Cluster Model Wave Functions Research Articles

A Nuclear Electric Quadrupole Moment Of 6Li Nucleus Using Hyperfine Structure In Atomic Spectra

The nuclear electric quadrupole moment (QM) is one of the most important bulk properties of the nucleus with which nuclear deformations can be investigated. In this paper the electric quadrupole moment for 6Li nucleus in the ground state is calculated from the hyperfine structure of atomic spectra. Initially a cluster model wavefunction is constructed in shell model using Resonating Group Method (RGM), Generate coordinate method (GCM) and Complex Generator Coordinate Technique (CGCT). CGCT transforms the cluster model wavefunction into antisymmetrized product of single particle wavefunction. The wavefunction is written with the total angular momentum, relative motion between alpha and deuteron clusters, definite parity and spin. The calculations have been made by taking Wood- Saxon potential. The results obtained are compared and found reasonably well in agreement with the experimental results.

Formulation of a shell–cluster overlap integral with the Gaussian expansion method

Abstract We formulate a computational method to evaluate the overlap integral of the shell-model and cluster-model wave functions. The framework is applied to the system of the core plus two neutrons, and the magnitude of the overlap of the shell-model configuration (core + $n$ + $n$) and the di-neutron cluster one (core + $2n$) is explored. We have found that the magnitude of the overlap integral is prominently enhanced when two neutrons occupy shell-model orbits with low orbital angular momenta, such as $s$- and $p$-wave orbits. The shell–cluster overlap is calculated in systems with $jj$-closed cores plus two neutrons, and the enhancement due to occupation of the $s$ or $p$ orbit also occurs in the systematic calculation. The effect of the configuration interaction on the shell–cluster overlap integrals is also discussed.

Root Mean Square Radius of 6Li Nucleus using Cluster Model Wavefunction

The calculation of root mean square nuclear charge radius is one of the most important nuclear parameters regarding the size and structure of the nucleus. In this paper calculations for root mean square (r.m.s.) radius of the ground state of 6Li nucleus using high energy electron scattering is presented. The initial work involves writing first the cluster model wavefunction employing the resonating group method, generator coordinate method and complex generator coordinate technique. The wavefunction is written with definite parity, spin, total angular momentum and relative motion between the alpha cluster and deuteron cluster and the center-of-mass of the two clusters. The application of complex generator coordinate technique transforms the cluster model wavefunction into antisymmetrized products of single particle wavefunction written in terms of single particle co-ordinates, the centerof-mass coordinates, parameter coordinates and generator coordinates. The width parameters of alpha and deuteron clusters are adjusted to obtain predictions close to experimental values.

CLUSTER SHELL MODEL WAVE FUNCTION: STRUCTURE OF 6LI NUCLEUS

In this paper cluster model wave function for 6Li using Shell Model with definite parity and angular momentum is written along with cluster co-ordinates, which are relative to the center-of-mass of various clusters and involve with parameters. These parameters can be adjusted to some extent to obtain predictions close to experimental properties. The cluster model wave function is written along with resonating group method (RGM) and the Complex Generator Coordinate Technique (CGCT). The Complex Generator Coordinate Technique allows the transformation of the cluster model wave function written in terms of cluster co-ordinates into anti-symmetrized product of single particle wave function. This wave function is written in terms of single particle co-ordinates, the center-of-mass co-ordinates, parameter coordinates and generator coordinates.

A rigorous non-empirical theoretical analysis of the 2p XPS of NiO: Is it necessary to invoke nonlocal screening?

Abstract The role of ligand field effects and many-body effects arising from angular momentum coupling and shake effects for the XPS of NiO, as a representative transition metal oxide, have been investigated using rigorous, non-empirical wave functions for single site cluster models. It is shown that important features of the XPS arise from proper treatment of angular momentum coupling and it is unnecessary to invoke non-local screening to explain the experimental XPS. Contrary to the usual understanding, it is shown that inclusion of shake excitations in the many body wavefunctions is responsible for the high BE satellites that are observed. The contribution of covalent mixing of metal and ligand orbitals in the closed shells to screening of core-holes is demonstrated.

