Nilsson Orbits Research Articles

Novel features of nuclear spectra of light mass Os isotopes

The (N=88-108) light mass Os isotopes exhibit level spectra different from the heavier (N > 108) isotopes. The spectral pattern, for ground, β- and γ- bands is illustrated empirically, and analyzed in terms of the collective rotation vibration model and the algebraic sd Interacting Boson Model IBM-1. The variation of the spectral features with neutron number N has been linked to the odd-A neighbors and to the occupation of the underlying Nilsson orbits. The quadrupole β vibration produces the axially symmetric shape of light mass Os isotopes, in contrast to the main role of the asymmetry parameter γ in the heavier Os isotopes. The energy level spectra, and the inter band B(E2) ratios in IBM-1 reproduce the shape variation from spherical to the X(5) symmetry status, and to the prolate deformed rotor, as in experiment. This reveals the interesting physics of the rotation vibration model by high lighting the special features of the light mass 164−184Os isotopes. The detailed band structure and their variation with neutron number N of the Os isotopes is presented empirically and in IBM-1.

Phase transition at N = 92 in 158Dy

Beyond the shape phase transition from the spherical vibrator to the deformed rotor regime at [Formula: see text], the interplay of [Formula: see text]- and [Formula: see text]-degrees of freedom becomes important, which affects the relative positions of the [Formula: see text]- and [Formula: see text]-bands. In the microscopic approach of the dynamic pairing plus quadrupole model, a correlation of the strength of the quadrupole force and the formation of the [Formula: see text]- and [Formula: see text]-bands in [Formula: see text]Dy is described. The role of the potential energy surface is illustrated. The [Formula: see text] transition rates in the lower three [Formula: see text]-bands and the multi-phonon bands with [Formula: see text] and [Formula: see text] are well reproduced. The absolute [Formula: see text] [Formula: see text] serves as a good measure of the quadrupole strength. The role of the single particle Nilsson orbits is also described.

Shell structure, emerging collectivity, and valence p-n interactions

The structure of atomic nuclei depends on the interactions of its constituents, protons and neutrons. These interactions play a key role in the development of configuration mixing and in the onset of collectivity and deformation, in changes to the single particle energies and magic numbers, and in the microscopic origins of phase transitional behavior. Particularly important are the valence proton-neutron interactions which can be studied experimentally using double differences of binding energies extracted from high-precision mass measurements. The resulting quantities, called δV pn , are average interaction strengths between the last two protons and the last two neutrons. Focusing on the Z=50-82, N=82-126 shells, we have considered a number of aspects of these interactions, ranging from their relation to the underlying orbits, their behaviour near close shells and throughout major shells, their relation to the onset of collectivity and deformation, and the appearance of unexpected spikes in δV pn values for a special set of heavy nuclei with nearly equal numbers of valence protons and neutrons. We have calculated spatial overlaps between proton and neutron Nilsson orbits and compared these with the experimental results. Finally we also address the relation between masses (separation energies), changes in structure and valence nucleon number.

Collective band structure of 166,168Hf in IBM and DPPQ models

166,168Hf are the lightest isotopes of Hf, for which the spectral information for non-yrast levels is now available from recent experiments. The algebraic Interacting Boson Model IBM-1 is employed to reproduce their level structures and to predict the E2 transition probabilities. The pairing plus quadrupole model is used to predict their spectra and E2 transition rates and the static moments in a microscopic approach. The spin assignments I π of new levels and their K-band structures are studied. The validity of the inclusion of 166,168Hf as members of a U(12) super group is studied using various empirical observables. The potential energy surfaces for the two isotopes are compared and the filling of the nucleons in Nilsson orbits is analyzed, to yield a consistent comprehensive view of the spectra of the two Z = 72 isotopes.

