Two-body Spin-orbit Interaction Research Articles

Simulating Spin–Orbit Coupling with Quasidegenerate N-Electron Valence Perturbation Theory

We present the first implementation of spin-orbit coupling effects in fully internally contracted second-order quasidegenerate N-electron valence perturbation theory (SO-QDNEVPT2). The SO-QDNEVPT2 approach enables the computations of ground- and excited-state energies and oscillator strengths combining the description of static electron correlation with an efficient treatment of dynamic correlation and spin-orbit coupling. In addition to SO-QDNEVPT2 with the full description of one- and two-body spin-orbit interactions at the level of two-component Breit-Pauli Hamiltonian, our implementation also features a simplified approach that takes advantage of spin-orbit mean-field approximation (SOMF-QDNEVPT2). The accuracy of these methods is tested for the group 14 and 16 hydrides, 3d and 4d transition metal ions, and two actinide dioxides (neptunyl and plutonyl dications). The zero-field splittings of group 14 and 16 molecules computed using SO-QDNEVPT2 and SOMF-QDNEVPT2 are in good agreement with the available experimental data. For the 3d transition metal ions, the SO-QDNEVPT2 method is significantly more accurate than SOMF-QDNEVPT2, while no substantial difference in the performance of two methods is observed for the 4d ions. Finally, we demonstrate that for the actinide dioxides the results of SO-QDNEVPT2 and SOMF-QDNEVPT2 are in good agreement with the data from previous theoretical studies of these systems. Overall, our results demonstrate that SO-QDNEVPT2 and SOMF-QDNEVPT2 are promising multireference methods for treating spin-orbit coupling with a relatively low computational cost.

Roles of NN-interaction components in shell-structure evolution

Since the importance of the monopole interaction was first emphasized in 1960s, roles of monopole strengths of two-body nucleon–nucleon interaction in shell structure have been discussed. Through the monopole strengths, we study the roles in shell-structure evolution, starting from explicit forms of the interaction. For the tensor component of the interaction, we show the derivation of the relation, (2j>+1)Vjj>+(2j<+1)Vjj<=0, with a detailed manipulation. We show that one-body spin–orbit term appears in the multipole expansion of two-body spin–orbit interaction. Only the spin–orbit components can affect the spin–orbit energy splitting between spin–orbit partners, when the spin–orbit partner orbits are fully occupied.

Proton and neutron correlations in10B

We investigate positive-parity states of $^{10}$B with the calculation of antisymmetrized molecular dynamics focusing on $pn$ pair correlations. We discuss effects of the spin-orbit interaction on energy spectra and $pn$ correlations of the $J^\pi T=1^+_10$, $=3^+_10$, and $0^+_11$ states. The $1^+_10$ state has almost no energy gain of the spin-orbit interaction, whereas the $3^+_10$ state gains the spin-orbit interaction energy largely to come down to the ground state. We interpret a part of the two-body spin-orbit interaction in the adopted effective interactions as a contribution of the genuine $NNN$ force, and find it to be essential for the level ordering of the $3^+_10$ and $1^+_10$ states in $^{10}$B. We also apply a $2\alpha+pn$ model to discuss effects of the spin-orbit interaction on $T=0$ and $T=1$ $pn$ pairs around the 2$\alpha$ core. In the spin-aligned $J^\pi T=3^+0$ state, the spin-orbit interaction affects the $(ST)=(10)$ pair attractively and keeps the pair close to the core, whereas, in the $1^+0$ state, it gives a minor effect to the $(ST)=(10)$ pair. In the $0^+1$ state, the $(ST)=(01)$ pair is somewhat dissociated by the spin-orbit interaction.

