Symmetry Potential Research Articles

Extracting the nuclear symmetry potential and energy from neutron–nucleus scattering data

According to the Hugenholtz–Van Hove theorem, the symmetry energy and its density slope can be decomposed uniquely in terms of the single-nucleon potential in asymmetric nuclear matter which, at normal density, can be constrained by the nucleon optical model potential extracted from analyzing the nucleon–nucleus scattering data. To more accurately extract information on the symmetry energy and its density slope at normal density from neutron–nucleus scattering data, going beyond the well-known Lane potential, we include consistently the second-order terms in isospin asymmetry in both the optical model potential and the symmetry energy decomposition. We find that the strength of the second-order symmetry potential Usym,2 in asymmetric nuclear matter is significant compared to the first-order one Usym,1, especially at high nucleon momentum. While the Usym,1 at normal density decreases with nucleon momentum, the Usym,2 is found to have the opposite behavior. Moreover, we discuss effects of the Usym,1 and Usym,2 on determining the density dependence of the symmetry energy, and we find that the available neutron–nucleus scattering data favor a softer density dependence of the symmetry energy at normal density with a negligible contribution from the Usym,2.

THE PROPERTIES OF NUCLEAR MATTER

The microscopic and bulk properties of nuclear matter at zero and finite temperatures are studied in the frame of the Brueckner theory. The results for the symmetry energy are also obtained using different potentials. The calculations are based on realistic nucleon-nucleon interactions which reproduce the nucleon-nucleon phase shifts. These microscopic approaches are supplemented by a density-dependent contact interaction to achieve the empirical saturation property of symmetric nuclear matter. Special attention is paid to behavior of the effective mass in asymmetric nuclear matter. The nuclear symmetry potential at fixed nuclear density is also calculated and its value decreases with increasing the nucleon energy. The hot properties of nuclear matter are also calculated using T2-approximation at low temperatures. Good agreement is obtained in comparison with previous works around the saturation point.

The effect of symmetry potential on the balance energy of light particles emitted from mass symmetric heavy-ion collisions with isotopes, isobars and isotones

Using the Ultrarelativistic Quantum Molecular Dynamics (UrQMD) model, the balance energies of free neutrons, free protons and Z=1 particles (including free protons, deuterons and tritons) from mass symmetric heavy-ion collisions with isotopes, isobars and isotones are studied. The influence of nuclear symmetry potential energy on the balance energy is emphasized. It is found that the balance energy of free neutrons is sensitive to the nuclear symmetry energy, while that of free protons is not. Particularly, the initial neutron/proton ratio dependence of the balance energy of free neutrons from Sn isotopes can be taken as a useful probe to constrain the stiffness of the nuclear symmetry energy.

Effects of neutron richness on the behavior of nuclear systems at intermediate energies

We discuss results concerning the behavior of hot nuclear sources formed in 40,48Ca+40,48Ca reactions at 25 MeV/nucleon. A correlation between the neutron-to-proton ratio of the total systems and heavy residue production is found. This correlation underlines the strong role played by the neutron-to-proton ratio degree of freedom on the behavior of hot nuclei at intermediate energies. Comparisons of our data with Constrained Molecular Dynamics calculations confirm that a moderately stiff behavior of the symmetry potential must be used to reproduce experimental data. These findings open the way for future investigations with exotic beams at radioactive ion beam facilities.Received 3 May 2012DOI:https://doi.org/10.1103/PhysRevC.85.064609©2012 American Physical Society

Symmetry Potential and Effective Mass with Consistent Three-Body Force

The momentum and isospin dependence of symmetry potential and effective mass in isospin asymmetric nuclear matter is investigated in the framework of the Brueckner—Hartree—Fock approach by adopting the realistic Bonn B two-body interaction in combination with a consistent microscopic three-body force. It is demonstrated that symmetry potentials at the normal density for different asymmetry parameters are in good agreement with the empirical value constrained by the experimental data, and the neutron and proton effective masses show strong isospin splitting with m*n > m*p on the whole range of asymmetry parameters. In the end, the density dependence of the Landau mass is displayed.

Effect of the momentum dependence of nuclear symmetry potential on the transverse and elliptic flows

In the framework of the isospin-dependent Boltzmann-Uehling-Uhlenbeck transport model, effect of the momentum dependence of nuclear symmetry potential on nuclear transverse and elliptic flows in the neutron-rich reaction $^{132}$Sn+$^{124}$Sn at a beam energy of 400 MeV/nucleon is studied. We find that the momentum dependence of nuclear symmetry potential affects the rapidity distribution of the free neutron to proton ratio, the neutron and the proton transverse flows as a function of rapidity. The momentum dependence of nuclear symmetry potential affects the neutron-proton differential transverse flow more evidently than the difference of neutron and proton transverse flows as well as the difference of proton and neutron elliptic flows. It is thus better to probe the symmetry energy by using the difference of neutron and proton flows since the momentum dependence of nuclear symmetry potential is still an open question. And it is better to probe the momentum dependence of nuclear symmetry potential by using the neutron-proton differential transverse flow and the rapidity distribution of the free neutron to proton ratio.

