Inclusion Of Three-body Forces Research Articles

Simulating The Residual Three-body Force in a Small System

The effect of residual three-body force on the small system was investigated in this paper. A one-level and two-level system were chosen as the model to study the feature of three-body force due to their simplicity and lower symmetry. In both systems, we found a consistent result, that the inclusion of three-body force will enhance and accelerate the pairing condensation, which is shown by a higher pairing gap and lower energy as compared to the system with only two-body pairing force. Some interesting results on the behavior of the residual three-body force were also presented here.

Hubbard-Stratonovich-like Transformations for Few-Body Interactions

Through the development of many-body methodology and algorithms, it has become possible to describe quantum systems composed of a large number of particles with great accuracy. Essential to all these methods is the application of auxiliary fields via the Hubbard-Stratonovich transformation. This transformation effectively reduces two-body interactions to interactions of one particle with the auxiliary field, thereby improving the computational scaling of the respective algorithms. The relevance of collective phenomena and interactions grows with the number of particles. For many theories, e.g. Chiral Perturbation Theory, the inclusion of three-body forces has become essential in order to further increase the accuracy on the many-body level. In this proceeding, the an-alytical framework for establishing a Hubbard-Stratonovich-like transformation, which allows for the systematic and controlled inclusion of contact three-and more-body inter-actions, is presented.

Nuclear pairing from two-body microscopic forces: analysis of the Cooper pair wavefunctions

In a recent paper we studied the behavior of the pairing gaps $\Delta_F$ as a function of the Fermi momentum $k_F$ for neutron and nuclear matter in all relevant angular momentum channels where superfluidity is believed to naturally emerge. The calculations employed realistic chiral nucleon-nucleon potentials with the inclusion of three-body forces and self-energy effects. In this contribution, after a detailed description of the numerical method we employed in the solution of the BCS equations, we will show a preliminary analysis of the Cooper pair wavefunctions.

Nuclear pairing from two-body microscopic forces: analysis of the Cooper pair wavefunctions

In a recent paper we studied the behavior of the pairing gaps $\Delta_F$ as a function of the Fermi momentum $k_F$ for neutron and nuclear matter in all relevant angular momentum channels where superfluidity is believed to naturally emerge. The calculations employed realistic chiral nucleon-nucleon potentials with the inclusion of three-body forces and self-energy effects. In this contribution, after a detailed description of the numerical method we employed in the solution of the BCS equations, we will show a preliminary analysis of the Cooper pair wavefunctions.

EOS of Neutron Matter and Neutron Star Properties

The Equation of State (EOS) of pure neutron matter at zero temperature is calculated up to five saturation densities within the Brueckner theory with the inclusion of three-body forces. Three different realistic and accurate two-body forces are considered to evaluate the G-matrix effective interaction for nuclear matter. These models are the chiral N 3 LO, the CD-Bonn and the Argonne V18, which give quite different EOS. Two types of three-body forces are included to the effective interaction, which might be important at densities several times that of nuclear matter density. Using a microscopic EOS for pure neutron matter, static properties of non-rotating neutron stars such as masses and radii are evaluated. The resulting maximum masses of neutron star using different interactions near 2 Mʘ are found to be in reasonable agreement with the measured ones PSR J1614−2230 (with Mmax = 1.97 ± 0.04 Mʘ) and PSR J0348+0432 (with Mmax = 2.01 ± 0.04 Mʘ).

Nuclear pairing from microscopic forces: Singlet channels and higher-partial waves

Background: An accurate description of nuclear pairing gaps is extremely important for understanding static and dynamic properties of the inner crusts of neutron stars and to explain their cooling process. Purpose: We plan to study the behavior of the pairing gaps $\Delta_F$ as a function of the Fermi momentum $k_F$ for neutron and nuclear matter in all relevant angular momentum channels where superfluidity is believed to naturally emerge. The calculations will employ realistic chiral nucleon-nucleon potentials with the inclusion of three-body forces and self-energy effects. Methods: The superfluid states of neutron and nuclear matter are studied by solving the BCS gap equation for chiral nuclear potentials using the method suggested by Khodel et al., where the original gap equation is replaced by a coupled set of equations for the dimensionless gap function $\chi(p)$ defined by $\Delta(p) = \Delta_F \chi(p)$ and a non-linear algebraic equation for the gap magnitude $\Delta_F = \Delta(p_F)$ at the Fermi surface. This method is numerically stable even for small pairing gaps, such as that encountered in the coupled $^3PF_2$ partial wave. Results: We have successfully applied Khodel's method to singlet ($S$) and coupled channel ($SD$ and $PF$) cases in neutron and nuclear matter. Our calculations agree with other ab-initio approaches, where available, and provide crucial inputs for future applications in superfluid systems.

