Bohr-Mottelson Model Research Articles

Deformation of nuclei as a function of angular momentum in the U(6) ⊃ SU(3) model

Moshińsky proposed recently a hybrid rotational model resulting from a comparison between the Gneuss and Greiner extension of the Bohr-Mottelson model and the interacting boson model. In this hybrid rotational model, we study the shape of nuclei by calculating the average of the expectation value of the square of the deformation parameter β with respect to the rotational states with the same angular momentum belonging to a given irreducible representation of SU(3). This work generalizes to three dimensions the corresponding analysis carried out in two dimensions by Chacón, Moshińsky, and Vanagas. We use the canonical chain of U(3) to obtain an analytical formula for the quantity studied. The overall stretching effect of the angular momentum on the shape of nuclei is demonstrated.

Comparison of the mass quadrupole and hydrodynamic collective models

Comparison is made between the mass quadrupole coordinate collective model and an appropriately extended version of the hydrodynamic collective model to ascertain whether or not the former is an exact microscopic formulation of the latter. We find that, while there are strong similarities, it is not.NUCLEAR STRUCTURE Bohr-Mottelson model, hydrodynamic model, mass quadrupole collective model; comparison, theory.

Relationship between the Bohr-Mottelson model and the interacting boson model

The interacting boson model was invented in two independent modes: The Schwinger mode using six bosons ($s$ and five $d$ bosons) and the Holstein-Primakoff mode using five quadrupole quasibosons. We show that the mathematical equivalence of the two modes can be used to define a number conserving quadrupole boson (the $b$ boson). Two equivalent bases, the usual $s\ensuremath{-}d$ basis and a new $s\ensuremath{-}b$ basis, are exhibited. By an exercise of (possibly objectionable) physical license, the result can be interpreted as a proof of equivalence of interacting boson model I with the Bohr-Mottelson model. In the $s\ensuremath{-}b$ basis, the Hamiltonian and other operators depend only on the $b$ boson. In this form, all the topics usually associated with the Bohr-Mottleson model can be discussed: potential energy surface, shape parameters, vibrations vs rotations, etc. The precise relationship of our method to that employed in previous work is exposed. The latter is shown to correspond to the use of the Dyson generators of SU(6).

Relationship Between the Interacting Boson Model and the Bohr Collective Hamiltonian

It is demonstrated that the Interacting Boson Model I (IBM), which is described by a Hamiltonian of six coupled harmonic oscillators, (one s-boson and 5 d-bosons), where the total number of bosons, N, is constant of the motion, can be transformed into a Bohr-Mottelson model (BMM) described by a Hamiltonian with the familiar five dimensional quadrupole oscillator degrees of freedom. The correspondence is one-to-one for a BMM acting in a sub-space of the full five-dimensional harmonic oscillator space. The proof depends on a well-known non-linear realization of the generators of SU(6) (which are the basic building blocks of the IBM) in terms of the five BMM bosons and the number N. The orthonormal basis vectors of the BMM are obtained. The relationship between the limiting symmetries of the IBM and the geometrically simple limits of the BMM is described with the aid of the concept of potential energy surface. The form of the BMM Hamiltonian is given for the adiabatic limit. Finally we show that previous work on this problem involving coherent states and the generator coordinate method is incomplete.

Microscopic derivation of nuclear collective variables

We give explicit expressions for the collective variables appearing in the Bohr-Mottelson model and in the oscillator part of an interacting boson approximation, as functions of the coordinates of an $A$ nucleon system when $A>>1$.NUCLEAR STRUCTURE Collective models, microscopic derivation.

On the relation between the interacting boson model of Arima and Iachello and the collective model of Bohr and Mottelson

The generator coordinate method is used to relate the interacting boson model of Arima and lachello and the collective model of Bohr and Mottelson through an isometric transformation. It associates complex parameters to the original boson operators whereas the ultimate collective variables are real. The absolute squares of the collective wave functions can be given a direct probability interpretation. The lowest order Bohr-Mottelson hamiltonian is obtained in the harmonic approximation to the interacting boson model; unharmonic coupling terms render the collective potential to be velocity dependent. The mapping of operators of the interacting boson model onto those of the Bohr-Mottelson model turns out to be of Holstein-Primakoff type.