Single-nucleon potential decomposition of the nuclear symmetry energy

The nuclear symmetry energy ${E}_{\mathrm{sym}}(\ensuremath{\rho})$ and its density slope $L(\ensuremath{\rho})$ can be decomposed analytically in terms of the single-nucleon potential in isospin asymmetric nuclear matter. Using three popular nuclear effective interaction models which have been extensively used in nuclear structure and reaction studies, namely, the isospin and momentum-dependent MDI interaction model, the Skyrme-Hartree-Fock approach, and the Gogny-Hartree-Fock approach, we analyze the contribution of different terms in the single-nucleon potential to ${E}_{\mathrm{sym}}(\ensuremath{\rho})$ and $L(\ensuremath{\rho})$. Our results show that the observed different density behaviors of ${E}_{\mathrm{sym}}(\ensuremath{\rho})$ for different interactions are essentially due to the variation of the symmetry potential ${U}_{\mathrm{sym},1}(\ensuremath{\rho},k)$. Furthermore, we find that the contribution of the second-order symmetry potential ${U}_{\mathrm{sym},2}(\ensuremath{\rho},k)$ to $L(\ensuremath{\rho})$ generally cannot be neglected. Moreover, our results demonstrate that the magnitude of ${U}_{\mathrm{sym},2}(\ensuremath{\rho},k)$ is generally comparable with that of ${U}_{\mathrm{sym},1}(\ensuremath{\rho},k)$, indicating that the second-order symmetry potential ${U}_{\mathrm{sym},2}(\ensuremath{\rho},k)$ may have significant corrections to the widely used Lane approximation for the single-nucleon potential in extremely neutron rich or proton rich nuclear matter.

Influence of in-mediumNNcross sections, symmetry potential, and impact parameter on isospin observables

We explore the influence of in-medium nucleon-nucleon cross section, symmetry potential and impact parameter on isospin sensitive observables in intermediate-energy heavy-ion collisions with the ImQMD05 code, a modified version of Quantum Molecular Dynamics model. At incident velocities above the Fermi velocity, we find that the density dependence of symmetry potential plays a more important role on the double neutron to proton ratio $DR(n/p)$ and the isospin transport ratio $R_i$ than the in-medium nucleon-nucleon cross sections, provided that the latter are constrained to a fixed total NN collision rate. We also explore both $DR(n/p)$ and $R_i$ as a function of the impact parameter. Since the copious production of intermediate mass fragments is a distinguishing feature of intermediate-energy heavy-ion collisions, we examine the isospin transport ratios constructed from different groups of fragments. We find that the values of the isospin transport ratios for projectile rapidity fragments with $Z\ge20$ are greater than those constructed from the entire projectile rapidity source. We believe experimental investigations of this phenomenon can be performed. These may provide significant tests of fragmentation time scales predicted by ImQMD calculations.

Effective mass splitting of neutron and proton and isospin emission in heavy-ion collisions

Within the framework of an isospin and momentum dependent transport model, the emissions of isospin particles (nucleons and light clusters) squeezed out in heavy-ion collisions are investigated as probes of the poorly known symmetry energy at high baryon density. Two different mass splittings of neutrons and protons in nuclear medium as mn⁎>mp⁎ and mn⁎<mp⁎ are used in the model and their influence on the isospin emission in heavy-ion collisions is discussed thoroughly. The competition between the stiffness and the momentum dependence of the symmetry potential on reaction dynamics are compared and systematically analyzed. It is found that the difference of the neutron and proton directed flows and the transverse momentum distribution of the neutron/proton ratio are sensitive to the stiffness of the symmetry energy, which cannot be changed with the controversial effective mass splitting. The elliptic flows of free nucleons at high transverse momentum within mid-rapidity emission are to be promising observables as distinguishing the nucleon effective mass splitting.