Properties of single-particle states in a fully self-consistent particle-vibration coupling approach

The properties of single-particle states in the magic nuclei $^{40}\mathrm{Ca}$ and $^{208}\mathrm{Pb}$, in particular the energies, spectroscopic factors, and the effective mass, have been studied in a fully self-consistent particle-vibration coupling (PVC) approach within the framework of Skyrme energy density functional theory. All selected phonons are obtained by the random phase approximation, and the same Skyrme interaction is also used in the PVC vertex. We focus on the effect of the noncentral two-body spin-orbit and tensor interactions on the single-particle properties. It has been found that the contributions of those terms are important to improve the results for $^{208}\mathrm{Pb}$. The calculated single-particle energies and spectroscopic factors are compared to available experimental data. The single-particle level density around the Fermi surface is significantly increased due to the effect of PVC.

Strength of reduced two-body spin-orbit interaction from a chiral three-nucleon force

The contribution of the chiral three-nucleon force to the strength of effective spin-orbit coupling is estimated. We construct a reduced two-body interaction by folding single-nucleon degrees of freedom of the three-nucleon force in nuclear matter. The spin-orbit strength is evaluated by the Scheerbaum factor obtained by a $G$-matrix calculation in nuclear matter with the two-nucleon interaction plus the reduced two-nucleon interaction. The inability of modern realistic two-nucleon interactions to account for the empirical spin-orbit strength is overcome. We found that spin-orbit coupling is weaker in a neutron-rich environment. Because the spin-orbit component of the three-nucleon force is determined by low-energy constants fixed in the two-nucleon sector, the present estimation has a small uncertainty.

Cluster correlations for low-lying intruder states of12Be

The formation of intruder states in the low-lying states of ${}^{12}\mathrm{Be}=\ensuremath{\alpha}+\ensuremath{\alpha}+4N$ is studied by applying the generalized two-center cluster model, which can optimize the excess neutrons' orbits depending on the $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ distance. The correlation energy for the intruder states is analyzed from the viewpoint of two different pictures based on the cluster structure: the covalent picture around two $\ensuremath{\alpha}$ clusters and the binary He-cluster picture. In the covalent picture, the binding energy of ${({\ensuremath{\pi}}_{32}^{\ensuremath{-}})}^{2}{({\ensuremath{\sigma}}_{12}^{+})}^{2}$, corresponding to $\ensuremath{\nu}{(0p)}^{4}{(1s0d)}^{2}$ in a naive shell model, gains largely owing to the spin-triplet pairing of the $0d$-wave neutrons, which is induced by the two-body spin-orbit interaction. The spin-triplet pairing gives rise to the reduction of the kinetic energy and the increase of the attractive spin-orbit interaction for the excess neutrons. As a result of these correlation energies, the $\ensuremath{\nu}{(0p)}^{4}{(1s0d)}^{2}$ configuration becomes dominant in the ground state. In the binary cluster picture, the correlation energy is investigated from the coupled channels among $\ensuremath{\alpha}+{}^{8}\mathrm{He}$, ${}^{6}\mathrm{He}+{}^{6}\mathrm{He}$, and ${}^{5}\mathrm{He}+{}^{7}\mathrm{He}$. The coupling to ${}^{5}\mathrm{He}+{}^{7}\mathrm{He}$, which is neglected in usual binary-cluster models, plays an important role for a large reduction of kinetic energy and the formation of a pair of the low-lying 0${}^{+}$ states with a close energy spacing recently observed in experiment. The rotational bands are also discussed from the viewpoint of these two cluster pictures.

Λ spin–orbit splitting in heavy hypernuclei as deduced from DWIA analyses of the [formula omitted] reaction

By taking nuclear core excitations into account, detailed structure calculations of 89 ΛY have been performed as a function of the Λ spin–orbit splitting. The obtained wave functions have been used to estimate the (π+,K+) reaction cross sections within the distorted-wave impulse approximation (DWIA). A theoretical explanation is given quantitatively for the first time of how to understand the doublet substructure of each major peak observed in medium-heavy hypernuclear production. A small Λ spin–orbit splitting of around δ(0fΛ)=0.2 MeV is deduced, a value consistent with the small ΛN two-body spin–orbit interactions deduced from γ-ray measurements in three light hypernuclei.