Thermal properties of nuclear matter in a variational framework with relativistic corrections

The properties of hot symmetric nuclear matter for a wide range of densities and temperatures are investigated by employing the $A{V}_{14}$ potential within the lowest order constrained variational (LOCV) method with the inclusion of a phenomenological three-body force as well as relativistic corrections. The relativistic corrections of many-body kinetic energies as well as the boot interaction corrections are presented for a wide range of densities and temperatures. The free energy, pressure, incompressibility, and other thermodynamic quantities of symmetric nuclear matter are obtained and discussed. The critical temperature is found, and the liquid-gas phase transition is analyzed both with and without the inclusion of three-body forces and relativistic corrections in the LOCV approach. It is shown that the critical temperature is strongly affected by the three-body forces but does not depend on the relativistic corrections. Finally, the results obtained in the present study are compared with other many-body calculations and experimental predictions.

Microscopic description of translationally invariant core+N+Noverlap functions

We derive expressions for core + N + N overlap integrals starting from microscopic wave functions obtained in the ab initio no-core shell model. These overlap integrals correspond to three-body channel form factors and can be used to investigate the clustering of many-body systems into a core plus two nucleons. We consider the case when the composite system and the core are described in Slater determinant, harmonic oscillator bases, and we show how to remove spurious center of mass components exactly in order to derive translationally-invariant overlap integrals. We study in particular the Borromean 6He nucleus using realistic chiral nuclear interactions, and we demonstrate that the observed clusterization in this system is a Pauli focusing effect. The inclusion of three-body forces has a small effect on this structure. In addition, we discuss the issue of absolute normalization for spectroscopic factors, which we show is larger than one. As part of this study we also perform extrapolations of ground-state observables and investigate the dependence of these results on the resolution scale of the interaction.

Microscopic optical potential with two and three body forces for nucleon–nucleus scattering

The proton - nucleus optical potentials generated by folding the calculated complex, density and energy dependent g- matrices (with and without three-body forces (TBF): Urbana IX (UVIX) and TNI) over the target nucleon density distributions obtained from the relativistic mean field theory, are used for the calculation of the differential cross section dσ / dθ , polarization Ay , spin rotation function (Q). for 65 and 200 MeV polarized proton incident on 40 Ca and 208 Pb . The agreement with the experiment is rather impressive. It is found that the inclusion of TBF (Urbana IX )UVIX) and TNI) reduces the strength of the central part of the optical potential in the nuclear interior and affects the calculated spin-orbit potential only marginally and leads to an improvement in the agreement with the corresponding experimental results.

A systematic analysis of microscopic nucleon–nucleus optical potential for p–Ni scattering

A microscopic proton–nucleus optical potential is calculated for all even isotopes 52–112Ni within the Brueckner–Hartee–Fock framework. The reaction matrices calculated using three (Argonne v18, Reid93 and NijmII) realistic inter-nucleon potentials with and without three-body forces (Urbana IX (UVIX) and the density dependent three-nucleon interaction (TNI) model of Lagaris, Friedman and Pandharipande), have been folded over the neutron and proton density distributions of the relevant targets obtained by using the relativistic mean field approach. It is observed that the inclusion of both models of the three-body forces somewhat reduces the strength of the central part of the optical potential in the nuclear interior and affects only marginally the spin–orbit part of the potential. Further, the calculated volume integral of the real part of the spin–orbit optical potential as well as its peak value are found to decrease systematically with the addition of neutrons. The calculated optical potentials reproduce remarkably well the existing experimental differential cross section (dσ/dθ) and the polarization (Ay) data for p–58–64Ni scattering at 39.6 and 65 MeV projectile energies. The inclusion of three-body forces (UVIX and TNI) does not lead to any change in the calculated observables (dσ/dθ and Ay) indicating that the p–Ni scattering data analyzed here are not sensitive to the nuclear interior. Identical observations are also found in the Zr, Sn and Pb isotopic chains. Therefore, these observed features of the microscopic optical potential seem general and may hold globally.