Influence of the symmetry energy on the balance energy of the directed flow

The influence of the density-dependent symmetry energy on the balance energy (Ebal) of directed flow from heavy ion collisions (HICs) at incident energies covered by INDRA and MSU experiments is studied, using the updated version of the ultrarelativistic quantum molecular dynamics (UrQMD) model, especially adapted to low-energy heavy ion collisions (HICs). Four mass-symmetric reactions with total mass numbers between 192 and 394 are chosen for investigating the influence of the symmetry energy on the system-mass dependence of Ebal. The results show that the uncertainty in the density dependence of the symmetry potential causes changes of Ebal of the order of several MeV, depending on the type of particle considered. The Ebal of neutrons from HICs is particularly sensitive to the density dependence of the symmetry potential energy, while the system-mass dependence of Ebal of Z = 1 particles is not.

Nuclear in-medium effects and collective flows in heavy-ion collisions at intermediate energies

A recent updated version of the Lanzhou quantum molecular dynamics is reviewed, in which the momentum dependence of symmetry potential and the effective mass splitting of proton and neutron in nuclear medium are included in the model. The in-medium nucleon-nucleon (NN) elastic cross sections are evaluated with the scaling approach according to the effective mass and their influence on collective flows in heavy-ion collisions are discussed. The inelastic cross sections, in particular for the process of the channel N$\Delta\rightarrow$NN, are parameterized in accordance with the available experimental data. It is found that the in-medium cross sections play a significant role on isospin emissions and result in a flat distribution for transverse flows and elliptic flows of free nucleons comparing with the in-vacuum ones. The rapidity distribution of the difference between neutron and proton transverse flows is sensitive to the stiffness of nuclear symmetry energy as a promising observable, which can not be influenced by the in-medium effect and collision centrality. Furthermore, the elliptic flow of free energetic nucleons in the case of the mass splitting of $m_{n}^{\ast}>m_{p}^{\ast}$ is also related to the symmetry energy. However, the pion flows weakly depend on the symmetry energy and the mass splitting.

Single-Particle Properties of Isospin Asymmetric Nuclear Matter

The single-particle potentials in isospin asymmetric nuclear matter is investigated in the framework of the Brueckner theory by adopting the realistic Bonn B two-body interaction in combination with a consistent microscopic three-body force. The rearrangement contribution induced by the ground state correlations to single-nucleon potentials is calculated in terms of the hole-line expansion of the mass operator. With the modification of this rearrangement term the symmetry potential is discussed as a function of momentum for several isospin asymmetries.

Isospin-dependent nucleon-nucleon cross section and the collective flow of nucleons in heavy-ion collision at intermediate energy

In this paper, the influences of the different forms of nucleon-nucleon (NN) cross sections on the rapidity distributions and the in-plane flows of nucleons in reactions 60Ca+60Ca and 60Ni+60Ni are studied by using IBUU04 transport model with the moment-dependent symmetry potential. The calculation results indicate that (1) the discrepancies of the rapidity distributions and in-plane flows corresponding to different forms of NN cross sections are large, (2) the sensitivity of the in-plane flow to the NN cross section still holds when the new form of moment-dependent symmetry energy is adopted, and (3) the influence of the N/P ratio of reaction systems on the transverse flow calculated with a certain NN cross section is more obvious when the incident energy is near the balance energy.

Momentum dependence of the symmetry potential and its influence on nuclear reactions

A Skyrme-type momentum-dependent nucleon-nucleon force distinguishing isospin effect is parameterized and further implemented in the Lanzhou Quantum Molecular Dynamics (LQMD) model for the first time, which leads to a splitting of nucleon effective mass in nuclear matter. Based on the isospin- and momentum-dependent transport model, we investigate the influence of momentum-dependent symmetry potential on several isospin-sensitive observables in heavy-ion collisions. It is found that symmetry potentials with and without the momentum dependence but corresponding to the same density dependence of the symmetry energy result in different distributions of the observables. The mid-rapidity neutron/proton ratios at high transverse momenta and the excitation functions of the total $\pi^{-}/\pi^{+}$ and $K^{0}/K^{+}$ yields are particularly sensitive to the momentum dependence of the symmetry potential.

Effect of the momentum dependence of the nuclear symmetry potential on theπ−/π+ratio in heavy-ion collisions

In the framework of the isospin-dependent Boltzmann-Uehling-Uhlenbeck transport model, the effect of the momentum dependence of the nuclear symmetry potential on the ${\ensuremath{\pi}}^{\ensuremath{-}}/{\ensuremath{\pi}}^{+}$ ratio in the neutron-rich reaction $^{132}\mathrm{Sn}$ $+$ $^{124}\mathrm{Sn}$ at a beam energy of $400$ MeV/nucleon is studied. We find that the momentum dependence of the nuclear symmetry potential affects the compressed density of colliding nuclei, the numbers of produced ${\ensuremath{\pi}}^{\ensuremath{-}}$ and ${\ensuremath{\pi}}^{+}$, as well as the value of the ${\ensuremath{\pi}}^{\ensuremath{-}}/{\ensuremath{\pi}}^{+}$ ratio. The momentum-dependent nuclear symmetry potential increases the compressed density of colliding nuclei, the number of produced resonances $\ensuremath{\Delta}(1232)$, ${N}^{*}(1440)$, ${\ensuremath{\pi}}^{\ensuremath{-}}$, and ${\ensuremath{\pi}}^{+}$, as well as the value of the ${\ensuremath{\pi}}^{\ensuremath{-}}/{\ensuremath{\pi}}^{+}$ ratio.