Effects of cluster–shell competition and BCS-like pairing in 12C

The antisymmetrized quasi-cluster model (AQCM) was proposed to describe {\alpha}-cluster and $jj$-coupling shell models on the same footing. In this model, the cluster-shell transition is characterized by two parameters; $R$ representing the distance between {\alpha} clusters and {\alpha} describing the breaking of {\alpha} clusters, and the contribution of the spin-orbit interaction, very important in the $jj$-coupling shell model, can be taken into account starting with the {\alpha} cluster model wave function. Not only the closure configurations of the major shells, but also the subclosure configurations of the $jj$-coupling shell model can be described starting with the {\alpha}-cluster model wave functions; however, the particle hole excitations of single particles have not been fully established yet. In this study we show that the framework of AQCM can be extended even to the states with the character of single particle excitations. For $^{12}$C, two particle two hole (2p2h) excitations from the subclosure configuration of $0p_{3/2}$ corresponding to BCS-like pairing are described, and these shell model states are coupled with the three {\alpha} cluster model wave functions. The correlation energy from the optimal configuration can be estimated not only in the cluster part but also in the shell model part. We try to pave the way to establish a generalized description of the nuclear structure.

The effect of symmetry on the U L3 NEXAFS of octahedral coordinated uranium(vi).

We describe a detailed theoretical analysis of how distortions from ideal cubic or Oh symmetry to tetrahedral, D4h, symmetry affect the shape, in particular the width, of the U L3-edge NEXAFS for U(vi) in octahedral coordination. The full-width-half-maximum (FWHM) of the L3-edge white line decreases with increasing distortion from Oh symmetry. In particular, the FWHM of the white line narrows whether the tetragonal distortion is to compression or to extension. The origin of this decrease in the FWHM is analyzed in terms of the electronic structure of the excited levels arising from the unoccupied U(6d). The relative importance of ligand field and of spin-orbit effects is examined, where the dominant role of ligand field effects is established. Especially at higher distortions, the ligand splittings decrease rapidly and lead to an accelerated, quadratic decrease in the FWHM with increasing distortion. This is related to the increase of covalent character in the appropriate component of the Oh derived eg orbitals. Our ab initio theory uses relativistic wavefunctions for cluster models of the structures; empirical or semi-empirical parameters were not used to adjust prediction to experiment. A major advantage is that it provides a transparent approach for determining how the character and extent of the covalent mixing of the relevant U and O orbitals affect the U L3-edge white line.

General transformation of α cluster model wave function to jj-coupling shell model in various 4N nuclei

General transformation of α cluster model wave function to jj-coupling shell model in various 4N nuclei

Container structure of alpha-alpha-Lambda clusters in 9-Lambda-Beryrium

A new concept of clustering is discussed in |$\Lambda$| hypernuclei using a new-type microscopic cluster model wave function, which has a structure in which constituent clusters are confined in a container, whose size is a variational parameter and which we refer to as a hyper-Tohsaki–Horiuchi–Schuck–Röpke (hyper-THSR) wave function. By using the hyper-THSR wave function, the |$2\alpha + \Lambda$|-cluster structure in |${{^9_\Lambda}\hbox{Be}}$| is investigated. We show that full microscopic solutions in the |$2\alpha + \Lambda$|-cluster system, which are given as |$2\alpha + \Lambda$| Brink-GCM (generator coordinate method) wave functions, are almost perfectly reproduced by the single configurations of the hyper-THSR wave function. The squared overlaps between both wave functions are calculated to be |$99.5$|%, |$99.4$|%, and |$97.7$|% for |$J^\pi =0^+ $|⁠, |$2^+ $|⁠, and |$4^+ $| states, respectively. We also simulate the structural change by adding the |$\Lambda$| particle, by varying the |$\Lambda N$| interaction artificially. With the increase of the |$\Lambda N$| interaction, the |$\Lambda$| particle gets to move more deeply inside the core and strongly invokes the spatial core shrinkage. Accordingly, distinct localized |$2\alpha$| clusters appear in the nucleonic intrinsic density, though, in the |${^8{\rm Be}}$| nucleus, a gaslike |$2\alpha$|-cluster structure is shown. The origin of the localization is associated with the strong effect of the Pauli principle. We conclude that the container picture of the |$2\alpha$| and |$\Lambda$| clusters is essential in understanding the cluster structure in |${^9_\Lambda{\rm Be}}$|⁠, in which the very compact spatial localization of clusters is shown in the density distribution.