Confrontations between the interacting boson approximation and the Bohr-Mottelson model

In the past year several papers were published discussing, from very different points of view, the relation between the interacting boson approximation and the Bohr-Mottelson model. We show that, under certain assumptions about the measure in the Hilbert space, these approaches are equivalent.

Confrontation of nuclear collective models

We discuss the main concepts used in the generalized Bohr-Mottelson model developed by the Frankfurt group and in the Interacting Boson Approximation (IBA) model. We show the equivalences that exist between these two models and between them and some of the current microscopic models.

An intrinsic state for the interacting boson model and its relationship to the Bohr-Mottelson model

Abstract Within the interacting boson model, an intrinsic state is defined, in terms of quadrupole shape variables, from which all eigenstates of the interacting boson model can be projected by appropriately averaging over these shape variables and the orientations of the intrinsic state. With this intrinsic state, an energy surface and a Bohr hamiltonian can be derived for any boson hamiltonian in terms of these quadrupole variables.

Present status of the VMI and related models

This article traces the evolution of the Variable Moment of Inertia model in its relation to the shell model, the Bohr-Mottelson model and the Interacting Boson Model. The discovery of a new type of spectrum, that of “pseudomagic” nuclei (isobars of doubly magic nuclei) is reported, and an explanation for their dynamics is suggested. The type of rotational motion underlying the ground state band of an e-e nucleus is shown to depend on whether the minimum number of valence nucleon pairs of one kind (neutrons or protons) is⩽ 2 or > 2. In the former case the “alphadumbbell model” holds; in the latter the “two-fluid model.”

On the algebraic formulation of collective models. I. The mass quadrupole collective model

This paper is the first in a series of three which together present a microscopic formulation of the Bohr-Mottelson (BM) collective model of the nucleus. In this article the mass quadrupole collective (MQC) model is defined and shown to be a generalization of the BM model. The MQC model eliminates the small oscillation assumption of BM and also yields the rotational and CM(3) submodels by holonomic constraints on the MQC configuration space. In addition, the MQC model is demonstrated to be an algebraic model, so that the state space of the MQC model carries an irrep of a Lie algebra of microscopic observables, the MQC algebra. An infinite class of new collective models is then given by the various inequivalent irreps of this algebra. A microscopic embedding of the BM model is achieved by decomposing the representation of the MQC algebra on many-particle state space into its irreducible components. In the second paper this decomposition is studied in detail. The third paper presents the symplectic model, which provides the realization of the collective model in the harmonic oscillator shell model.

Improved eikonal approximation. II. The study of electron-oriented nucleus scattering and polarisation of e-, nuclei and emitted gamma rays

For pt.I see ibid., vol.4, no.3, p.375 (1978). The improved eikonal approximation method developed in the previous paper is applied to the investigation of high-energy elastic and inelastic electron scattering from oriented deformed nuclei. Using the Bohr-Mottelson model, a very general expression for the cross section of polarised electron scattering from oriented nuclei is derived. These 'new terms' cause a difference in the cross section for polarised electron scattering from an oriented nucleus at beta (angle between orientation direction and recoil momentum) and pi - beta . A sum rule and some branching ratios involving transition probabilities to different rotational levels of oriented nuclei are obtained. Expressions are given for the statistical tensors for (a) the final scattered states of the nucleus, involving those for the initial state, and also for (b) the nuclear state after gamma emission. Using these expressions the circular polarisations of the emitted gamma rays are also investigated.

