Single-j Shell Research Articles

A test of the interacting boson-fermion model in a single j-shell of neutrons and protons

An isospin invariant form of the interacting boson-fermion model is shown to reproduce realistic shell-model results for odd nuclei in a single j= 7 2 shell. Some low-lying states of mixed symmetry are found. The boson-fermion interaction is found to have a simple structure and its relation with the shell-model interaction is studied.

Collective magnetic multipole excitations in open shells: 48Ti.

The magnetic multipole even-parity transitions are calculated in $^{48}\mathrm{Ti}$ with a mind to finding interesting collective behavior and to study the effects of increasing the model space. The single-j shell signature selection rules are compared to those in the neutron-proton interacting boson model with good F spin. There are similarities but also differences. Some behavior of the single-j shell calculation survives as the model space increases, e.g., low lying collective M1 and M7 modes. Configuration mixing is, however, vital to describe the M3 and M5 modes as well as the spin flip M1 modes. The distribution of strength between lower and higher isospins is discussed. Results are compared with calculated transitions in $^{42}\mathrm{Sc}$. It appears that allowing one-particle excitations from the single-j shell gives the pattern of the strength distribution; the further addition of two-particle excitations leads to a quenching of the strength distribution.

Quadrupole and higher-order pairing forces

Recent work suggests that the quadrupole pair field may be important in describing certain properties of the crossings of high-spin rotational bands (e.g. crossing frequencies) in quantitative detail. We investigate its relative importance by performing an exact diagonalisation of a cranked deformed hamiltonian containing short-range residual interactions in a single j-shell. The major diflerence between odd and even particle-number crossing frequencies is seen to be described by blocking plus quadrupole pairing ( J = 2) along with a significant hexadecapole pairing effect ( J = 4). Absolute crossing frequencies are also sensitive to higher J-values. The interactions between the crossing levels are also discussed.

Collective magnetic dipole transitions: Dependence of the energies and rates on the nuclear effective interaction

The dependence on the properties of the effective interaction of the energies of and excitation strengths to magnetic dipole states in open shell nuclei is studied. In particular the single j-shell for 48Ti is used as an example. In this case there are two 1 + states with isospin T = 2 and one with isospin T = 3. The conditions for having a strong low-lying collective 1 + state are examined. Focus is also given on the excitation strength to the analog T = 3 state since this is of relevance also to β + Gamow-Teller reactions and to double beta decay. It is found that in the rotational limit there is no strength to the T = 3 state and there is an overly strong low-lying 1 + T = 2 state. This is almost also true for a quadrupole-quadrupole interaction. At the other extreme, as shown by Halse, with a pairing interaction all the strength goes to the T = 3 state. Other interactions considered are pairing plus quadrupole, spin-dependent delta and Kuo-Brown bare and renormalized, and matrix elements taken from the spectrum of 42Sc. It is found that the β + strength can be reduced either by making the two-body J = 2 T = 1 matrix element or J = 1 T = 0 matrix element more attractive, just as was shown by others in heavier nuclei. However such parameter changes have effects on other properties of the 1 + spectrum, which can serve as indicators as to whether or not these changes are justified.

Number projection method.

A relationship between the number projection and the shell model methods is investigated in the case of a single-j shell. We can find a one-to-one correspondence between the number projected and the shell model states.

Properties of Terminating Bands in Nuclei

General properties of terminating bands are briefly reviewed and exemplified on the observed high-spin properties of 158Er and 156Er. The very similar features of the positive parity high-spin spectra of the N = 88 isotones of Dy, Er and Yb are pointed out and discussed. The possibility to estimate B(E2)-values in terminating bands from measured feeding times is explored. The spin contribution from different orbitals is calculated in terminating bands as well as more collective bands. One aim is to get some idea of the interaction between different configurations and to this end we also consider a single-j shell model. The limits of terminating bands in the yrast region when the number of valence nucleons increases is discussed. The usefulness of these results when planning future experimental studies is considered.

Application of the Kishimoto-Tamura boson expansion theory to a single-j shell model.

The boson expansion theory of Kishimoto and Tamura is applied to a single-j shell model. It is shown that this theory is quite accurate, giving results that agree very closely with those of the exact fermion calculations. The fast convergence of the boson expansion is also demonstrated. A critical discussion is then made of an earlier paper by Arima, in which he stated that the Kishimoto-Tamura theory gives rise to very poor numerical results. The source of the trouble encountered by Arima is unmasked.

An IBM analysis of a single j-shell of neutrons and protons

A subset of states of a single j-shell of neutrons and protons with good isospin is mapped into those of a system of isovector s- and d-bosons (IBM-3). The shell-model analogue of the boson U(6) and U(5) labels is established and a shell-model calculation for nuclei in the f 7 2 shell with a realistic effective interaction confirms the physical significance of these labels.

Bose-fermi [formula omitted] supersymmetries and high-spin anomalies

A supersymmetric extension of the interacting boson model (IBM) is constructed to describe high-spin anomalies in both even- and odd-mass spectra of the Hg, Pt region (190 ⩽ A ⩽ 200). Supergroup chains such as U( 6 2j + 1 ) ⊃ Osp( 6 2j + 1 ) ⊃ O(6) × Sp( 6 2j + 1 )… U( 6 2j + 1 ) ⊃ U( 5 2j + 1 ) ⊃ Osp( 5 2j + 1 )… incorporate a single j-shell fermion in addition to the usual “s” and “d” bosons ( L = 0 and L = 2). The orthosympletic supergroup reflects the strong pairing force in the subspace of the fermion intruder level. The model agrees favourably with experiment and microscopic calculation.

