Energy levels of A = 21−44 nuclei (V)

Coexistence in odd-mass nuclei

The concept of boson realization (or mapping) of Lie algebras appeared first in nuclear physics in 1962 as the idea of expanding bilinear forms in fermion creation and annihilation operators in Taylor series of boson operators, with the object of converting the study of nuclear vibrational motion into a problem of coupled oscillators. The physical situations of interest are quite diverse, depending, for instance, on whether excitations for fixed- or variable-particle number are being studied, on how total angular momentum is decomposed into orbital and spin parts, and on whether isotopic spin and other intrinsic degrees of freedom enter. As a consequence, all of the semisimple algebras other than the exceptional ones have proved to be of interest at one time or another, and all are studied in this review. Though the salient historical facts are presented in the introduction, in the body of the review the progression is (generally) from the simplest algebras to the more complex ones. With a sufficiently broad view of the physics requirements, the mathematical problem is the realization of an arbitrary representation of a Lie algebra in a subspace of a suitably chosen Hilbert space of bosons (Heisenberg-Weyl algebra). Indeed, if one includes the study of odd nuclei, one is forced to consider the mappings to spaces that are direct-product spaces of bosons and (quasi)fermions. Though all the methods that have been used for these problems are reviewed, emphasis is placed on a relatively new algebraic method that has emerged over the past decade. Many of the classic results are rederived, and some new results are obtained for odd systems. The major application of these ideas is to the derivation, starting from the shell model, of the phenomenological models of nuclear collective motion, in particular, the geometric model of Bohr and Mottelson and the more recently developed interacting boson model of Arima and Iachello. A critical discussion of those applications is interwoven with the theoretical developments on which they are based; many other applications are included, some of practical interest, some simply to illustrate the concepts, and some to suggest new lines of inquiry.

Boson realizations of Lie algebras with applications to nuclear physics

Overview of neutron-proton pairing

The role of neutron-proton pairing correlations on the structure of nuclei along the $N=Z$ line is reviewed. Particular emphasis is placed on the competition between isovector ($T=1$) and isoscalar $(T=0$) pair fields. The expected properties of these systems, in terms of pairing collective motion, are assessed by different theoretical frameworks including schematic models, realistic Shell Model and mean field approaches. The results are contrasted with experimental data with the goal of establishing clear signals for the existence of neutron-proton ($np$) condensates. We will show that there is clear evidence for an isovector $np$ condensate as expected from isospin invariance. However, and contrary to early expectations, a condensate of deuteron-like pairs appears quite elusive and pairing collectivity in the $T=0$ channel may only show in the form of a phonon. Arguments are presented for the use of direct reactions, adding or removing an $np$ pair, as the most promising tool to provide a definite answer to this intriguing question.

Overview of Neutron-Proton Pairing

A more useful number conserving basis which takes the place of the BCS quasi-particle basis is constructed. The nucleon operator is expressed in terms of two kinds of operators which represent J =0-coupled nucleon pairs and unpaired nucleons. The pairing states are treated by a linearization technique which makes use of the boson-like property of operators. The motion of unpaired nucleons is described by a modified quasi-particle. The method is applicable not only to the super-conducting phase but also to the normal one. It gives number conserving treatments of the pairing vibration and of the coupling with quadrupole phonon.

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Modification of Quasi-Particle Theory in Spherical Nuclei. I Method

We study global features of proton-neutron (p-n) interactions using double differences of binding energies. The p-n interactions are microscopically investigated using the quasiparticle shell model calculations in an extended P+QQ model proposed recently. It is shown that the isoscalar monopole force and the Q p Q n force are important in determining the behavior of the p-n interaction energies. The relations of respective terms in the empirical mass formula to the mass formula obtained using nuclear interaction are discussed. The symmetry energy originates dominantly in the monopole-type isoscalar p-n force and the J = 0 pairing force.

Global Features of Proton-Neutron Interactions

We study single and double β decay for N ≈ Z nuclei, using an extended P + QQ interaction with four force strengths. The Gamov-Teller (GT) transition is examined using the proton-neutron QRPA (pn-QRPA) in many j shells and using a single j shell model calculation. First, we calculate the single β-decay strength. The integrated strengths of N ≈ Z nuclei in the fp shell region are reproduced well by the pn-QRPA, although some quenching factor is necessary for several cases. The model accounts for the experimental observation that the GT strength increases with decreasing |N − Z|. The single j shell model calculation indicates that the GT strength becomes very large at N = Z. Second, we perform calculations for two-neutrino double β-decay transition strengths for 76 Ge and 82 Se. The results suggest fairly good applicability of the extended P + QQ interaction to single and double β decay.

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Single and Double β-Decay in N≍Z Nuclei

Modification of Quasi-Particle Theory in Spherical Nuclei. II Numerical Calculations in Nuclei with Z or N ≃ 40

The P0+ P2+ QQ force has been completely revised, and in its new form, it is capable of describing nuclear structure of N ≈ Z nuclei well. This paper investigates the applicability of the extended P +QQ model in the upper part of the f7/2 shell using shell model calculations. It is shown that the extended P + QQ model can account for the collective bands of most 52 ≤ A ≤ 56 nuclei as well as the lower part of the f7/2 shell. However, considerable discrepancy with experiment exists for 52 Fe, which suggests the limit of the applicability of this force.

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Munetake Hasegawa

Papers

Modification of Quasi-Particle Theory in Spherical Nuclei. I Method

Global Features of Proton-Neutron Interactions

Single and Double β-Decay in N≍Z Nuclei

Modification of Quasi-Particle Theory in Spherical Nuclei. II Numerical Calculations in Nuclei with Z or N ≃ 40

Applicability of the Extended P+QQ Model in the Upper Part of the f7/2 Shell.