A microscopic explanation of the isotonic multiplet at N = 90 and of the F-spin multiplet in Dy-Hf

The N = 90 isotones of Nd, Sm, Gd and Dy, with almost similar deformed level structure and associated with X (5) symmetry, form the isotonic multiplet in Z = 50–66, N = 82–104 quadrant. This is explained microscopically in terms of the Nilsson level diagram. Using the Dynamic Pairing-Plus-Quadrupole model of Kumar-Baranger, the quadrupole deformation and the occupancies of the neutrons and protons in these nuclei in the Nilsson orbits have been calculated, which support the formation of N = 88, 90 isotonic multiplets. The existence of the F-spin multiplets, with almost identical spectra, recognized in earlier works, in Z = 66–82, N = 82–104 quadrant, is also explained microscopically in our study.

Pair correlation in deformed neutron-drip-line nuclei: The eigenphase formalism and asymptotic behavior

The Hartree-Fock-Bogoliubov (HFB) equation for deformed nuclei in a simplified model is solved in coordinate space with correct asymptotic boundary conditions, in order to study the pair correlation in nuclei close to the neutron drip line. The eigenphase formalism is applied, when the upper components of HFB radial wave functions are continuum wave functions. Calculated occupation probabilities of various Nilsson orbits in the HFB ground state vary smoothly from the region of the upper components being bound wave functions to that of those being continuum wave functions. It is shown that weakly-bound or resonance-like Omega(pi)=1/2(+) Nilsson orbits contribute little to the occupation probability of the HFB ground state, while the contribution by the orbits with a large value of Omega, of which the smallest possible orbital-angular-momentum is neither 0 nor 1, may be approximately estimated using the BCS formula. (Less)

Dominance of low-ℓcomponent in weakly bound deformed single-neutron orbits

Calculating single-particle (Nilsson) levels in axially symmetric quadrupole-deformed potentials in coordinate space, the structure of weakly bound neutron orbits is studied in the absence of pair correlation. It is shown that in the wave functions of ${\ensuremath{\Omega}}^{\ensuremath{\pi}}=1∕{2}^{+}$ orbits, where $\ensuremath{\Omega}$ expresses the projection of the particle angular momentum along the symmetry axis, the $\ensuremath{\ell}=0\phantom{\rule{0.3em}{0ex}}({s}_{1∕2})$ component becomes overwhelmingly dominant as the binding energy of the orbits approaches zero, irrespective of the size of the deformation and the kind of Nilsson orbits. Consequently, all ${\ensuremath{\Omega}}^{\ensuremath{\pi}}=1∕{2}^{+}$ levels become practically unavailable for both deformation and many-body pair correlation, when the levels approach continuum or lie in the continuum.

Two-quasiparticle bands in neutron-rich nuclei

The large multi-detector arrays such as EUROBALL and GAMMASPHERE have made possible the study of the spectroscopy of neutron-rich nuclei through the observation of discrete, prompt $\gamma$ -rays emitted following fission. Most of the information obtained has concerned yrast states and collective excitations. In the present work, a search has been made for excited bands based upon two-quasiparticle intrinsic structures. Such bands have been found in several even-even nuclei from 96Sr to 112Pd. Careful analysis of triple-coincidence spectra has been performed in order to determine branching ratios within the bands. These branching ratios are then used to establish the magnetic properties of the intrinsic structure, permitting, in most cases, the determination of which Nilsson orbits (and whether they are neutron or proton states) are contributing to the excitation. Some example results from this search are presented.

Shell effects on rotational damping in superdeformed nuclei

Damping of collective rotational motion in A ∼ 190 and A ∼ 150 superdeformed nuclei is studied by means of the cranked shell model with two-body residual force. Numerical calculations predict that in a typical A ∼ 190 superdeformed nucleus, 192Hg, the rotational damping width is significantly small, Γ rot ∼ 30 keV, and that the number of superdeformed bands in the offyrast region amounts up to 150 at a given rotational frequency. These features are quite different from the prediction for A ∼ 150 superdeformed nuclei and rare-earth normally deformed nuclei. It is shown that the single-particle alignments of the cranked Nilsson orbits have strong shell oscillation. It affects significantly the properties of rotational damping in superdeformed 192Hg.

Nilsson-based truncation of the nuclear shell model.