Rotational motion in nuclei

A review is given of the historical development of the study and interpretation of rotational states in nuclei. The explanation provided by the Bohr-Mottelson model of observed spectra and moments of inertia is emphasized. (AIP)

Indication ofE4 giant-resonance strength

The giant-multipole-resonanee propert ies in the mass 90 region have been studied by a var ie ty of experiments. In part icular , electron (~) and hadron (2,3) scattering have consistently demonstrated the presence of a giant quadrupolc resonance (GQR) in the exci tat ion region between 10 and 17 MeV. Theoretical studies (4), using the RPA, have shown tha t in this excitat ion region not only can a GQR exist but also tha t there is some strength of a giant hexadecapole resonance (GHR). This expectat ion of GHR strength is suggested not only from the Bohr-Mottelson collective model, but also from a naive shell model (5) which predicts energy ccntroids of higher giant multipole resonances (L, AT) according to

Exact and approximate angular momentum projection for light nuclei (II)

Abstract The Peierls-Yoccoz angular momentum projection method is applied to axially symmetric Hartree-Fock wave functions for several sd-shell nuclei. Excitation energies, rms radii and B (E2; 2 + → 0 + ) values are calculated and compared with the results of two different approximation methods. This is also done for some 12 C wave functions considered earlier. A discussion of the accuracy of these approximations is given. An approximate formula for the B (E2; 2 + → 0 + ) value is presented, which gives much better agreement with the exact angular momentum projection results than the formula from the Bohr-Mottelson model.

A Nilsson model for deformed phonons

Vibrations of deformed even nuclei are treated as vibrations in the intrinsic system projected out into rotational bands in the laboratory. A vibrational quantum is viewed as a quantized surface ripple, with a good K but not J, in analogy to a nucleon in the Nilsson well. The intrinsic Hamiltonian is taken to be of the Bohr-Mottelson type, with the parameters B lm , C tm and β 0 defining the deformed field. The intrinsic problem is solved in terms of the phonons of the spherical field, first in perturbation theory and then exactly. The Bogoyubov-Valatin transformation for bosons is used in the latter. By the Peierls-Yoccoz projection method, the physical states are found as certain superpositions of spherical-phonon states with various phonon numbers. The present model reproduces the usual results of the Bohr-Mottelson model; in addition, it suggest a K = 1 quadrupole-vibrational band. Finally, the coherence, or collectivity, of the projected states is discussed in terms of the B(E2) value between the two lowest states. The expected qualitative behaviour is established. The B(E2) value increases as deformation sets in; slow convergence encumbers the study of large deformations. Altogether, the present work continues work continue the phonon descriptiopn of spherical nuclei to the permanently deformed ones; it forms a rather detailed examination of the long-alleged equivalence of deformation and vibration.

Nuclear spectroscopy in the neutron-deficient 173 ≦ A ≦ 185 region

The internal conversion spectra of deformed nuclei in the mass interval 173 ≦ A ≦ 185 were investigated with permanent magnet spectrographs. By combining the conversion results with complementary data from independent photon studies, partial decay schemes are deduced from energy sums, multipolarities and intensities of the transitions. The quantum numbers of nuclear energy levels are associated with intrinsic and collective modes of excitation in the framework of the Nilsson and Bohr-Mottelson models. Parameters relevant to rotational configurations and branching ratios of the de-exciting transitions are evaluated. The energy systematics of proton and neutron particle states in odd-mass nuclei is correlated with the particle-coupled spectra in adjacent doubly odd nuclei. More complete conversion electron measurements permits extension of nuclear level structures of 148, 146Sm.

The linked-cluster expansion for the transition amplitude for deuteron stripping reaction: W. F. Junkin and F. Villars. Laboratory for Nuclear Science, M.I.T., Cambridge, Massachusetts

