gap) which is ∼ 1.7 times larger than that predicted by the particle aligned model. The quadrupole moment predicted by the pair aligned model at maximum deformation is ∼ 1.4 smaller than that predicted by the particle aligned model. The pair aligned model and the particle aligned model thus describe two many-body systems which are rather different. In the first pairing correlations are as important as quadrupole correlations while in the second the quadrupole deformation plays the dominant role. The effect of allowing pairs of particles to couple to angular momenta λ > 2 is found to be essential to achieve quantitative agreement between the two models.