The origin of the spin–orbit splitting in some light nuclei using a modified interaction

The spin–orbit splittings in the spectra of nuclei with A = 5, 15, and 17 have been restudied, with a new set of matrix elements, within the framework of the shell model with core excitations. Investigated are the dependences of these splittings on core-excitation and on some of the different components of the nucleon- nucleon interaction. Our work confirms that the two-body spin–orbit interaction has a major effect whose magnitude depends on the size of the model space that is used. The effect of the tensor interaction is again seen to be considerably less important.

Variation after parity projection calculation with the Skyrme interaction for light nuclei

A self-consistent calculation with variation after parity projection is proposed to study both ground and excited states of light nuclei. This procedure provides description of the ground state incorporating some correlation effects, and self-consistent solutions for the excited states of negative parity. For flexible description of nuclear shapes, single particle orbitals are represented on a uniform grid in the three-dimensional Cartesian coordinates. The angular momentum projection is performed after variation to calculate rotational spectra. To demonstrate the usefulness of the method, results are shown for two $N=Z$ nuclei, $^{20}\mathrm{Ne}$ and $^{12}\mathrm{C}$, for which clustering correlations are known to be important. In the $^{20}\mathrm{Ne}$ nucleus, both cluster-like and shell-model-like states are described simultaneously in the present framework. For $^{12}\mathrm{C}$ nucleus, the appearance of three-alpha clustering correlation in the ground state is investigated in relation to the strength of the two-body spin-orbit interaction.

Dependence of the shell-model single-particle energies on different components of the nucleon-nucleon interaction

The spin-orbit splittings in the spectra of nuclei with mass numbers 5, 15, and 17 are studied within the framework of shell-model configuration mixing calculations including $2\ensuremath{\Elzxh}\ensuremath{\omega}$ excitations. The contributions of the two-body spin-orbit and tensor components of the nucleon-nucleon interaction are studied in various model spaces. It is found that the effects of the two-body spin-orbit interaction are dominant and quite sensitive to the size of the model space considered. The effects of the tensor interaction are weaker. The correlations effects which are included in the larger (0+2) $\ensuremath{\Elzxh}\ensuremath{\omega}$ shell-model space reduce the spin-orbit splitting in the case of $A=5$ by 20%, and enhance it for $A=15$ by about the same 20%. However, it is found that the correlations have a very small effect on the ${d}_{3/2}\ensuremath{-}{d}_{5/2}$ splitting in $A=17.$

Molecular orbital structures in10Be

The structure of ${}^{10}\mathrm{Be}$ is investigated using a microscopic $\ensuremath{\alpha}+\ensuremath{\alpha}+n+n$ model based on the molecular orbit (MO) model. The low-lying states are characterized by several configurations of valence neutrons, which are constructed as combinations of three basic orbits. The model space employed is extended from the traditional MO models, and the orbits are expressed as linear combinations of local Gaussians. Their positions are determined variationally. Using this model, we reanalyze the structure of ${}^{9}\mathrm{Be}$ and show that this extension enables us to use the original two-body spin-orbit interaction determined from a scattering phase-shift analysis of $\ensuremath{\alpha}\ensuremath{-}n.$ In ${}^{10}\mathrm{Be},$ all of the observed positive-parity bands and the negative-parity bands are described within the model. The ${0}^{+}$ ground state of ${}^{10}\mathrm{Be}$ is described by a dominant ${(3/2}^{\ensuremath{-}}{)}^{2}$ configuration. The state has a rather large binding energy (8.38 MeV from the $\ensuremath{\alpha}+\ensuremath{\alpha}+n+n$ threshold experimentally), and the mechanism leading to binding, such as a pairing effect and reduction of the kinetic energy between two clusters, is discussed in detail. In spite of this large binding, the $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ clustering in the ground state persists due to a coupling effect between the ${}^{6}\mathrm{He}+\ensuremath{\alpha}$ configuration and the ${}^{5}\mathrm{He}{+}^{5}\mathrm{He}$ configuration, which provides a smooth potential for the valence neutrons. The second ${0}^{+}$ state of ${}^{10}\mathrm{Be}$ has a large $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ structure with a ${(1/2}^{+}{)}^{2}$ configuration. An enlargement of the $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ distance due to two-valence neutrons along the $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ axis makes their wave function smooth and reduces the kinetic energy drastically. Furthermore, the contribution of the spin-orbit interaction due to coupling between the ${S}_{z}=0$ and the ${S}_{z}=1$ configurations is important. We also show the mediation effect of two valence neutrons between two $\ensuremath{\alpha}$ clusters.