N/Z effects on evaporation residue emission near fragmentation threshold

We will discuss results concerning the fate of hot nuclear systems populated in 40,48 Ca+ 40,48 Ca reactions at 25 MeV/nucleon. Due to the complexity of events, we used the Chimera 4π device as detection system. By selecting central events of reaction, we found that the interplay between binary-like and evaporation residue emissions is strongly sensitive on the N/Z of the entrance channels. In particular, evaporation residue emission increases at increasing the neutron content of colliding system. By comparing or data with CoMD-II model calculations, we can extract information about the density depend part of Symmetry Potential in the Nuclear Equation of State at near-saturation densities.

Slow-roll inflation with the Gauss-Bonnet and Chern-Simons corrections

We study slow-roll inflation with the Gauss-Bonnet and Chern-Simons corrections. We obtain general formulas for the observables: spectral indices, tensor-to-scalar ratio and circular polarization of gravitational waves. The Gauss-Bonnet term violates the consistency relation r = −8nT. Particularly, blue spectrum nT > 0 and scale invariant spectrum |8nT|/r << 1 of tensor modes are possible. These cases require the Gauss-Bonnet coupling function of ξ,ϕ ∼ 108/MPl. We use examples to show new-inflation-type potential with 10MPl symmetry breaking scale and potential with flat region in ϕ≳10MPl lead to observationally consistent blue and scale invariant spectra, respectively. Hence, these interesting cases can actually be realized. The Chern-Simons term produce circularly polarized tensor modes. We show an observation of these signals supports existence of the Chern-Simons coupling function of ω,ϕ ∼ 108/MPl. Thus, with future observations, we can fix or constrain the value of these coupling functions, at the CMB scale.

Finding a Probe for Extracting Information on the Momentum Dependent-Interaction in Heavy Ion Collisions

The effect of momentum-dependent interaction on the kinetic energy spectrum of the neutron-proton ratio ⟨(n/p)gas⟩b (Ek) for 64Zn + 64Zn is studied. It is found that ⟨(n/p)gas⟩b (Ek) sensitively depends on the momentum-dependent interaction and weakly on the in-medium nucleon-nucleon cross section and symmetry potential. Therefore ⟨(n/p)gas⟩b(Ek) is a possible probe for extracting information on the momentum-dependent interaction in heavy ion collisions.

Can the nuclear symmetry potential at supra-saturation densities be negative?

In the framework of an isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model, for the central $^{197}\mathrm{Au}+^{197}\mathrm{Au}$ reaction at an incident beam energy of $400$ MeV/nucleon, the effect of nuclear symmetry potential at supra-saturation densities on the preequilibrium clusters emission is studied. It is found that for the positive symmetry potential at supra-saturation densities the neutron-to-proton ratio of lighter clusters with mass number $A\ensuremath{\leqslant}3$ [$(n/p){}_{A\ensuremath{\leqslant}3}$] is larger than that of the heavier clusters with mass number $A&gt;3$ [$(n/p){}_{A&gt;3}$], whereas for the negative symmetry potential at supra-saturation densities the $(n/p){}_{A\ensuremath{\leqslant}3}$ is smaller than the $(n/p){}_{A&gt;3}$. This may be considered as a probe of the negative symmetry potential at supra-saturation densities.

Isospin emission and flow at high baryon density: A test of the symmetry potential

High-energy heavy-ion collisions are studied to access nuclear-matter properties at high density. Particular attention is paid to the selection of observables sensitive to the poorly known symmetry energy at high baryon density, of large fundamental interest, even for the astrophysics implications. Using fully consistent transport simulations built on effective theories, we test isospin observables ranging from nucleon and cluster emissions to collective flows (in particular the elliptic, squeeze-out, part). The effects of the competition between stiffness and momentum dependence of the symmetry potential on the reaction dynamics are thoroughly analyzed. In this way we try to shed light on the controversial neutron and proton effective mass splitting at high baryon and isospin densities. New, more exclusive, experiments are suggested.