Symmetry Energy Coefficient Research Articles

Surface and symmetry energies in isoscaling for multifragmentation reactions

The possibilities of modifications for symmetry and surface energy coefficients of nuclear matter at freeze-out density are investigated theoretically by means of isoscaling (isotopic scaling) on the basis of a statistical multifragmentation model. It is seen that while the isoscaling parameters as predicted by Markov chain calculations for the single sources 124Sn and 112Sn are very sensitive to the symmetry term, the surface energy variation slightly affects the isoscaling parameters in the multifragmentation region. The influence of these changes on the mean neutron to proton ratios of light fragments for 2 ⩽ Z ⩽ 10 is also shown. Comparing our results with MSU experimental data, it is confirmed that a significant reduction of the symmetry-term coefficient is found necessary to reproduce the isoscaling parameters of Z ⩽ 8 fragments.

Symmetry coefficients and incompressibility of clusterized supernova matter

The symmetry energy coefficients, incompressibility, and single-particle and isovector potentials of clusterized dilute nuclear matter are calculated at different temperatures employing the S-matrix approach to the evaluation of the equation of state. Calculations have been extended to understand the aforesaid properties of homogeneous and clusterized supernova matter in the subnuclear density region. A comparison of the results in the S-matrix and mean-field approach reveals some subtle differences in the density and temperature region we explore.

Temperature dependence of nuclear matter generalized isovector symmetry energy with Skyrme-type interactions

The temperature dependence of the nuclear matter isovector symmetry energy coefficient (A{sub 0,1}) is investigated in the framework of the generalized nuclear polarizability with Skyrme interactions, as worked out in previous articles [F. L. Braghin, Nucl. Phys. A665, 13 (2000); Phys Rev. C 71, 064303 (2005); Int J. Mod. Phys. E 12, 755 (2003)]. The variation of A{sub 0,1}(T) is very small (of the order of 1 MeV) for temperatures (T) in the range of 0 and 18 MeV. Different behaviors with temperature are found to strongly depend on the Skyrme parametrization, in particular at densities lower than the saturation density {rho}{sub 0}.

Erratum: Nuclear matter symmetry energy from generalized polarizabilities: Dependences on momentum, isospin, density, and temperature [Phys. Rev. C71, 064303 (2005)

Symmetry energy terms from macroscopic mass formulae are investigated as generalized polarizabilities of nuclear matter. Besides the neutron-proton (n-p) symmetry energy the spin dependent symmetry energies and a scalar one are also defined. They depend on the nuclear densities ($\rho$), neutron-proton asymmetry ($b$), temperature ($T$) and exchanged energy and momentum ($q$). Based on a standard expression for the generalized polarizabilities, a differential equation is proposed to constrain the dependence of the symmetry energy on the n-p asymmetry and on the density. Some solutions are discussed. The q-dependence (zero frequence) of the symmetry energy coefficients with Skyrme-type forces is investigated in the four channels of the particle-hole interaction. Spin dependent symmetry energies are also investigated indicating much stronger differences in behavior with $q$ for each Skyrme force than the results for the neutron-proton one.

Isoscaling in statistical fragment emission in an extended compound nucleus model

Based on an extended compound nucleus model, isospin effects in statistical fragment emission from excited nuclear systems are investigated. An experimentally observed scaling behavior of the ratio of isotope yields Yi(N,Z) from two similar emitting sources with different neutron-to-proton ratios is predicted theoretically, i.e., the relationship of Y2/Y1∝exp(αN+βZ) is demonstrated. The symmetry energy coefficient Csym extracted from the simulation results is ∼27 MeV which is consistent with realistic theoretical estimates and recent experimental data. The influence of the surface entropy on the isoscaling behavior is discussed in detail. It is found that although the surface entropy increases the numerical values of isoscaling parameters α and β, it does not affect the isoscaling behavior qualitatively and has only a minor effect on the extracted symmetry energy coefficient.

Nuclear Symmetry Energy Probed by Neutron Skin Thickness of Nuclei

We describe a relation between the symmetry energy coefficients c(sym)(rho) of nuclear matter and a(sym)(A) of finite nuclei that accommodates other correlations of nuclear properties with the low-density behavior of c(sym)(rho). Here, we take advantage of this relation to explore the prospects for constraining c(sym)(rho) of systematic measurements of neutron skin sizes across the mass table, using as example present data from antiprotonic atoms. The found constraints from neutron skins are in harmony with the recent determinations from reactions and giant resonances.