IBA mapping of microscopic hamiltonians

Using the seniority-conserving mapping of a realistic fermion hamiltonian into a boson space it is shown that this leads to serious problems for quadrupole forces in a single j-shell. The lack of agreement of the exact spectrum with the mapped one persists if realistic forces are used. This observation invalidates many “microscopic justifications” of the interacting boson model by lowest-order boson mapping.

Self-Consistent Treatment of the Pairing Plus Quadrupole Force in the Nilsson Plus BCS Model and in the Interacting Boson Model

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.

On the exchange term of the interacting Boson-Fermion Hamiltonian

The exchange term of the Interacting Boson Fermion Model is investigated by using I. Talmi's method based on the shell model. A quadrupole operator of a three-proton system is formed; the protons are quadrupole-coupled to the neutron-bosons. Seniority conserving and seniority non conserving terms are considered. The particle number dependence of the parameters is investigated for the single-j shell. The relation between exchange and direct, seniority non conserving terms is examined. Approximate formulas are given for the multi-j shell.

Structure of collective states in even-even systems with a proton single j-shell and a neutron single j-shell

A relationship between the shell model, the quasi-particle treatment and the interacting boson model is presented in terms of two independent operators standing for J = 0 coupled nucleon pairs and unpaired nucleons. Shell model calculations are carried out in a model space of a proton single j-shell and a neutron one, to obtain a refined picture of the collective states in spherical nuclei. The calculated results are analysed from the following viewpoints: (a) What characterizes the collective states? (b) What part of the phonon picture persists? (c) How can we truncate the space? (d) What interaction may we use?

Dynamical Interplay of Pairing and Quadrupole Modes in Transitional Nuclei. III

Using the single j-shell model with the pairing plus quadrupole force, we show that l) the transition from vibrational to rotational excitation structure can be reproduced within the collective model space built up from the J =0- and J =2- coupled nucleon pairs, 2) change of the monopole pair field due to many-quasiparticle excitations does accelerate the transition; this fact indicates the importance of taking explicit account of the cou­ pling between pairing rotation and many-phonon excitations in transitional nuclei.

Effect of Higher Multipole Nucleon Pairs on the Band Structure in Transitional Nuclei

Using the single j-shell model with the pairing plus quadrupole force, we show that the higher multipole nucleon pairs bring about the spin dependent changes in the band structure. The collectivity associated with the E2 transition is found to persist.

Relation of the interacting boson model to the shell model

A microscopic foundation of the interacting boson model is discussed from the shell model point of view. The Ginocchio model and the single j-shell model are taken to study various approximations for deriving the boson Hamiltonian. In the neutron-proton interacting boson model, we show the importance of the coupling to non-collective states for renormalizing some parameters in the boson Hamiltonian.

On the Applicability of Generalized Schwinger Representation of the Fermion Pairs to the Anharmonicity. II

The generalized Schwinger representation reformulated in Part I is applied to the description of the anharmonicity. The relations to RPA, the SU(6)-model and he boson expansion are discussed. Numerical results concerning E2-transitions and energies based on the Tamm-Dancoff approximation are shown in the case of single-j shell model with j = 11/2.

An orthonormalization method: Theory and application to the j- j coupling of fermions in a single j-shell

An orthonormalization method suitable to deal with an overcomplete and non-orthogonal set of vectors is derived. It is then used to obtain the orthonormal set of states connected with the angular momentum coupling of N fermions in a single j-shell.

A Semi-Microscopic Derivation of Coupling between Rotational and Intrinsic Motions. I

Euler angles prescribing a body-fixed coordinate system are introduced by the use of the degeneracy of Hartree-Bogolyubov field in space orientation. To express such a degeneracy, a method of pair operators is applied. The Euler angles are quantized so as to recover the breaking of rotation symmetry. Collective rotational and intrinsic excitation modes are separated in a similar way to an RPA treatment. The importance of the non-canonical nature of a transformation to the body-fixed coordinate system is remarked. For simplicity, a single-j shell model with pairing-plus-quadrupole force is adopted.

Collective model description of transitional odd- A nuclei : (I). The triaxial-rotor-plus-particle model

The energy spectrum of an odd nucléon coupled to a triaxial rotating core has been calculated as a function of the deformation β, the asymmetry γ and the Fermi energy λ f . Results are presented in a series of plots with the odd nucleon restricted to a single j-shell and the parameters covering the area which is of most interest for transitional odd- A nuclei in the A = 135 and the A = 190 mass regions. The results apply to unique-parity spectra which are based either on particle or hole states in a. j = 11 2 shell, but hold also for j = 9 2 and j = 13 2 . In addition, results on moments and transition probabilities are given. The quasiparticle-rotor Hamiltonian is derived with special emphasis on the particle-hole symmetry. The analytic solution for the even triaxial rotor at γ = 30° is given. Concerning the odd- A spectrum, characteristics which can be tested experimentally are discussed, and a qualitative physical interpretation is given. In particular, band structures in the triaxial region and their approximate classification are pointed out.