The isospin-dependent quadrupole-quadrupole interaction used in shell model calculations – the effects of including ΔN = 2 excitations and a two-body spin-orbit interaction term

The effect of the “B” term in the interaction −χQ(1) · Q(2)[1+Bτ(1) ·τ(2)] was previously considered in the 0p shell (small space). It is now studied in a larger space which additionally includes ΔN = 2 excitations. When B is made sufficiently negative we still obtain for 10Be, even in the larger space, an unphysical collapse of some of the low-lying states so that their energies are less than the energy of the conventional J = 0+ ground state. This effect, however, occurs for values of B considerably more negative than was the case in the smaller space. It is shown that the inclusion of an additional two-body spin-orbit interaction term prevents this unrealistic collapse in both the large and small spaces.

The isospin-dependent quadrupole-quadrupole interaction used in shell model calculations – the effects of including ΔN = 2 excitations and a two-body spin-orbit interaction term

The isospin-dependent quadrupole-quadrupole interaction used in shell model calculations – the effects of including ΔN = 2 excitations and a two-body spin-orbit interaction term

The effects of the spin-orbit and tensor interactions in nuclei

We devise a nucleon-nucleon interaction V c + xV so + yV t to study the effects of the spin-orbit and tensor interactions by varying x and y. For negative parity 1 p−1 h states in 16O, we find that in a TDA calculation, the weighted sum of the eigenenergies Σ J (2 J + 1) Σ i E i ( J, T) is zero for a pure two-body spin-orbit interaction or a pure tensor interaction. If a minor closed shell nucleus like 12C is taken as the core, the tensor interaction lowers the p 1 2 single-particle energy relative to p 3 2 . This is precisely opposite to what the two-body spin-orbit interaction does. For the 1 + states in 12C, a 1 p−1 h diagonalization leads to a near collapse with E ∗(T = 0) ∼ 0.9 MeV and E ∗(T = 1) ∼ 3.7 MeV as compared with experiment (12.7 MeV for T = 0 and 15.1 MeV for T = 1). Only a full scalar shell model calculation brings these states up to a respectable energy. We find that in a full shell model calculation, the excitation energies of these 1 + states are surprisingly insensitive to the spin-orbit interaction over a rather wide range of the parameter x and there does not appear to be any phase transition as we vary x. We reconsider the old problem of the effect of the spin-orbit and tensor interactions on the nearly vanishing Gamow-Teller matrix element 14C( J = 0, T = 1) → 14N( J = 1, T = 0). We calculate the single-particle energies with the same interaction that is used for the residual particle-particle or particle-hole matrix elements.

Anisotropic two-body orbit-orbit and spin-orbit interactions for paramagnetic ions in crystals

Specific mechanisms involving conventional crystal fields and excitations to other sites are presented which lead to effective anisotropic orbit-orbit interactions. The method of relativistic operator equivalents then leads to contributions to the Hamiltonian which are not only nonscalar but also spin dependent. A comparison is made between the matrix elements of one-body and two-body operators with the same resultant spin and orbital ranks for f7 and d5 configurations. In a number of cases two-body operators possess matrix elements between spectroscopic terms whereas one-body operators with the same resultant ranks do not.

The giant-dipole and spin-dipole resonances: A coexistence problem

We present numerical and analytical results both indicating that giant-dipole states have no appreciable transition strengths for spin-dipole operators if these states exhaust almost all their sum rule. We also show that two-body spin-orbit and tensor interaction components have a fairly large influence on spin-dipole states but a very limited one on ( S = 0) giant-dipole states. Implications regarding the relative scale of cross section for S = 0 and S = 1 dipole excitations are also discussed.