DYNAMICAL AND SEQUENTIAL DECAY EFFECTS ON ISOSCALING AND DENSITY DEPENDENCE OF THE SYMMETRY ENERGY

Isoscaling properties of the primary and final products are studied via isospin dependent quantum molecular dynamics (IQMD) model and the followed sequential decay model GEMINI, respectively. Both primary and final products isoscaling parameter α keeps no significant change for light fragments, but increases with the mass for intermediate and heavy products. The dynamical effect on isoscaling is reflected on the α decreasing a little with the evolution time of the system, and opposite trend for the heavy products. The secondary decay effect on isoscaling is reflected on the increasing of the α value for the final products which experienced secondary decay process. Furthermore the density dependence of the symmetry energy has also been explored for the primary and secondary products, the symmetry energy coefficient can be expressed by the form of C sym (ρ) ~ C0(ρ/ρ0)γ, C0 and γ extracted from the primary products are consistent with the input parameters, but C0 and γ extracted from the final products deviate the input values. In the paper we also suggest that it might be more reasonable to describe the density dependence of the symmetry energy coefficient by the C sym (ρ/ρ0) ≈ C1(ρ/ρ0)γ soft + C2(ρ/ρ0)γstiff with γ soft ≤ 1, γstiff ≥ 1 and C1, C2 constant parameters.

ISOSPIN EFFECT AND ISOSCALING PHENOMENON IN PROJECTILE FRAGMENTATION

We have studied the isospin effect and isoscaling behavior in projectile fragmentation using a modified statistical abrasion-ablation (SAA) model. The relationship between neutron skin thickness (δnp) and neutron abrasion cross section (σ nabr ) has also been investigated. We find the normalized peak differences and reduced isoscaling parameters decrease with (Z proj - Z)/Z proj or the excitation energy per nucleon and have no significant dependence on the size of the reaction systems. The excitation energy dependence of the symmetry energy coefficients are tentatively extracted from α and β. Assuming Fermi-type density distributions for proton and neutron and introducing a parameter to adjust the diffuseness of the neutron density distribution for neutron-rich nucleus, δnp and σ nabr for neutron-rich nucleus were found to have a linear correlation. It is suggested that σ nabr could be used as an observable to extract the neutron skin thickness for neutron-rich nucleus.

Isoscaling, symmetry energy and thermodynamic models

The isoscaling parameter usually denoted by α depends upon both the symmetry energy coefficient and the isotopic contents of the dissociating systems. We compute α in theoretical models: first in a simple mean field model and then in thermodynamic models using both grand canonical and canonical ensembles. For finite systems the canonical ensemble is much more appropriate. The model values of α are compared with a much used standard formula. Next we turn to cases where in experiments, there are significant deviations from isoscaling. We show that in such cases, although the grand canonical model fails, the canonical model is capable of explaining the data.

The Nuclear Symmetry Energy in Self-Consistent Green-Function Calculations

In this paper we have reported the study of symmetry energy within the self-consistent Green-function approach. For the sake of comparison, the same calculations are performed using Brueckner–Hartree–Fock approximation. The symmetry energy is calculated for different densities and discussed in comparison with other predictions. The self-consistent Green's function leads to slightly larger energies as compared to the Brueckner–Hartree–Fock approach. This effect increases with density and thereby leads to smaller symmetry energies. The calculation of the symmetry energy in the Brueckner–Hartree–Fock approach shows a monotonic increase as a function of baryonic density. We extract the symmetry energy coefficient at density ρ= 0.16 fm -3 to be about 28.3 MeV for self-consistent Green-function approach using CD-Bonn potential, which is in good agreement with the empirical value of 30 ±4 MeV. The dependence of the equation of state on the neutron excess parameter is clearly linear as function of α 2 for self-co...

Erratum: Isospin dependence of incompressibility in relativistic and nonrelativistic mean field calculations [Phys. Rev. C 76, 034327 (2007)

The isospin dependence of incompressibility is investigated in the Skyrme Hartree-Fock (SHF) and relativistic mean field (RMF) models. The correlations between the nuclear matter incompressibility and the isospin dependent term of the finite nucleus incompressibility is elucidated by using the Thomas-Fermi approximation. The Coulomb term is also studied by using various different Skyrme Hamiltonians and RMF Lagrangians. The symmetry energy coefficient of incompressibility is extracted to be K_{\tau}=-(500\pm50) MeV from the recent experimental data of isoscalar giant monopole resonances (ISGMR) in Sn isotopes. Microscopic HF+random phase approximation (RPA) calculations are also performed with Skyrme interactions for ^{208}Pb and Sn isotopes to study the strength distributions of ISGMR. .

Global systematics of the Wigner energy

Following Zeldes, double-beta decay Q values are used as a filter for extracting symmetry energy and Wigner energy coefficients across the full range of nuclei, from A=10 to A=246. The symmetry coefficient extracted is found to vary smoothly with A and mass formula coefficients can be determined for the corresponding symmetry and surface symmetry terms. However, the extracted Wigner coefficient has large standard errors and fluctuates dramatically with A, even as regards its sign. Shell corrections remove most of the fluctuations and allow the determination of a reliable Wigner coefficient for the mass formula.

