ar X iv : n uc l - th / 9 90 70 72 v 1 1 9 Ju l 1 99 9 An Intrinsic State for IBM - 4

An intrinsic-state formalism for IBM-4 is presented. A basis of deformed bosons is introduced which allows the construction of a general trial wave function which has Wigner’s spin–isospin SU(4) symmetry as a particular limit. Intrinsic-state calculations are compared with exact ones showing good agreement. Typeset using REVTEX 1 The Interacting Boson Model (IBM) was originally proposed to describe collective lowlying states in even-even nuclei. The model building blocks are monopolar (s) and quadrupolar (d) bosons. In the original formulation of the model (IBM-1) no distinction was made between neutrons and protons [1]. Later, connections with the nuclear shell model were investigated [2,3] and a new version was proposed in terms of neutron (sν , dν) and proton (sπ, dπ) bosons, known as IBM-2 [1]. The model has been widely applied to medium-mass and heavy nuclei, where neutrons and protons are filling different major shells. In lighter nuclei with N ≈ Z, however, neutrons and protons are in the same shell and a boson made of one neutron and one proton (known as a δ boson) should be included. This version of the boson model, called IBM-3 [4], is the simplest isospin invariant formulation of IBM. The three types of bosons (ν, π, and δ) form an isospin T = 1 triplet and correspond, microscopically, to spatially symmetric nucleon pairs with S = 0. In particular, the δ boson corresponds to a spatially symmetric S = 0 neutron-proton pair. A further extension of the IBM introduces the neutron-proton boson with T = 0 or σ boson, corresponding to a spatially symmetric nucleon pair with S = 1. This version is known as IBM-4 [5] and gives a proper description of even-even as well as odd-odd N ≈ Z nuclei. The IBM-3 and IBM-4 are appropriate models for N ≈ Z nuclei approaching the proton drip line. Such nuclei are studied intensively at the moment in particular with radioactive nuclear beams. Also, the IBM-4 is a reasonably simple, yet detailed model to study the competition between T = 0 and T = 1 pairing, one of the hot topics in current-day nuclear structure physics. All versions of IBM are algebraic in nature and do not have a direct geometrical interpretation. Such interpretation can be achieved, however, by introducing an intrinsic state which provides a connection to geometric models such as that of Bohr and Mottelson [6]. Intrinsic states have been proposed for IBM-1 [7–10], for IBM-2 [11–13], and for IBM-3 [14,15]. Their primary use is to provide a geometric visualization of the model. In addition, a considerable reduction is achieved in the complexity of calculations, which leaves room for the inclusion of extra degrees of freedom. 2 The purpose of this letter is to propose an intrinsic state for IBM-4. In the limit of strong isovector pairing it reduces to the intrinsic state for IBM-3; in general, it can be used for studying the competition between T = 0 and T = 1 pairing in N ≈ Z nuclei. First, the mean-field formalism for IBM-4 is presented. This formalism is subsequently checked against the results of an exact calculation. The ensemble of bosons in the IBM-4 consists of isovector T = 1 and isoscalar T = 0 bosons which have intrinsic spin S = 0 and S = 1, respectively, to ensure spatial symmetry. The allowed spin-isospin combinations are thus (T, S) = (1, 0) and (T, S) = (0, 1). These, together with the orbital angular momenta l = 0, 2, give rise to 36 different bosons. The corresponding boson creation and annihilation operators are γ lm,Tτ,Sσ and γlm,Tτ,Sσ where l is the orbital angular momentum, m is its projection, T is the isospin, τ is its projection, S is the spin, and σ is its projection. The operators γ̃lm,Tτ,Sσ = (−1) γl−m,T−τ,S−σ are introduced for having appropriate tensor transformation properties. The construction of an intrinsic state requires two ingredients. First, it needs a basis of deformed bosons and secondly, it requires a trial wave function. The deformed bosons are defined in terms of the spherical ones through a unitary Hartree-Bose transformation, Ω†p,T τ,Sσ = ∑ lm λ lm γ † lm,Tτ,Sσ, γ † lm,Tτ,Sσ = ∑ p λ lm Ω † p,T τ,Sσ, (1) and their hermitian conjugates. The deformation parameters λ in these equations verify the following orthonormalization relations: