Three-body Forces Research Articles

Spin-polarized neutron-rich matter at different orders of chiral effective field theory

Spin-polarized neutron matter is studied using chiral two- and three-body forces. We focus, in particular, on predictions of the energy per particle in ferromagnetic neutron matter at different orders of chiral effective field theory and for different choices of the resolution scale. We discuss the convergence pattern of the predictions and their cutoff dependence. We explore to which extent fully polarized neutron matter behaves (nearly) like a free Fermi gas. We also consider the more general case of partial polarization in neutron matter as well as the presence of a small proton fraction. In other words, in our calculations, we vary both spin and isospin asymmetries. Confirming the findings of other microscopic calculations performed with different approaches, we report no evidence for a transition to a polarized phase of neutron matter5

Unified equation of state for neutron stars on a microscopic basis

We derive a new equation of state (EoS) for neutron stars (NS) from the outer crust to the core based on modern microscopic Brueckner-Hartree-Fock (BHF) calculations using the Argonne $v_{18}$ potential plus three-body forces computed with the Urbana model. To deal with the inhomogeneous structures of matter in the NS crust, we use the recent Barcelona-Catania-Paris-Madrid (BCPM) nuclear energy density functional that is directly based on the same microscopic BHF calculations, and which is able to reproduce the ground-state properties of nuclei along the periodic table. The EoS of the outer crust requires the masses of neutron-rich nuclei, which are obtained through Hartree-Fock-Bogoliubov calculations with the BCPM functional when they are unknown experimentally. To compute the inner crust, Thomas-Fermi calculations in Wigner-Seitz cells are performed with the same functional. Existence of nuclear pasta is predicted in a range of average baryon densities between $\simeq$0.067 fm$^{-3}$ and $\simeq$0.0825 fm$^{-3}$, where the transition to the core takes place. The NS core is computed from the nuclear EoS of the BHF calculation assuming non-exotic constituents (core of $npe\mu$ matter). In each region of the star, we discuss the comparison of the new EoS with previous EoSes for the complete NS structure, in particular, with the Lattimer-Swesty EoS and with the Shen et al. EoS widely used in astrophysical calculations. The new microscopically derived EoS fulfills at the same time a NS maximum mass of 2~$M_\odot$ with a radius of 10 km, and a 1.5~$M_\odot$ NS with a radius of 11.7 km.

Dipole response inPb208within a self-consistent multiphonon approach

Background: The electric dipole strength detected around the particle threshold and commonly associated to the pygmy dipole resonance offers a unique information on neutron skin and symmetry energy, and is of astrophysical interest. The nature of such a resonance is still under debate. Purpose: We intend to describe the giant and pygmy resonances in 208 Pb by enhancing their fragmentation with respect to the random-phase approximation. Method: We adopt the equation of motion phonon method to perform a fully self-consistent calculation in a space spanned by one-phonon and two-phonon basis states using an optimized chiral two-body potential. A phenomenological density dependent term, derived from a contact three-body force, is added in order to get single-particle spectra more realistic than the ones obtained by using the chiral potential only. The calculation takes into full account the Pauli principle and is free of spurious center of mass admixtures. Results: We obtain a fair description of the giant resonance and obtain a dense low-lying spectrum in qualitative agreement with the experimental data. The transition densities as well as the phonon and particle-hole composi- tion of the most strongly excited states support the pygmy nature of the low-lying resonance. Finally, we obtain realistic values for the dipole polarizability and the neutron skin radius. Conclusions: The results emphasize the role of the two-phonon states in enhancing the fragmentation of the strength in the giant resonance region and at low energy, consistently with experiments. For a more detailed agreement with the data, the calculation suggests the inclusion of the three-phonon states as well as a fine tuning of the single-particle spectrum to be obtained by a refinement of the nuclear potential.

Equation of State and Symmetry Energy at High Densities for Zero and Finite Temperatures

The equation of state of isospin asymmetric nuclear matter within the framework of the Brueckner theory is used to calculate the energy per particle for nuclear and neutron matter. Pressure and symmetry energy are also presented. Here we extended our work to include Skyrme-like zero-range density-dependent two-body forces, which could mimic three-body forces. A three-body forces are shown to be necessary for reproducing the empirical saturation properties of symmetric nuclear matter. We also studied the effect of extending the calculation to finite temperatures.