Structure of16C: Testing shell model andab initioapproaches

Excited states in C-16 were populated via the Be-9(N-17,C-16+gamma)X one-proton knockout reaction. The lifetime of the 2(1)(+) state in C-16 was measured using the recoil distance method. The extracted lifetime of tau(+)(21) = 11.4(-0.9)(+0.8)(stat) +/- 0.7(syst(B rho))(-1.5)(+0.0)(syst(feeding)) ps yields a deduced B(E2;2(1)(+) -> 0(1)(+)) = 4.21(-0.26)(+0.34)(stat)(-0.24)(+0.28)(syst(B rho))(-0.00)(+0.64)(syst(feeding)) e(2)fm(4) value in good agreement with a previous measurement. The one-proton knockout cross section is used to extract the proton amplitude of the C-16 2(1)(+) state, which confirms the neutron dominant character of this state. Gamma-ray branching ratios between the 2(2)(+) state and the 2(1)(+) and ground states were also determined. The results are compared with p-sd shell model and no-core shell model (with NN and NN + NNN) calculations. The inclusion of three-body forces are essential in order for the no-core shell model calculations to reproduce the experimental findings on the gamma-ray branching ratios.

Excited-state transition-rate measurements in18C

Excited states in ${}^{18}$C were populated by the one-proton knockout reaction of an intermediate energy radioactive ${}^{19}$N beam. The lifetime of the first ${2}^{+}$ state was measured with the K\"oln/NSCL plunger via the recoil distance method to be $\ensuremath{\tau}({2}_{1}^{+})=22.4\ifmmode\pm\else\textpm\fi{}0.9{(\mathrm{stat})}_{\ensuremath{-}2.2}^{+3.3}(\mathrm{syst})$ ps, which corresponds to a reduced quadrupole transition strength of $B(E2;{2}_{1}^{+}\ensuremath{\rightarrow}{0}_{1}^{+})=3.{64}_{\ensuremath{-}0.14}^{+0.15}{(\mathrm{stat})}_{\ensuremath{-}0.47}^{+0.40}(\mathrm{syst})$ e${}^{2}$fm${}^{4}$. In addition, an upper limit on the lifetime of a higher-lying state feeding the ${2}_{1}^{+}$ state was measured to be $\ensuremath{\tau}<4.6$ ps. The results are compared to large-scale ab initio no-core shell model calculations using two accurate nucleon-nucleon interactions and the importance-truncation scheme. The comparison provides strong evidence that the inclusion of three-body forces is needed to describe the low-lying excited-state properties of this $A=18$ system.

Solutions of the bound-state Faddeev-Yakubovsky equations in three dimensions by usingNNand3Npotential models

A recently developed three-dimensional approach (without partial-wave decomposition) is considered to investigate solutions of Faddeev-Yakubovsky integral equations in momentum space for three- and four-body bound states, with the inclusion of three-body forces. In the calculations of the binding energies, spin-dependent nucleon-nucleon ($\mathit{NN}$) potential models [soft-core potential S3, Malfliet-Tjon (MT) I-III, Yamaguchi-type potentials (YS), and ${P}_{5.5}$-model of Gibson-Lehman (P${}_{55}$GL)] are considered along with the scalar two-meson exchange three-body potential. The presently reported results agree well with the ones obtained by other techniques, demonstrating the advantage of an approach in which the formalism is much more simplified and easy to manage for direct computation.

Bound-state calculations of the three-dimensional Yakubovsky equations with the inclusion of three-body forces

The four-body Yakubovsky equations in a three-dimensional approach with the inclusion of the three-body forces are proposed. The four-body bound state with two- and three-body interactions is formulated in the three-dimensional approach for identical particles as a function of vector Jacobi momenta, specifically, the magnitudes of the momenta and the angles between them. The modified three-dimensional Yakubovsky integral equations are successfully solved with the scalar two-meson exchange three-body force, where the Malfliet-Tjon-type two-body force is implemented. The three-body force effects on the energy eigenvalue and the four-body wave function, as well as accuracy of our numerical calculations are presented.