A method for truncating the nuclear shell model in terms of Nilsson orbits is developed and tested in the context of a simple model that admits coexistence between deformed and spherical configurations. The method reproduces the exact-diagonalization spectra of both the spherical and deformed configurations if angular momentum projection is carried out prior to diagonalization. Implementation of this method in [ital m]-scheme shell-model codes is discussed.

High- j bands in 173Ir

High-spin states in rotational bands associated with the 11 2 −[505] , 1 2 −[541] and 1 2 +[660] Nilsson orbits have been observed in the previously unknown 173Ir nucleus using the 144Sm( 32S,p2n) reaction at 163 MeV. The B(M1; I → I − 1)/ B(E2; I → I −2) ratios have been extracted for the 11 2 −[505] band and compared with predictions of the semiempirical model by Dönau and Frauendorf. The quasineutron band crossing frequencies and alignment gains are discussed in terms of the CSM model. Calculations for equilibrium deformations suggest different deformations for the three bands observed.

Electromagnetic transition strengths between high spin states inSr79andSr80

High spin states in $^{79}\mathrm{Sr}$ were populated via the reaction $^{58}\mathrm{Mg}$,2pn) at 80 and 85 MeV beam energy. Twelve lifetimes in the range from 0.5 to 150 ps and four lifetime limits were measured with Doppler-shift methods. By comparing lifetimes in $^{79}$,80,82Sr obtained with the recoil distance and Doppler-shift attenuation methods, the appropriate electronic stopping power of Sr ions recoiling in Ta was determined. In addition, lifetimes of yrast states in $^{80}\mathrm{Sr}$ and $^{79}\mathrm{Rb}$ were determined in the concurrent reactions $^{58}\mathrm{Mg}$,2p) and $^{58}\mathrm{Mg}$,3p). The \ensuremath{\Delta}I=2 E2 transition strengths and the E2/M1 mixing ratios of the \ensuremath{\Delta}I=1 transitions in $^{79}\mathrm{Sr}$ confirm the picture of extremely prolate deformed rotational bands (${\ensuremath{\beta}}_{2}$\ensuremath{\approxeq}0.42), based on the [301]${(3/2}^{\mathrm{\ensuremath{-}}}$ and [422]${(5/2}^{+}$ Nilsson orbits. A reduction in B(E2) values is observed at the onset of (3qp) alignment. The band structure of $^{79}\mathrm{Sr}$ is discussed in terms of the cranked shell and the Woods-Saxon cranking model with pairing.

Two-and four-particle surface clusterization in heavy deformed nuclei

We study the problem of the influence of the pairing interaction on two- and four-particle surface correlations in deformed nuclei. Taking the overlap between the two- or four-particle wave function and a di-nucleon or alpha-particle wave function, respectively, one obtains cluster probability distributions, as functions of the center-of-mass coordinate of the considered particles. For particles moving in pure Nilsson orbits the probability is localized in the intrinsic frame in different region of the nuclear surface, according to the K-quantum numbers of the considered orbitals. The inclusion of the pairing interaction leads to a probability distributed over the entire surface but at the same time to values of the total spectroscopic factors orders of magnitude larger than those associated with pure Nilsson orbits.

The [formula omitted] bands in the odd- A = 19–31 nuclei

The systematics of excitation energies, single-particle spectroscopic factors and electromagnetic transition rates of levels belonging to the K π = 1 2 − bands based on the Nilsson orbits 4 and 14 in the odd- A = 19–31 nuclei have been studied and compared to weak-coupling and particle-rotor model predictions. Both bands are characterized by large decoupling parameters. The E1 transition rates are found to be typically of the order of 10 −4 to 10 −3 W.u., which is normal for this mass region. The transitions within the 1 2 −[101] band are enhanced, stretched E2 transitions. The over-all agreement with the model predictions is good.

Levels in179W studied in the181Ta(p, 3n) reaction by on-line electron and gamma ray spectroscopy

Levels in179W have been deduced from in-beam gamma and conversion electron studies of the181Ta(p, 3n)179W reaction. The gamma-ray spectrum was studied with Ge(Li) detectors and a crystal diffraction spectrometer; the conversion electrons were measured with a broad range transport solenoid Si(Li) spectrometer. The multipolarities of some 50 transitions could be determined. Coriolis mixing of the Nilsson orbits withN=6 was calculated.