Alpha-particle condensation in light hypernuclei

Abstract Gas-like α-cluster states are investigated in light Λ hypernuclei. The α condensate-type microscopic cluster model wave function is introduced to describe Λ hypernuclei. In particular, the analogous state to the famous Hoyle state, the second 0 + state in 12C, is investigated in 13ΛC, with the use of the new type wave function. The Hoyle state is known to have a gas-like 3α cluster structure, where the 3α particles are condensed into an identical S orbit. The second 1 / 2 + state in 13ΛC is shown to have the 3 α + Λ structure. A strong shrinkage effect by adding the Λ particle is seen for the state, reducing the rms radius from 3.8 fm in the Hoyle state to 2.8 fm. In spite of the shrinkage effect, the α condensate fraction of about 60% still survives, though it is reduced by about 20% from the Hoyle state.

Necessary conditions for accurate computations of three-body partial decay widths

The partial width for decay of a resonance into three fragments is largely determined at distances where the energy is smaller than the effective potential producing the corresponding wave function. At short distances the many-body properties are accounted for by preformation or spectroscopic factors. We use the adiabatic expansion method combined with the WKB approximation to obtain the indispensable cluster model wave functions at intermediate and larger distances. We test the concept by deriving conditions for the minimal basis expressed in terms of partial waves and radial nodes. We compare results for different effective interactions and methods. Agreement is found with experimental values for a sufficiently large basis. We illustrate the ideas with realistic examples from a emission of 12 C and two-proton emission of 17 Ne. Basis requirements for accurate momentum distributions are briefly discussed.

Cluster structure in nuclei

Two topics of the cluster structure study are discussed. They are (1) α cluster condened states, and (2) superdeformation and molecular states in32S In the first topic, I show that the 3α microscopic cluster model wave functions which were obtained long time ago for the12C second 0+ state are almost completely the same as single 3α condensate model wave functions. The α condensate model wave functions we used for this study are obtained by projecting out good angular momenta from intrinsic a condensate wave functions which are spatially deformed. It is found that the effect of deformation is not large for the present case of the12C O2+ states in the sense that the obtained wave functions contain the spherical condensate components more than 90 %. In the second topic, I show from the AMD + GCM study that the superdeformed excited rotational band in32S is the same as the16O +16O lowest molecular band. I explain that the AMD + GCM calculation gives not only the16O +16O lowest molecular band which is identified as the superdeformed excited band but also the16O +16O second and third lowest molecular bands whose excitation energies are higher than the lowest band by about 10 MeV and 20 MeV, respectively. I point out that the excitation energies of these three lowest bands are almost the same as those of the three lowest bands generated by the so-called “unique optical potential” of the16O +16O scattering.

Cluster structure in nuclei

Two topics of the cluster structure study are discussed. They are (1) α cluster condened states, and (2) superdeformation and molecular states in 32S In the first topic, I show that the 3α microscopic cluster model wave functions which were obtained long time ago for the 12C second 0+ state are almost completely the same as single 3α condensate model wave functions. The α condensate model wave functions we used for this study are obtained by projecting out good angular momenta from intrinsic a condensate wave functions which are spatially deformed. It is found that the effect of deformation is not large for the present case of the 12C O 2 + states in the sense that the obtained wave functions contain the spherical condensate components more than 90 %. In the second topic, I show from the AMD + GCM study that the superdeformed excited rotational band in 32S is the same as the 16O + 16O lowest molecular band. I explain that the AMD + GCM calculation gives not only the 16O + 16O lowest molecular band which is identified as the superdeformed excited band but also the 16O + 16O second and third lowest molecular bands whose excitation energies are higher than the lowest band by about 10 MeV and 20 MeV, respectively. I point out that the excitation energies of these three lowest bands are almost the same as those of the three lowest bands generated by the so-called “unique optical potential” of the 16O + 16O scattering.

