Contribution Of Three-body Force Research Articles

Energy of Phonons and Zero-Point Vibrations in Compressed Rare-Gas Crystals

A dynamic matrix of rare-gas crystals is constructed within the model of deformable and polarizable atoms based on the nonempirical short-range repulsion potential taking into account the three-body interaction and deformation of electron shells of dipole-type atoms within the pair and three-body approximations. Ab initio calculations of the phonon energy for compressed rare-gas crystals are performed at two and ten mean-value points of the Chadi–Kohen method in a wide pressure range. It is shown that the contribution of three-body forces to the phonon frequencies due to overlap of electron shells of neighboring atoms is small as compared to the pair interaction even at a high pressure and most pronounced for Xe. The contributions of the deformation of electron shells within the pair and three-body approximations differ for different mean-value points and increase with an increase in pressure. The zero-point energies calculated by the Chadi–Kohen method for crystals of the Ne–Xe series are compared with the known experimental data at p = 0 and calculation results obtained by other researchers.

Contribution of chiral three-body forces to the monopole component of the effective shell-model Hamiltonian

We present a study of the role played by realistic three-body forces in providing a reliable monopole component of the effective shell-model Hamiltonian. To this end, starting from a nuclear potential built up within the chiral perturbation theory, we derive effective shell-model Hamiltonians with and without the contribution of the three-body potential and compare the results of shell-model calculations with a set of observables that evidence shell-evolution properties. The testing ground of our investigation are nuclei belonging to fp shell, since the shell evolution towards shell closures in 48Ca and 56Ni provides a paradigm for shell-model Hamiltonians. Our analysis shows that only by including contributions of the three-body force the monopole component of the effective shell-model Hamiltonian is then able to reproduce the experimental shell evolution towards and beyond the closure at N=28.

Энергия фононов и нулевых колебаний в сжатых кристаллических инертных газах

A dynamic matrix of rare-gas crystals is constructed on the basis of a nonempirical short-range repulsion potential taking into account the three-body interaction and dipole-type deformation of the electron shells of atoms in the two- and three-body approximations in the model of deformable and polorizable atoms. Ab initio calculations of the phonon energy for compressed rare-gas crystals were performed at the two and ten mean-value points of the Chadi-Cohen method in a wide pressure range. It is shown that the contribution of three-body forces associated with the overlap of the electron shells of nearest-neighbor atoms in the phonon frequencies is small against the background of pair interaction, even at high pressure and most noticeable in Xe. The contribution of the deformation of the electron shells in the two- and three-body approximations is different for the different mean-value points and increases with increasing pressure. Comparison of the zero-point energy calculated by the Chadi-Cohen method for compressed crystals of the Ne–Xe series was performed with the available experiment at p=0 and the results of other authors.

Analysis of proton and neutron pair breakings: High-spin structures of 124–127Te isotopes

In the present work recently available experimental data for high-spin states of four nuclei, Te52124, Te52125, Te52126, and Te52127, have been interpreted using state-of-the-art shell model calculations. The calculations have been performed in the 50–82 valence shell composed of 1g7/2, 2d5/2, 1h11/2, 3s1/2, and 2d3/2 orbitals. We have compared our results with the available experimental data for excitation energies and transition probabilities, including high-spin states. The results are in reasonable agreement with the available experimental data. The wave functions, particularly, the specific proton and neutron configurations which are involved to generate the angular momentum along the yrast lines are discussed. We have also estimated overall contribution of three-body forces in the energy level shifting. Finally, results with modified effective interaction are also reported.

Three-body-force effect on nucleus-nucleus elastic scattering

The three-body force (TBF) effect is studied in nucleus-nucleus elastic scattering with the use of a new complex $G$-matrix interaction that reproduces well proton-nucleus scattering observables. The TBF contribution is included in the $G$-matrix so as to give a reasonable nuclear-matter saturation curve. The $^{16}\mathrm{O}$$+$$^{16}\mathrm{O}$ potential is obtained by the double-folding procedure with the $G$-matrix interaction. The observed cross sections for $^{16}\mathrm{O}$$+$$^{16}\mathrm{O}$ elastic scattering at $E/A=70$ MeV are nicely reproduced up to the most backward scattering angles when the TBF effect is included.