Number-conserving treatment of the BCS-Tamm-Dancoff approximation for deformed rare-earth nuclei

The formulation for the exact number-conserving treatment of the BCS-Tamm-Dancoff approximation (NTDA) in deformed even nuclei is given. It is applied to theKπ=0− octupole vibrations of rare-earth nuclei and is also compared with the ordinary TDA or RPA method based on the Bogoliubov transformation which has the defect of mixing of the particle number in the wave functions. The excitation energies andB(E3) calculated by NTDA method show rather stronger dependence on the Nilsson orbits than those calculated by the usual TDA or RPA methods, especially atN=94 nuclei.

Mean lives of states in 51Cr

Mean nuclear lifetimes of 24 levels in 51Cr, below 4.1 MeV excitation, have been measured by the Doppler-shift attenuation method. The 48Ti(α, nγ) reaction was used at bombarding energies of 6.5, 7.5 and 8.5 MeV. Compton suppressed γ-ray spectra were collected with a 44 cm 3 Ge(Li) detector at 0°, 60°, 90° and 120° to the beam. The electromagnetic properties of the low-lying 51Cr levels indicate a rotational structure built on K π = 7 2 − and 1 2 − Nilsson orbits.

Theory of a self-consistent static field in the presence of two-particle-two-hole admixtures

The problem of two-particle-two-hole (2p-2h) admixtures is studied and applied to doubly even N = Z 2s-1d shell nuclei. A proper diagonalization corresponding to a resummation of Goldstone diagrams is derived. The problem of spurious effects, such as the non-conservation of angular momentum, is studied and a method is devised to check the validity of approximate eliminations of spurious effects. The effect of 2p-2h admixtures on the equilibrium deformation is first studied in a simplified calculation using Nilsson orbits. Then a fully self-consistent theory is formulated of the static average field in the presence of 2p-2h admixtures. A topological analysis of Goldstone diagrams, due to Bloch, yields a simple stationary expression for the ground state energy. The theory is illustrated by a complete numerical application in the 2s-1d shell.

Analysis of the Low-Lying Level Structure ofSi31

The strong-coupling collective model has been applied to the low-lying level structure of ${\mathrm{Si}}^{31}$. Two rotational bands are identified corresponding to the Nilsson orbits 8 ($K=\frac{3}{2}$) and 9 ($K=\frac{1}{2}$), with the ground and first excited (760-keV) levels being the respective bandheads. Detailed comparisons are made concerning the properties of the low-lying levels, such as spectroscopic factors for neutron capture, $\ensuremath{\gamma}$-ray multipole mixing ratios, the $\ensuremath{\gamma}$-ray branching ratio for the 1.70-MeV level, and the $log\mathrm{ft}$ value for $\ensuremath{\beta}$ decay. The results indicate that ${\mathrm{Si}}^{31}$ is an oblate spheroid with a deformation on the order of $\ensuremath{\eta}\ensuremath{\approx}\ensuremath{-}2.0$. With the parameters $\ensuremath{\kappa}\ensuremath{\approx}0.10$, ${g}_{R}\ensuremath{\approx}0.32$, and $\ensuremath{\mu}\ensuremath{\approx}0.2$, the model is found to give reasonable results for many of the above properties. In particular, the model is found to predict the anomalously high $\frac{T(E2)}{T(M1)}$ multipole intensity ratio observed for the 1.70\ensuremath{\rightarrow}0-MeV $\ensuremath{\gamma}$-ray transition.

Les bandes de rotation K=0 −ET K=1 − dans le 234Pa

The Pa 234 nucleus is interpreted as a rotator plus two odd nucleons. The energies and wave functions of the various states of the K=0 and K=1 rotational bands arising from the coupling of two Ω= 1 2 Nilsson orbits are found by calculating and numerically diagonalizing the Hamiltonian matrix.