Core exciton energies of bulk MgO,Al2O3,andSiO2from explicitly correlatedab initiocluster model calculations

Ab initio cluster model wave functions are used to predict the existence of localized excited states in MgO, ${\mathrm{Al}}_{2}{\mathrm{O}}_{3},$ and ${\mathrm{SiO}}_{2}$ arising from metal $2p$ core-level excitations. Theoretical values obtained at different levels of theory result in a quantitative agreement with experiment, and the use of different models permits us to quantify the different contributions to the final excitation energy. The most important contribution is atomic in nature; a meaningful zero-order approximation is that in MgO and ${\mathrm{Al}}_{2}{\mathrm{O}}_{3}$ the exciton can be assigned to a ${M(2p}^{6})\ensuremath{\rightarrow}{M(2p}^{5}{3s}^{1})$-like excitation, where $M=\mathrm{Mg}$ or Al. For the atomic models, the singlet-triplet exchange in the excited configuration is in good agreement with experiment. In addition, the solid-state effects on this exchange energy predicted by experiment are well reproduced by the cluster models representing MgO and ${\mathrm{SiO}}_{2},$ whereas a less clear situation appears in ${\mathrm{Al}}_{2}{\mathrm{O}}_{3}.$ The open-shell orbital in the final state has, however, important contributions from the ions near the atomic site where excitation occurs. Nevertheless, the final state appears to be localized in space without any a priori assumption, the localization following from the hole-particle interaction implicitly induced in the final-state wave function. The Madelung field reduces the excitation energy with respect to the atomic value; the effect of neighboring atoms, mainly Pauli repulsion, acts in the opposite way; and electronic correlation effects decrease it again. In agreement with the covalent nature of ${\mathrm{SiO}}_{2},$ the exciton cannot be simply understood as arising from a $\mathrm{Si}{(2p}^{6})\ensuremath{\rightarrow}\mathrm{Si}{(2p}^{5}{3s}^{1})$ in a fully oxidized Si cation.

Heavy-Ion Resonance and Multi-Cluster Structure of Nuclei

New kind of heavy-ion resonances recently discovered in nuclear collisions of C+C and O+O systems leading to various multi-cluster exit channels, such as C+C →3α+12C, 3α+3α, 2α+O and O+O → (α+C)+O, (α+C)+(α+C), are studied theoretically by the coupled-channel (CC) and coupled-channel Born approximation (CCBA) calculations using microscopic cluster-model wave functions of the colliding nuclei, C and O, and the nucleus-nucleus interactions based on the realistic G-matrix interaction. These resonance states correspond to highly excited states of the compound systems, Mg and S nuclei, at excitation energies more than 40∼55 MeV, having the associated multi-cluster structures and large decay widths to these reaction channels. In this talk, we show that characteristic features of these newly-found resonances as well as those of the classical resonances of di-nucleus molecular nature are well reproduced by the microscopic CC and CCBA calculations. The reaction mechanism leading to the formation of these exotic resonances is also discussed in some detail for the O+O → (α+C)+O, (α+C)+(α+C) and C+C → Be+O reactions.

Calculation of the complete Glauber amplitude for p+ 6He scattering

A method for calculating the complete expansion of the Glauber amplitude is applied to the elastic scattering of protons on a 6He target using correlated wave functions. The angular distribution measured at T p = 700 MeV is very well reproduced by a no parameter calculation using a fully microscopic, 6-nucleon α + n + n-cluster wave function. The accuracy of various approximations for the Glauber amplitude is compared with the complete expansion calculation. The matter radius of 6He consistent with the analysis is 2.51 fm. Simple shell-model or di-neutron cluster-model wave functions fail in reproducing the angular distribution.

