Forces In Nuclear Matter Research Articles

Exact solution of the Brueckner-Bethe-Goldstone equation with three-body forces in nuclear matter

An exact treatment of the operators $Q/e(\ensuremath{\omega})$ and the total momentum is adopted to solve the nuclear matter Bruecker-Bethe-Goldstone equation with two- and three-body forces. The single-particle potential, equation of state, and nucleon effective mass are calculated from the exact $G$ matrix. The results are compared with those obtained under the angle-average approximation and the angle-average approximation with total momentum approximation. It is found that the angle-average procedure, whereas preventing huge calculations of coupled channels, nevertheless provides a fairly accurate approximation. On the contrary, the total momentum approximation turns out to be quite inaccurate compared to its exact counterpart.

Relativistic effects and three-nucleon forces in nuclear matter and nuclei

We review a large body of predictions obtained within the framework of relativistic meson theory together with the Dirac–Brueckner–Hartree–Fock approach to nuclear matter and finite nuclei. The success of this method has been largely related to its ability to take into account important three-body effects. Therefore, the overarching theme of this paper is the interpretation of the so-called “Dirac effects” as an effective three-nucleon force. We address the equation of state (EoS) of isospin symmetric and asymmetric nucleonic matter and related issues, ranging from proton and neutron density distributions to momentum distributions and short-range correlations. A central part of the discussion is devoted to the optical model potential for nucleon–nucleus scattering. We also take the opportunity to explore similarities and differences with predictions based on the increasingly popular chiral effective field theory.

Three nucleon forces in nuclear matter in QCD sum rules

We calculate the single-particle nucleon characteristics in symmetric nuclear matter with inclusion of the 3N interactions. The contributions of the 3N forces to nucleon self energies are expressed in terms of the nonlocal scalar condensate (d=3) and of the configuration of the four-quark condensates (d=6) in which two diquark operators act on two different nucleons of the matter. The most important part of the contribution of the four-quark condensate is calculated in a model-independent way. We employed a relativistic quark model of nucleon for calculation of the other parts. The density dependence of the vector and scalar nucleon self energies and of the single-particle potential energy are obtained. Estimations on contributions of the 4N forces to the nucleon self energies are made.

Regulator artifacts in uniform matter for chiral interactions

Regulator functions applied to two- and three-nucleon forces are a necessary ingredient in many-body calculations based on chiral effective field theory interactions. These interactions have been developed recently with a variety of different cutoff forms, including regulating both the momentum transfer (local) and the relative momentum (nonlocal). While in principle any regulator that suppresses high momentum modes can be employed, in practice artifacts are inevitable in current power counting schemes. Artifacts from particular regulators may cause significant distortions of the physics or may affect many-body convergence rates, so understanding their nature is important. Here we characterize the differences between cutoff effects using uniform matter at Hartree-Fock and second-order in the interaction as a testbed. This provides a clean laboratory to isolate phase-space effects of various regulators on both two- and three-nucleon interactions. We test the normal-ordering approximation for three-nucleon forces in nuclear matter and find that the relative size of the residual 3N contributions is sensitive to the employed regularization scheme.

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.

Density functional theory with a separable pairing force in finite nuclei

A separable pairing interaction is introduced in the covariant density functional theory. The parameters of the separable force are adjusted to reproduce the pairing properties of the Gogny force in nuclear matter. Applying well known techniques of Talmi and Moshinsky this finite range separable pairing interaction is adopted to study the pairing properties of finite nuclei in the relativistic Hartree-Bogoliubov and quasiparticle random phase approximation. We conclude that with this new pairing force the superfluid of finite nuclei can be well depicted on almost the same footing as with the original Gogny pairing interaction.