At the Technische Hochschule in Darmstadt the group of Richter and coworkers found in 1983/84 in deformed rare earth nuclei low-lying isovector 1+ states. Such states have been predicted in the generalized Bohr-Mottelson model and in the interacting boson model no. 2 (IBA2). In the generalized Bohr-Mottelson model one allows for proton and neutron quadrupole deformations separately. If one includes only static proton and neutron deformations the generalized Bohr-Mottelson model reduces to the two rotor model. It describes the excitation energy of these states in good agreement with the data but overestimates the magnetic dipole transition probabilities by a factor 5. In the interacting boson model (IBA2) where only the outermost nucleons participate in the excitation the magnetic dipole transition probability is only overestimated by a factor 2. The too large collectivity in both models results from the fact that they concentrate the whole strength of the scissors vibrations into one state. A microscopic description is needed to describe the spreading of the scissors strength over several states. For a microscopic determination of these scissors states one uses the Quasi-particle Random Phase Approximation (QRPA). But this approach has a serious difficulty. Since one rotates for the calculation the nucleus into the intrinsic system the state corresponding to the rotation of the whole nucleus is a spurious state. The usual procedure to remove this spuriosity is to use the Thouless theorem which says that a spurious state created by an operator which commutes with the total hamiltonian (here the total angular momentum, corresponding to a rotation of the whole system) produces the spurious state if applied to the ground state. It says further the energy of this spurious state lies at zero excitation energy (it is degenerate with the ground state) and is orthogonal to all physical states. Thus the usual approach is to vary the quadrupole-quadrupole force strength so that a state lies at zero excitation energy and to identify that with the spuríous state. This procedure assumes that a total angular momentum commutes with a total hamiltonian. But this is not the case since the total hamiltonian contains a deformed Saxon-Woods potential. Thus one has to take care explicitly that the spurious state is removed. This we do in our approach by introducing Lagrange multipliers for each excited states and requesting that these states are orthogonal to the spurious state which is explicitly constructed by applying the total angular momentum operator to the ground state. To reduce the number of free parameters in the hamiltonian we take the Saxon-Woods potential for the deformed nuclei from the literature (with minor adjustments) and determine the proton-proton, neutron-neutron and the proton-neutron quadrupole force constant by requesting that the hamiltonian commutes with the total angular momentum in the (QRPA) ground state. This yields equations fixing all three coupling constants for the quadrupole-quadrupole force allowing even for isospin symmetry violation. The spin-spin force is taken from the Reid soft core potential. A possible spin-quadrupole force has been taken from the work of Soloviev but it turns out that this is not important. The calculation shows that the strength of the scissors vibrations are spread over many states. The main 1+ state at around 3 MeV has an overlap of the order of 14 % of the scissors state. 50% of that state are spread over the physical states up to an excitation energy of 6 MeV. The rest is distributed over higher lying states. The expectation value of the many-body hamiltonian in the scissors vibrational state shows roughly an excitation energy of 7 MeV above the ground state. The results also support the experimental findings that these states are mainly orbital excitations. States are not very collective. Normally only a proton and neutron particle-hole pair are with a large amplitude participating in forming these states. But those protons and neutrons which are excited perform scissors type vibrations.

Nature of the K = 2 states in even deformed nuclei

A crucial test to discriminate between the Bohr-Mottelson model and the Davydov-Filippov model is proposed. The analysis of the beta transition rates from 186Re and 188Re to the K = 2 bands in 186Os and 188Os disfavours the asymmetric-rotor model. The beta transition rate to the K = 2 state is evaluated on basis of the microscopic description of the gamma vibration. Presently available data on M1 transitions from 1 + states to the K = 2 bands in 156Gd and 188Os also disfavour the asymmetric-rotor model.

The 28Si nucleus in the Hartree-Fock model

An estimate of the energies of all the low-lying states of 28Si has been made using the Hartree-Fock and Tamm-Dancoff methods. A one-one correspondence with the correct ordering has been obtained for all observed levels below 8 MeV, however the calculated spectrum is too dense by about a factor of two. Possible reasons for this compression are discussed. Relative E2 transition rates are in fair agreement with experiment. Extensive comparisons are made with a parallel investigation using the SU 3 model, and the failure of the latter model in this case is discussed. An interpretation is made of some of the low-lying states in terms of the Bohr-Mottelson model of a rotating-vibrating nucleus.