Dynamic polarization potential of 12C+12C system at molecular-resonance energies

The nuclear reaction dynamics leading to the formation of recently discovered resonance in the mutual-02+ channel of the 12C+12C inelastic scattering around Ec.m.≃ 32 MeV is studied in terms of the dynamic polarization potential (DPP) induced by the channel coupling among various excited states in 12C. The microscopic 3α cluster-model wave functions are used to generate the 12C−12C diagonal and coupling potentials in the double-folding model. It is found that DPP for the 02++ 02+ channel is an unusually strong attractive potential which even exceeds the zeroth-order folding-model potential of this channel around the nuclear surface region and that the strong coupling between the 02+ and 22+ states is predominantly responsible for the unusual DPP in this channel. The effective potential, the sum of the original folding-model potential and the attractive DPP, is found to generates resonance states in the same energy region as that of the resonance states generated by the original folding-model potential but the former states are found to be higher-nodal states having four additional radial nodes. Similar but more moderate property of DPP is also found in the entrance (elastic) channel. These results suggest that the reaction dynamics of generating the resonance in the 12C(02+) +12C(02+) channel may rather differ from that of the simple crossing of the zeroth-order molecular band generated by the potentials in the entrance and exit channels suggested by the standard band-crossing model.

A new analysis of image charge theory

Ab initio cluster model wavefunctions for Al and Pt surfaces have been used to investigate the response of metal surfaces to the presence of point charges above the surface. The dipole moment curves, μ( z), as a function of the distance of the point charge above the surface have been used to characterize the metallic response. There are three regions where the behavior of μ( z) is different. In the region where z>∼15–20 bohr, the asymptotic behavior given by image charge theory holds. In the region where z<∼4 bohr, the well known failure of the asymptotic formula is found; this reflects the fact that the point charge penetrates the metal charge density. In an intermediate region, ∼4< z<∼15–20 bohr, a new and unexpected behavior has been found. The magnitude of the slope of the dipole moment curve is significantly larger than the value of the point charge! In this region the metallic response is strongly dependent on the distance of the point charge from the metal surface. Further, the magnitude of the slope is quite different for positive and negative point charges indicating that the metallic response has important contributions from non-linear terms.

Ab Initio Modeling of the Metal−Support Interface: The Interaction of Ni, Pd, and Pt on MgO(100)

The interaction of Ni, Pd, and Pt on MgO has been investigated using cluster models and a variety of ab initio techniques. The case of a single Pd atom with acidic and basic sites of the MgO(100) surface has been chosen to test the adequacy of different ab initio wave functions and DFT techniques. Both approaches result in a similar description but only if large basis sets, including high angular momentum terms, and extensive treatment of the many-body effects are accounted for in the ab initio cluster model wave function. The DFT results appear to be more stable and less dependent on the basis sets. When properly compared, both approaches lead to the same conclusion. Pd interacts in a rather weak way with the acidic sites, but a moderately large interaction is predicted for interaction above the basic sites. Next, the interaction of Ni, Pd, and Pt on the basic sites of the MgO(100) surface is considered. The general trend is that the interaction energy increases from Ni to Pt. The nature of the bond is rationalized by analysis of the dipole moment curves and of the molecular orbitals.

Bonding of Atomic S to Pt(111) from ab Initio Explicitly Correlated Cluster Model Wave Functions

The ab initio cluster model approach has been applied to the study of chemisorption of atomic S on Pt(111). Hartree−Fock and explicitly correlated wave functions predict reasonable results for the equilibrium distance perpendicular to the surface and its corresponding vibrational frequency. However, the interaction energy has been shown to be largely affected by electronic correlation effects. Use of MR-MP2 and MRCI wave functions with large reference spaces results in a De value which is surprisingly close to the experimental estimate of the interaction energy. All approaches suggest that S is strongly bonded to the Pt(111) surface through covalent interactions but also predict that chemisorbed S exhibits a rather large negative charge. Therefore, it is predicted that S chemisorption will strongly affect the electronic properties of the Pt(111) surface. This situation will be specially valid in the low-coverage limit.