Revisiting the monopole components of effective interactions for the shell model

In this paper, we revisit the monopole components of effective interactions for the shell model. Without going through specific nuclei or shell gaps, universal roles of central, tensor, and spin–orbit forces can be proved, reflecting the intrinsic features of shell model effective interactions. For monopole matrix elements, even and odd channels of central force often have a canceling effect. However, for the contributions to the shell evolution, its even and odd channels could have both positive or negative contributions, enhancing the role of central force on the shell structure. Tensor force is generally weaker than central force. However, for the effect on shell evolutions, tensor force can dominate or play a competitive role. A different systematics has been discovered between T = 1 and 0 channels. For example, tensor force, well established in the T = 0 channel, becomes uncertain in the T = 1 channel. We calculate the properties of neutron-rich oxygen and calcium isotopes in order to study T = 1 channel interactions further. It is learned that the main improvements of empirical interactions are traced to the central force. For non-central forces, antisymmetric spin–orbit (ALS) force, originated from many-body perturbations or three-body force, could also play an explicit role. T = 1 tensor forces are less constrained so their effect can differ in different empirical interactions. The influence of tensor force may sometimes be canceled by many-body effects. For T = 0 channels of effective interactions, which is the main source of neutron–proton correlations, central and tensor forces are the leading components. For T = 1 channels, which can act between like-particles, the request for many-body correlations could be more demanding, so that the monopole anomaly of the T = 1 channel might be more serious.

Effects of repulsive three-body force in 12C + 12C scattering at 100A MeV

The angular distribution of 12C + 12C scattering at an incident energy of 100A MeV has been measured. The elastic and inelastic scatterings in 12C to the excitation energies of up to ∼45 MeV were measured simultaneously for the first time with the high-resolution Grand Raiden spectrometer at the Research Center for Nuclear Physics (RCNP). The angular distributions of the elastic scattering to the ground state (01+) and inelastic scattering to the 4.44 MeV (21+) excited state were precisely obtained in the angular range of 1.0°–7.5° with a step of 0.1°. Additionally, the angular distribution was obtained for the sum of the cross sections for excitation energies above the 4.44 MeV state up to 11 MeV, which includes the 7.65 MeV (02+), 9.64 MeV (31−), and 10.30 MeV (22+) states, in addition to probably the simultaneous excitation of the 4.44 MeV state in the projectile and the target nuclei. Those combined data provide a means to study the effects of channel coupling on the elastic cross section. The observed angular distributions are compared with theoretical calculations based on three double-folding models with complex G-matrix interactions, the CEG07b, MPa, and ESC models. The importance of three-body repulsive forces included in the CEG07b and MPa models will be discussed.

Mass Spectrum of Baryons in a Relativistic Hypercentral Constituent Quark Model

We have investigated the light and strange baryons as a relativistic three-body bound system. Inspired by lattice QCD calculations, we treated the baryons as a spin-independent three-quark system. For this purpose we introduced a spin independent relativistic description for the SU (6) invariant part of the spectrum by presenting the analytical solution of the three-particle Klein-Gordon equation. We carried out calculations within the constituent quark model based on a hypercentral approach, which takes into account three-body force effects and standard two-body potential contributions. Herewith the average energy values of the nonstrange and strange resonances are reproduced. Then we used the generalized Gursey-Radicati mass formula as a SU (6) breaking interaction to describe the splittings within the SU (6)-multiplets and the hyperfine structure of the baryon. The considered SU (6)-invariant potential is the well-known ＂Coulomb-plus-linear＂ potential. The resulting description of the baryon spectrum is fairly good and comparable with the experimental spectrum.

136Sn and three-body forces

New experimental data on 2+ energies of 136,138Sn confirm the trend of lower 2+ excitation energies of even–even tin isotopes with N>82 compared to those with N<82. However, none of the theoretical predictions using both realistic and empirical interactions can reproduce experimental data on excitation energies as well as the transition probabilities ( $B(E2; 6^{+} \rightarrow 4^{+}$ )) of these nuclei, simultaneously, apart from the ones whose matrix elements have been changed empirically to produce mixed seniority states by weakening the pairing. We have shown that the experimental result also shows good agreement with the theory in which three-body forces have been included in a realistic interaction. The new theoretical results on transition probabilities are discussed to identify the experimental quantities which will clearly distinguish between different views.

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.