Axially deformed relativistic Hartree Bogoliubov theory with a separable pairing force

A separable form of pairing interaction in the ${}^{1}{S}_{0}$ channel has been introduced and successfully applied in the description of both static and dynamic properties of superfluid nuclei. By adjusting the parameters to reproduce the pairing properties of the Gogny force in nuclear matter, this separable pairing force is successful in depicting the pairing properties of ground states and vibrational excitations of spherical nuclei on almost the same footing as the original Gogny force. In this article, we extend these investigations for relativistic Hartree-Bogoliubov theory in deformed nuclei with axial symmetry (RHBZ) using the same separable pairing interaction. To preserve translational invariance we construct one- and two-dimensional Talmi-Moshinsky brackets for the cylindrical harmonic oscillator basis. We show that the matrix elements of this force can then be expanded in a series of separable terms. The convergence of this expansion is investigated for various deformations. We observe a relatively fast convergence. This allows for a considerable reduction in computing time as compared to RHBZ calculations with the full Gogny force in the pairing channel. As an example we solve the RHBZ equations with this separable pairing force for the ground states of the chain of Sm isotopes. Good agreement with the experimental data as well as with other theoretical results is achieved.

Separable pairing force for relativistic quasiparticle random-phase approximation

We have introduced a separable pairing force, which was adjusted to reproduce the pairing properties of the Gogny force in nuclear matter. This separable pairing force is able to describe in relativistic Hartree-Bogoliubov (RHB) calculations the pairing properties in the ground state of finite nuclei on almost the same footing as the original Gogny interaction. In this work we investigate excited states using the Relativistic Quasiparticle Random-Phase Approximation (RQRPA) with the same separable pairing force. For consistency the Goldstone modes and the convergence with various cutoff parameters in this version of RQRPA are studied. The first excited ${2}^{+}$ states for the chain of Sn isotopes with $Z=50$ and the chain of isotones with $N=82$ isotones are calculated in RQRPA together with the ${3}^{\ensuremath{-}}$ states of Sn isotopes. By comparing our results with experimental data and with the results of the original Gogny force we find that this simple separable pairing interaction is very successful in depicting the pairing properties of vibrational excitations.

Relativistic Quasiparticle Random Phase Approximation with a Separable Pairing Force

In our previous work [Phys. Lett. (to be published), Chin. Phys. Lett. 23(2006) 3226], we introduced a separable pairing force for relativistic Hartree–Bogoliubov calculations. This force was adjusted to reproduce the pairing properties of the Gogny force in nuclear matter. By using the well known techniques of Talmi and Moshinsky it can be expanded in a series of separable terms and converges quickly after a few terms. It was found that the pairing properties can be depicted on almost the same footing as the original pairing interaction, not only in nuclear matter, but also in unite nuclei. In this study, we construct a relativistic quasiparticle random phase approximation (RQRPA) with this separable pairing interaction and calculate the excitation energies of the first excited 2+ states and reduced B(E2; 0+ → 2+) transition rates for a chain of Sn isotopes in RQRPA. Compared with the results of the full Gogny force, we find that this simple separable pairing interaction can describe the pairing properties of the excited vibrational states as well as the original pairing interaction.

A Separable Pairing Force in Nuclear Matter

The method introduced by Duguet is adopted to derive aseparable form of the pairing interaction in the 1S0 channel from a bare or aneffective nucleon–nucleon (NN) interaction in nuclear matter. With this approachthe separable pairing interaction reproduces the pairing properties provided by itscorresponding NN interaction. In this work, separable forms of pairing interactionsin the 1S0 channel for the bare NN interaction, Bonn potential and the Gognyeffective interaction are obtained. It is found that the separable force of theGogny effective interaction in the 1S0 channel has a clear link with the bare NNinteraction. With such a simple separable form pairing properties provided by theGogny force in nuclear matter can be reproduced.

Temperature-dependent mean field and its effect on heavy-ion reactions

Temperature-dependent mean field potentials of nucleons are obtained by solving the Bethe-Goldstone equation for a realistic force in nuclear matter at finite temperature. For a more efficient utilization of these potentials in studying the heavy-ion reactions using a transport theory, the density and temperature dependence of these potentials is parametrized in a Skyrme type form. These parametrized temperature-dependent potentials are implemented in quantum molecular dynamics. The temperature during the simulations is deduced using a hot Thomas-Fermi approach generalized for the case of two interpenetrating pieces of nuclear matter. First of all, we show that our formalism works well in the nuclear matter limit. In order to study the effect of temperature dependence in the mean-field potential in heavy-ion reactions, the reactions 40Ca+ 40Ca and 93Nb+ 93Nb are simulated using both a finite temperature-dependent potential and a temperature-independent (i.e. zero temperature) potential. Our detailed investigation shows that the temperature dependence of the mean field affects the heavy-ion reaction dynamics to a significant amount. These effects are stronger in case of heavier nuclei and are of the same order as the differences between the usual “soft” and “hard” equation of state. An analytical parametrization of the temperature dependence of the self-consistent field is given in a Skyrme type form.