HIGH RESOLUTION STUDY OF RYDBERG STATES OF BARIUM

We have studied the 6snl S-, P-, D- and F-Rydbergseries of Barium using CW dye lasers. The structure of the F-series is analyzed in terms of the electrostatic repulsion between the two valence electrons, the spin-orbit interaction and the hyperfine interaction. However, an effective two-body spin-orbit interaction has to be included. The structure of the Rydbergstates is sensitive to the presence of perturbing configurations, as shown in the S- and F-series.

Application of the ivv-w formalism to the continuum shell model

Energy-dependent radial functions v c and w c are employed to achieve good channel convergence in continuum shell-model calculations. All required many-body matrix elements are expressed with respect to standard shell-model basis states constructed from oscillator eigenfunctions. Target recoil and the Pauli principle are treated correctly. Test calculations for nucleon scattering from LS closed targets demonstrate the effectiveness of this approach. Target recoil corrections are shown to have a significant effect on the calculated energies of single-particle bound states and resonances. Other calculations illustrate the use of a finite-range two-body spin-orbit (in addition to central) interaction and the influence of target recoil corrections on 16O + p and 40Ca + p scattering in the 40–50 MeV region.

The effective interaction between nucleons deduced from nuclear spectra

Two-body matrix elements of the residual nucleon-nucleon interaction are extracted from experimental data throughout the periodic table and are used to determine the ranges and well depths of various components of a local interaction. The $T=1$ even and odd components of the central interaction both definitely require two wells with different ranges; a shorter-range attractive well with a longer-range repulsive one. The need for a tensor interaction and a two-body spin-orbit interaction is also explored and their inclusion improves the fit slightly.

Spin-orbit splitting in nuclei near closed shells: (I). Contribution of the two-body spin-orbit interaction

We have studied the spin-orbit (s.o.) splitting arising from the effective two-body s.o. interaction in the Brueckner-Hartree-Fock (BHF) approximation, considering all of the most commonly used realistic nucleon-nucleon (NN) interactions and three different nuclear models. We conclude that the s.o. interaction alone cannot account for the observed s.o. splitting in nuclei. We derive the one-body s.o. potential arising from the effective NN s.o. interaction in the BHF approximation. The derivation applies to the interaction of a nucleon with a spin-saturated (s.s.) core of nucleons. The expression we obtain is more than a factor of two stronger than some similar expressions obtained by previous investigators. The enhancement results from the exchange term, which had previously been ignored. We discuss, and obtain expressions for, the effective s.o. interaction arising, in the local-density approximation, from the most commonly used realistic NN potentials, namely the Reid soft core (RSC), Reid hard core (RHC), Hamada-Johnston (HJ), Yale, and Gammel-Thaler (GT) potentials. We find that they yield two distinct sets of values for the strength, − 1 2 πS 30 , of the one-body potential: the RSC, RHC, and HJ potentials give − 1 2 πS 30 ≈ 59 MeV · fm 5 while the Yale and GT potentials give − 1 2 πS 30 ≈ 73 MeV · fm 5 . We have calculated the s.o. splitting for three different nuclear models: (a) the spherical harmonic oscillator (h.o.) shell model, (b) a hybrid model which uses a Fermi distribution to describe the nucleons in the s.s. core and h.o. orbitals to describe the single-particle levels, and (c) the self-consistent Brueckner model of Negele. For each of these, quantitative results for the splitting are obtained for the five modern NN potentials listed above. Significant differences between the three models are found, models (b) and (c) giving better agreement with experiment than (a) for the heavier nuclei. We have also used the h.o. model to verify the numerical accuracy of our approach. We find that the effective s.o. interaction gives splittings which are roughly compatible with experiment for the normally unoccupied levels of the entirely s.s. nuclei 16O and 40Ca. For all the other levels of the nuclei studied, it appears that the s.o. interaction alone lacks sufficient strength to account for the entire s.o. splitting.