Microscopic Three-Body Force Effect on Nucleon-Nucleon Cross Sections in Symmetric Nuclear Matter

We provide a microscopic calculation of neutron-proton and proton-proton cross sections in symmetric nuclear matter at various densities, using the Brueckner–Hartree–Fock approximation scheme with the Argonne V14 potential including the contribution of microscopic three-body force. We investigate separately the effects of three-body force on the effective mass and on the scattering amplitude. In the present calculation, the rearrangement contribution of three-body force is considered, which will reduce the neutron and proton effective mass, and depress the amplitude of cross section. The effect of three body force is shown to be repulsive, especially in high densities and large momenta, which will suppress the cross section markedly.

Nucleon-nucleon cross sections in dense nuclear matter

We present microscopic calculations of cross sections for scattering of identical and nonidentical nucleons in symmetric nuclear matter at various densities, using the Brueckner-Hartree-Fock approximation scheme with the Argonne ${v}_{14}$ potential including the contribution of microscopic three-body forces. We investigate separately the effects of three-body forces on the effective mass and on the scattering amplitude. In the present calculation, the rearrangement contribution of the three-body force is considered, which reduces the neutron and proton effective mass and suppresses the magnitude of the cross section. The presence of ``Z diagrams'' in the three-body force enables us to make a comparison with the medium effects on the nucleon-nucleon cross sections obtained with the Dirac-Brueckner-Hartree-Fock approximation.

Temperature dependence of single-particle properties in nuclear matter

The single-nucleon potential in hot nuclear matter is investigated in the framework of the Brueckner theory by adopting the realistic Argonne ${V}_{18}$ or Nijmegen 93 two-body nucleon-nucleon interaction supplemented by a microscopic three-body force. The rearrangement contribution to the single-particle potential induced by the ground state correlations is calculated in terms of the hole-line expansion of the mass operator and provides a significant repulsive contribution in the low-momentum region around and below the Fermi surface. Increasing temperature leads to a reduction of the effect, while increasing density makes it become stronger. The three-body force suppresses somewhat the ground state correlations due to its strong short-range repulsion, increasing with density. Inclusion of the three-body force contribution results in a quite different temperature dependence of the single-particle potential at high enough densities as compared to that adopting the pure two-body force. The effects of three-body force and ground state correlations on the nucleon effective mass are also discussed.

Single nucleon potential and effective mass with ground state correlations in hot nuclear matter

In the framework of the finite temperature Brueckner-Hartree-Fock approach including the contribution of the microscopic three-body force, the single nuclear potential and the nucleon effective mass in hot nuclear matter at various temperatures and densities have been calculated by using the hole-line expansion for mass operator, and the effects of the three-body forces and the ground state correlations on the single nucleon potential have been investigated. It is shown that both the ground state correlations and the three-body force affect considerably the density and temperature dependence of the single nucleon potential. The rearrangement correction in the single nucleon potential is repulsive and it reduces remarkably the attraction of the single nucleon potential in the low-momentum region. The rearrangement contribution due to the ground state correlations becomes smaller as the temperature rises up and becomes larger as the density increases. The effect of the three-body force on the ground state correlations is to reduce the contribution of rearrangement. At high densities, the single nucleon potential containing both the rearrangement correction and the contribution of the three-body force becomes more repulsive as the temperature increases.

Nuclear three-body force effect on a kaon condensate in neutron star matter

We explore the effects of a microscopic nuclear three-body force on the threshold baryon density for kaon condensation in chemical equilibrium neutron star matter and on the composition of the kaon condensed phase in the framework of the Brueckner-Hartree-Fock approach. Our results show that the nuclear three-body force affects strongly the high-density behavior of nuclear symmetry energy and consequently reduces considerably the critical density for kaon condensation provided that the proton strangeness content is not very large. The dependence of the threshold density on the symmetry energy becomes weaker as the proton strangeness content increases. The kaon condensed phase of neutron star matter turns out to be proton rich instead of neutron rich. The three-body force has an important influence on the composition of the kaon condensed phase. Inclusion of the three-body force contribution in the nuclear symmetry energy results in a significant reduction of the proton and kaon fractions in the kaon condensed phase which is more proton-rich in the case of no three-body force. Our results are compared to other theoretical predictions by adopting different models for the nuclear symmetry energy. The possible implications of our results for the neutron star structure are also briefly discussed.