Proton-three-nucleon nuclei elastic scattering: three-body force effect

Elastic scattering differential cross sections at 0.6 GeV for p-3H and at 0.6, 0.715 and 1 GeV for p-3He are calculated in the framework of the multiple-scattering theory of Glauber. A simple approach for the three-body force effect is used. We have found that the three-body force effect appears in the range of momentum transfer squared $$ q^{2} \ge 0.2\; ( {\text{GeV/c)}}^{ 2} $$ and plays an important role to obtain a good fit with the experimental data at the first minimum region.

Effects of three-body forces on the maximum mass of neutron stars in the lowest-order constrained variational formalism

The equation of state of neutron star matter is calculated within the lowest order constrained variational (LOCV) method using different two-body interactions and an Urbana type three-body force. The core of the neutron star is modelled by $\ensuremath{\beta}$-stable matter. The corresponding neutron star mass-radius relations are presented and the effect of using a three-body interaction on the maximum gravitational mass of the star is also discussed. It is shown that including the three-body force generally increases the maximum gravitational mass.

Relativistic three-body quark model of light baryons based on hypercentral approach

In this paper, we have treated the light baryons as a relativistic three-body bound system. Inspired by lattice QCD calculations, we treated baryons as a spin-independent three-quark system within a relativistic three-quark model based on the three-particle Klein–Gordon equation. We presented the analytical solution of three-body Klein–Gordon equation with employing the constituent quark model based on a hypercentral approach through which two- and three-body forces are taken into account. Herewith the average energy values of the up, down and strange quarks containing multiplets are reproduced. To describe the hyperfine structure of the baryon, the splittings within the SU(6)-multiplets are produced by the generalized Gürsey Radicati mass formula. The considered SU(6)-invariant potential is popular "Coulomb-plus-linear" potential and the strange and non-strange baryons spectra are in general well reproduced.

Convergence of the hyperspherical-harmonics expansion with increasing number of particles for bosonic systems. II. Inclusion of the three-body force

This paper presents a numerical convergence study of a hyperspherical-harmonics expansion for binding energies of a system of $4\ensuremath{\le}N\ensuremath{\le}728$ helium atoms using a phenomenological soft attractive two-body He-He potential and a repulsive three-body force aimed at compensating for the absence of the two-body repulsive core. Earlier calculations with such a potential have shown an improved convergence when $N$ increases from four to six. The present study reveals that the improved convergence occurs only for a limited range of $N$ determined by the range of the three-body repulsion. For a soft repulsive three-body force, the convergence is fast for $N\ensuremath{\le}20$, while for a short-range three-body repulsion it deteriorates at $N\ensuremath{\ge}10$. The reasons for this deterioration are discussed. The range of the three-body force also determines the binding energy behavior with $N$, and it is also responsible for binding the excited states. The long-range force binds all first excited ${0}^{+}$ states but strongly underbinds the systems of $N$ helium atoms at large $N$. The short-range force does not bind the first ${0}^{+}$ states for $A\ensuremath{\le}7$ but gives better predictions of binding energies as compared to the calculations of other authors though overestimating them. Some options to improve both the description of the binding energies and the convergence of the hyperspherical-harmonics expansion using phenomenological forces are discussed. It is pointed out that a fast convergence is very much needed for the reliable predictions of states with nonzero angular momentum, examples of which are also given.

Three-body critical Casimir forces

Within mean-field theory we calculate universal scaling functions associated with critical Casimir forces for a system consisting of three parallel cylindrical colloids immersed in a near-critical binary liquid mixture. For several geometrical arrangements and boundary conditions at the surfaces of the colloids we study the force between two colloidal particles in the direction normal to their axes, analyzing the influence of the presence of a third particle on that force. Upon changing temperature or the relative positions of the particles we observe interesting features such as a change of sign of this force caused by the presence of the third particle. We determine the three-body component of the forces acting on one of the colloids by subtracting the pairwise forces from the total force. The three-body contribution to the total critical Casimir force turns out to be more pronounced for small surface-to-surface distances between the colloids as well as for temperatures close to criticality. Moreover, we compare our results with similar ones for other physical systems such as three atoms interacting via van der Waals forces.

Hyperons in neutron stars

Using the Dirac–Brueckner–Hartree–Fock approach, the properties of neutron-star matter including hyperons are investigated. In the calculation, we consider both time and space components of the vector self-energies of baryons as well as the scalar ones. Furthermore, the effect of negative-energy states of baryons is partly taken into account. We obtain the maximum neutron-star mass of 2.08M⊙, which is consistent with the recently observed, massive neutron stars. We discuss a universal, repulsive three-body force for hyperons in matter.