The temperature dependence of the Skyrme forces

The temperature dependence of the Skyrme forces in nuclear matter is investigated under the assumption that the effective forces can be derived as a local density expansion on the two-body reaction matrix. We obtain a simple analytic expression for the full equation of state. The chemical potential and the pressure are the main quantities which are affected by the temperature dependence of the Skyrme forces, while the energetics is essentially unaffected. The inclusion of the temperature dependence lowers the flash temperature by ≈ 2 MeV and the critical temperature by ≈ 3 MeV.

Heavy-ion optical potentials at finite temperature calculated using a complex effective interaction derived from a realistic force

Hot density distributions of heavy ions generated by a modified Thomas-Fermi calculation at finite temperature are used to calculate the optical potential using the double-folding method and a complex effective energy and density-dependent interaction deduced from a realistic NN (Reid soft-core) force in nuclear matter. The real and the imaginary part of the optical potential become more attractive when the temperature increases.

Three-body forces in nuclear matter from intermediate Δ-states in three-nucleon clusters

Those three-body force contributions in nuclear matter usually generated through a πN scattering amplitude dominated by the Δ(1236) resonance, are here treated as a three-nucleon cluster, in which one of the nucleons becomes, in an intermediate state, a Δ-resonance. All exchange diagrams are calculated and found to significantly reduce the energy per particle from the direct graph. This is contrary to earlier estimates of the exchanges, using more approximate approaches. The resulting attractive contribution is rather small, −1.1 MeV at κ F = 1.4 fm −1, but the roughly linear density dependence has a crucial effect on the saturation properties. The sensitivity of the results to the correlations used, and to the two-body force spin structure, is displayed. The energy per particle from clusters with three intermediate Δ-resonances is also estimated.

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

Three-body force in nuclear matter

The simplest three-body force, due to two-pion exchange, contributes only about 0.8 MeV/A to the binding energy of nuclear matter. This is to be contrasted with previous estimates of 2 MeV and 6 MeV. A proper Brueckner theory calculation was done in place of the perturbation theory employed previously.

Exchange effects andn-body forces in nuclear matter

Exchange effects and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>n</mml:mi></mml:math>-body forces in nuclear matter

Four-body forces in nuclear matter

The binding energy contributions to nuclear matter from four-body forces have been estimated. The four nucleon potential is written in coordinate space, and two intermediate nucleons are treated as spectators to give an effective two-body potential. Two-body correlations are included between all four nucleons and, following a coordinate transformation, two of the six dimensions for the spectator nucleons can be integrated analytically. Spin-isospin dependence is extracted and the remaining integrations are done numerically to give a one-pion-exchange-potential-like effective potential similar to the three-body force effective potentials of Loiseau, Nogami, and Ross. Binding energy calculations can then be done in the usual manner. As the number of nucleons increases the effect of exchanges of the spectator nucleons must also be considered. When this is taken into account, the over-all contribution to the energy of nuclear matter is approximately 0.1 MeV attraction.NUCLEAR STRUCTURE Many-body forces, calculated binding $E$ of four-body force; includes correlations; effective two-body potentials.

Three-body force in nuclear matter

We report calculations on the contribution of the three-body potential $W$ to the energy of nuclear matter which include realistic correlations in all internucleon distances which are consistent with realistic two-body potentials ($V$) in the calculation of the dominant second order [$〈V(\frac{Q}{e})W〉$] contribution to the binding energy. For the Reid soft core potential we get a binding energy contribution of 6 MeV/nucleon. This is to be compared with the binding energy of 1.7 MeV/nucleon obtained using hard core potentials. The latter result is consistent with the calculations of Loiseau, Nogami, and Ross. The physical reason for the large difference between the results obtained from hard core and soft core two-body potentials is clarified.

Soft-core correlations and three body forces in nuclear matter

We show that, when nucleon-nucleon correlations are taken into account using correlation functions derived. from the Reid soft core potential, the contribution of three body forces to the binding energy of nuclear matter is enhanced. The two pion exchange three body forces contributes 6 MeV attraction.