Dependence of the nuclear equation of state on two-body and three-body forces

The nuclear equation of state is calculated within the Bethe–Brueckner–Goldstone expansion with the inclusion of three-body forces. Two different realistic and accurate two-body forces are considered, the CD Bonn and the Argonne v18, which give quite different EOS. Surprisingly, the inclusion of the same three-body forces in the calculation produces two EOS's which are close to each other not only around saturation but also up to density as large as five times the saturation one. The results suggest that the nuclear EOS is, to some extent, unique within the microscopic many-body theory with accurate realistic two-body forces.

Searh for three-body force effects in elastic proton-deuteron scattering

The effect of three-body forces (TBF) has been investigated in elastic proton-deuteron scattering. Measurements of cross sections and analyzing powers have resulted in very precise data over a large range of scattering angles for incident proton energies of 108, 120, 135, 150, 170 and 190 MeV. The results have been compared with the state-of-the-art Faddeev calculations including various two and three-body forces and very recent calculations in the framework of chiral-pertubation theory. Even though the inclusion of three-body forces in the calculation reduces the discrepancy between the calculations and the data, there remains large differences, especially at backward angles, which increases as a function of increasing beam energy.

Intermolecular potential of carbon dioxide dimer from symmetry-adapted perturbation theory

A four-dimensional intermolecular potential energy surface for the carbon dioxide dimer has been computed using the many-body symmetry-adapted perturbation theory (SAPT) and a large 5s3p2d1f basis set including bond functions. The SAPT level applied is approximately equivalent to the supermolecular many-body perturbation theory at the second-order level. An accurate fit to the computed data has been obtained in a form of an angular expansion incorporating the asymptotic coefficients computed ab initio at the level consistent with the applied SAPT theory. A simpler site-site fit has also been developed to facilitate the use of the potential in molecular dynamics and Monte Carlo simulations. The quality of the new potential has been tested by computing the values of the second virial coefficient which agree very well with the experimental data over a wide range of temperatures. Our potential energy surface turns out to be substantially deeper than previous ab initio potentials. The minimum of −484 cm−1 has been found for the slipped parallel geometry at the intermolecular separation R=3.54 Å and a saddle point at −412 cm−1 for the T-shaped configuration and R=4.14 Å. Three minima and two first-order saddle points have been located on the pairwise-additive potential energy surface of the CO2 trimer. The nonplanar structure of C2 symmetry has been found to be 48.8 cm−1 more stable than the cyclic planar form of C3h symmetry, in disagreement with experimental observation. It is suggested that the relative stability of the two isomers cannot be reliably determined by pairwise-additive potential and inclusion of three-body forces is necessary for this purpose.

Density-functional theory of three-body interactions in liquids and glasses

The integral equation for the correlation functions of liquids and glasses is derived in the presence of a weak three-body potential. The inclusion of three-body forces is essential to describe the covalent interactions. The formulation is based on a generalization of the density-functional theory in which the free energy is regarded as a functional not only of the density but also of the two-body distribution function. In the first-order calculation, the structure of the two-body theory is not modified by virtue of the variational property; the three-body effects are fully taken into account through renormalization of the pair potential. An approximate expression for the three-body correction which is amenable to the numerical computations is obtained.

A comment on the paper by Fielek on 'The inclusion of three-body forces and the condition of equilibrium into the models of Krebs and Cheveau'

For original work see ibid., vol.8, p.577 (1978). Several mathematical and physical errors contained in the model of Fielek should be rectified before it is applied to actual lattice dynamical studies.

A reply to the comments of Shukla on 'The inclusion of three-body forces and the condition of equilibrium into the models of Krebs and Cheveau'

For comments see ibid., vol.8, p.L289 (1978) and for original work see ibid., vol.8, p.577 (1978). This reply shows, by explicitly explaining the physics involved, that the errors referred to in Shukla's letter are not mistakes. Hence, the referred to paper is correct as published.