Hot nuclear matter equation of state with a three-body force

The finite temperature Brueckner-Hartree-Fock approach is extended by introducing a microscopic three-body force. In the framework of the extended model, the equation of state of hot asymmetric nuclear matter and its isospin dependence have been investigated. The critical temperature of liquid-gas phase transition for symmetric nuclear matter has been calculated and compared with other predictions. It turns out that the three-body force gives a repulsive contribution to the equation of state which is stronger at higher density and as a consequence reduces the critical temperature of liquid-gas phase transition. The calculated energy per nucleon of hot asymmetric nuclear matter is shown to satisfy a simple quadratic dependence on asymmetric parameter $\ensuremath{\beta}$ as in the zero-temperature case. The symmetry energy and its density dependence have been obtained and discussed. Our results show that the three-body force affects strongly the high-density behavior of the symmetry energy and makes the symmetry energy more sensitive to the variation of temperature. The temperature dependence and the isospin dependence of other physical quantities, such as the proton and neutron single particle potentials and effective masses, are also studied. Due to the additional repulsion produced by the three-body force contribution, the proton and neutron single particle potentials are correspondingly enhanced as similar to the zero-temperature case.

Finite-Temperature Equation of State with Three-Body ForceCorrelations

The equation of state of asymmetric nuclear matter atfinite temperature is studied in the framework of theBrueckner–Hartree–Fock approximation which is extended to includethree-body force correlations. Due to the additional repulsion producedby three-body force contribution, the proton or neutron potentialenergy and effective mass are correspondingly enhanced. The symmetryenergy gradually becomes more sensitive to the temperature with theincreasing density compared with the result that only includes the twohole-line contribution. The asymmetry parameter dependence of symmetryenergy is also indicated. It is shown that there is the same linearrelation between symmetry energy and β2 as the zerotemperature case with β being the asymmetry parameter.

On the contribution of three-body forces to Nd interaction at intermediate energies

Available data on large-angle nucleon-deuteron elastic scattering Nd→dN below the pion threshold give a signal for three-body forces. The problem exists of the separation of possible subtle aspects of these forces from off-shell effects in two-nucleon potentials. By considering the main mechanisms of the process Nd→dN, we show qualitatively that in the quasi-binary reaction N+d→(NN)+N with the final spin singlet nucleon-nucleon pair in the S-state, the relative contribution of the three-nucleon forces differs substantially from the elastic channel. It gives a new testing basis for the problem in question.

Folding model with three-body force

A folding model for the nucleon-nucleus interaction potential involving a three-body force is presented. Well known models for the nuclear two-body density have been used to derive, under the zero-range approximation, an expression for the two-body effective interaction which includes the three-body force contribution as an additional density dependence. The roles of the centre of mass and Pauli pair correlations in shaping the density dependence of the two-body effective interaction are discussed.

Two- and three-body forces in the interaction of He atoms with Xe overlayers adsorbed on (0001) graphite

In order to address the problem of three-body interactions in gas–surface scattering, we considered the collision of a He atom with the (0001) surface of graphite coated by a monolayer of Xe. To eliminate the uncertainties connected with errors in the two-body He–Xe interaction, we determined the latter by crossed-beam differential collision cross-section measurements performed at two energies (67.2 and 22.35 meV). These scattering data together with room-temperature bulk diffusion data are then fitted with a Hartree–Fock–dispersion–type function to yield an interaction potential that explains most of the properties of this system within the experimental errors and represents an improvement on previously published He–Xe potentials. Helium diffraction measurements are then carried out from the Xe overlayer and the dependence of the specular intensity from the angle of incidence is carefully determined. Further, a He–surface potential is constructed by adding together the following terms: (1) the He–Xe pairwise sum, (2) the long-range He–(0001)C interaction, (3) the three-body contribution generated by the Axilrod–Teller–Muto term, (4) the so-called surface-mediated three-body interaction He–Xe–(0001)C first considered by A. D. McLachlan [Mol. Phys. 7, 381 (1964)], and finally (5) a small correction which is meant to take into account the nonstationary nature of the surface. Using this potential, well-converged close-coupling scattering calculations are carried out, and their results compared with the data. In general, good agreement is obtained. The agreement can, however, be improved by (a) an increase of about 30% in the contribution of three-body forces, (b) the lowering of the He–graphite long-range attraction coefficient by about 15%, or (c) a reduction of the two-body interaction well depth of 1.6% (the experimental error) together with any combination of the factors under (a) and (b) reduced by an adequate amount. Elimination of the contribution of the graphite surface by studying Xe multilayers is hindered by the uncertainties in the ‘‘thermal correction’’ [point (5) above] which, due to the multilayer increased ‘‘softness,’’ becomes an appreciable source of uncertainty.