Hyperon puzzle: hints from quantum Monte Carlo calculations.

The onset of hyperons in the core of neutron stars and the consequent softening of the equation of state have been questioned for a long time. Controversial theoretical predictions and recent astrophysical observations of neutron stars are the grounds for the so-called hyperon puzzle. We calculate the equation of state and the neutron star mass-radius relation of an infinite systems of neutrons and Λ particles by using the auxiliary field diffusion MonteCarlo algorithm. We find that the three-body hyperon-nucleon interaction plays a fundamental role in the softening of the equation of state and for the consequent reduction of the predicted maximum mass. We have considered two different models of three-body force that successfully describe the binding energy of medium mass hypernuclei. Our results indicate that they give dramatically different results on the maximum mass of neutron stars, not necessarily incompatible with the recent observation of very massive neutron stars. We conclude that stronger constraints on the hyperon-neutron force are necessary in order to properly assess the role of hyperons in neutron stars.

Shell-model study of spin modes in nuclei and nuclear forces

Spin-dependent modes in nuclei are studied by shell-model method with the use of new shell-model Hamiltonians which properly take into account important roles of tensor interactions. New Hamiltonians can describe spin degrees of freedom in nuclei remarkably well. Nuclear weak processes at stellar environments are investigated based on these successes. New neutrino-nucleus reaction cross sections on 12C are applied to light-element synthesis in supernova explosions. The production rate for 11B/7Li is pointed out to be useful to determine v-oscillation parameters, in particular, v-mass hierarchy. New e-capture rates in Ni isotopes are obtained and implications for element synthesis are discussed. The monopole-based universal interaction is applied to study structure of p-sd shell nuclei and 40Ar as well as v-induced reactions on 40Ar. Repulsive corrections in the isospin T=1 monopoles are shown to be important for proper shell evolutions in neutron-rich carbon isotopes. The repulsive correction is pointed out to be due to three-body forces, in particular, the Fujita-Miyazawa force. Roles of the three-body forces on the shell evolution of neutron-rich calcium isotopes, the closed- shell nature of 48 Ca and M1 transition in 48 Ca are studied on top of the two-body G-matrix obtained by including core-polarization effects in larger spaces (≤24ℏω). Effects of the inclusion of g9/2-shell are also discussed.

Self-consistent studies of the dipole response in neutron rich nuclei using realistic potentials

The dipole response in neutron rich nuclei is investigated within self-consistent approaches which make direct use of a nucleon-nucleon optimized chiral potential complemented with a density dependent term simulating a three-body force. Hartree-Fock-Bogoliubov plus Tamm-Dancoff and random-phase approximations show that such a potential improves the description of the dipole modes with respect to other realistic interactions. The inclusion of the two-phonon states within an equation of motion method induces a pronounced fragmentation of both giant and pygmy resonances in agreement with recent experiments.

Simulation of 200–400 MeV/u 12C+12C elastic scattering on SHARAQ spectrometer**Supported by National Nature Science Foundation of China (11035007, 11235002, 11175011)

In order to further obtain the information of three-body force (TBF) from 200–400 MeV/u 12C+12C elastic scattering, we plan to perform this experiment on a SHARAQ spectrometer. Based on the experimental condition of the Radioactive Ion Beam Factory (RIBF)-SHARAQ facility, a simulation is given to find a compromise between the better energy and angular resolutions, and higher yield by optimizing the target thickness, beam transport mode, beam intensity and angular step. From the simulation, we found that the beam quality mainly limits the improvements of energy and angular resolutions. A beam tracking system as well as a lateral and angular dispersion-matching technique are adopted to reduce the influence of beam quality. According to the two angular settings of SHARAQ as well as the expected cross sections on the basis of the theoretical model, the energy and angular resolutions, and statistical accuracy are estimated.

Effect of three-body forces on response functions in infinite neutron matter

We study the impact of three-body forces on the response functions of cold neutron matter. These response functions are determined in the random phase approximation (RPA) from a residual interaction expressed in terms of Landau parameters. Special attention is paid to the non-central part, including all terms allowed by the relevant symmetries. Using Landau parameters derived from realistic nuclear two- and three-body forces grounded in chiral effective field theory, we find that the three-body term has a strong impact on the excited states of the system and in the static and long-wavelength limit of the response functions for which a new exact formula is established.