Contribution of three-body force to the trinucleon problem by an essentially exact calculation

The hyperspherical harmonic method has been used to calculate the effect of two-pion exchange three-body force (Fujita-Miyazawa type) on trinucleon systems, with the $N\ensuremath{-}N$ Afnan- Tang S3 potential. The Coulomb force and three-body force have been taken into account nonperturbatively. Simplification in the numerical calculation has been achieved through the use of an adiabatic approximation in order to decouple the system of differential equations. Binding energy, charge form factor, and pointlike proton density of both $^{3}\mathrm{H}$ and $^{3}\mathrm{He}$ have been calculated for various values of the cutoff parameter in the three-body force. Results indicate that the inclusion of the three-body force improves agreement with experiment. An interesting feature is the possible appearance of nodes near origin in the hyper-radial wave function.NUCLEAR STRUCTURE Trinucleon systems; three-body force; bound states of $^{3}\mathrm{H}$ and $^{3}\mathrm{He}$ using hyperspherical harmonic method; adiabatic approximation; charge form factors.

Three-body force contribution in nuclear matter via perturbation and proper Brueckner calculations

In a previous paper (~), the energy contribution of the simplest three-body force in nuclear matter was studied via a proper Brueckner theory calculation. Versions I I and I I I of the three-body force of Lois~Atr, NOGAMI and Ross (2) (LNR) were found to contribute respectively 0.8 and 0.7 MeV/A to the binding of the infinite system at normal density, in contrast with BLATT and McK~LLAR'S recent value of 6 MeV/A (3), which is in error due to an incorrect calculation of the effective two-body force 1/3 der ivedfrom the three-body force W(BM'spr ivatecommunicat ion) . Furthermore, a sLNR, they used the perturbation theory with an effective mass m* 1 which we show here to over-estimate the contribution main ly in the 3S~ partial wave. We report on a partial wave per part ial wave comparison between results of perturbation theory and results of the proper Brueckner calculation with the Reid soft core the potential V 2 plus the effective force V 3. In this way we avoid treating V 3 as a perturbation and a meaningful comparison between the two methods is therefore possible. From Mc KELLAR and RAZA~A~AN (4), the contribution to the binding energy of nuclear matter due to the three-body force W only is given by

The saturating effect of the Δ(1236) in nuclear matter

Abstract It is shown that the explicit introduction of the Δ(1236) into the two-nucleon interaction gives rise in nuclear matter to a repulsive contribution to the binding energy of 5 MeV per particle at k F = 1.4 fm −1 . This contribution increases approximately as the square of the density. The effect is found using both the Pandharipande and Brueckner approaches with quantitative agreement between the two. A brief discussion is made of the compensating three-body force contribution.

Effect of Correlations on the Contribution of Three-Body Forces to the Binding Energy of Nuclear Matter

We investigate the contribution of three-body forces to the binding energy of nuclear matter, including the effects of two- and three-body correlations induced by the two-nucleon forces, and find that it may by approximated using plane-wave three-nucleon states cut off when any interparticle distance is less than about 0.9 fm. Existing calculations can then be used to estimate that three-body forces contribute about 1 MeV to the binding energy.

Contribution of Three-Body Forces to the Theory of Liquids

Received 22 January 1968DOI:https://doi.org/10.1103/PhysRevLett.20.529©1968 American Physical Society