Isotope analysis in central heavy ion collisions at intermediate energies

Symmetry energy is a key quantity in the study of the equation of state of asymmetric nuclear matter. Heavy ion collisions at low and intermediate energies, performed at Laboratori Nazionali di Legnaro and Laboratori Nazionali del Sud, can be used to extract information on the symmetry energy coefficient Csym, which is currently poorly known but relevant both for astrophysics and for structure of exotic nuclei.

Excitation energy dependence of the symmetry energy of finite nuclei

A finite-range density and momentum-dependent effective interaction is used to calculate the density and temperature dependence of the symmetry energy coefficient ${C}_{\mathrm{sym}}(\ensuremath{\rho},T)$ of infinite nuclear matter. This symmetry energy is then used in the local density approximation to evaluate the excitation energy dependence of the symmetry energy coefficient of finite nuclei in a microcanonical formulation that accounts for thermal and expansion effects. The results are in good harmony with the recently reported experimental data from energetic nucleus-nucleus collisions.

Systematic study of isoscaling behavior in projectile fragmentation by the statistical abrasion–ablation model

The isospin effect and isoscaling behavior in projectile fragmentation have been systematically investigated by a modified statistical abrasion–ablation (SAA) model. The normalized peak differences and reduced isoscaling parameters are found to decrease with (Zproj − Z)/Zproj or the excitation energy per nucleon and have no significant dependence on the size of reaction systems. Assuming a Fermi-gas behavior, the excitation energy dependence of the symmetry energy coefficients is tentatively extracted from α and β which looks consistent with the experimental data. It is pointed out that the reduced isoscaling parameters can be used as an observable to study excitation extent of system and asymmetric nuclear equation of state in heavy ion collisions.

Isospin effect in statistical sequential decay

Isospin effect of the statistical emission fragments from the equilibrated source is investigated in the frame of statistical binary decay implemented into the GEMINI code, isoscaling behavior is observed and the dependences of isoscaling parameters \ensuremath{\alpha} and \ensuremath{\beta} on emission fragment size, source size, source isospin asymmetry and excitation energies are studied. Results show that \ensuremath{\alpha} and \ensuremath{\beta} neither depends on light fragment size nor on source size. A good linear dependence of \ensuremath{\alpha} and \ensuremath{\beta} on the inverse of temperature $T$ is manifested and the relationship of $\ensuremath{\alpha}=4{C}_{\mathrm{sym}}[({Z}_{s}/{A}_{s}){}_{1}^{2}\ensuremath{-}({Z}_{s}/{A}_{s}){}_{2}^{2}]/T$ and $\ensuremath{\beta}=4{C}_{\mathrm{sym}}[({N}_{s}/{A}_{s}){}_{1}^{2}\ensuremath{-}({N}_{s}/{A}_{s}){}_{2}^{2}]/T$ from different isospin asymmetry sources is satisfied. The symmetry energy coefficient ${C}_{\mathrm{sym}}$ extracted from simulation results is \ensuremath{\sim} 23 MeV which includes both the volume and surface term contributions, of which the surface effect seems to play a significant role in the symmetry energy.

Properties of hot nuclear fragments formed in multifragmentation and their astrophysical implications

The process of nuclear multifragmentation leads to thermodynamic conditions akin to those encountered in the collapse/explosion of massive stars. Thus, the knowledge obtained from studying multifragmentation may be applied to describe matter distribution in supernova. Along these lines, the present study focuses on the symmetry energy of hot fragments produced in the multifragmentation of neutron-rich systems following collisions near the Fermi energy. The experiments consist of high-resolution mass spectrometric measurements of the isotopic and velocity distributions of heavy projectile fragments. The data were systematically compared to calculations with the Statistical Multifragmentation Model (SMM). The study indicates a gradual decrease of the symmetry energy coefficient from the standard value of ∼25MeV in the compound nucleus regime (E∗/A<2MeV) towards ∼15MeV in the bulk multifragmentation regime (E∗/A>4MeV). The isotopic distributions of the hot fragments are very wide and extend towards or beyond the neutron drip-line. These findings may have important implications to the composition and evolution of hot astrophysical environments, such as core-collapse supernova and the course of the ensuing nucleosynthesis processes, such as the r-process.

Nuclear symmetry energy from effective interaction and masses of isospin asymmetric nuclei

Binding energy of isospin asymmetric nuclei can be accessed with minimally modified formula along the lines of the liquid droplet model by partitioning the symmetry term into volume and surface terms. The volume symmetry energy coefficient extracted from finite nuclei provides a constraint on the nuclear symmetry energy. This approach also yields the neutron skin of a finite nucleus through its relationship with the volume and surface symmetry terms and the Coulomb energy coefficient. The symmetry energy at saturation density obtained from the isoscalar as well as isovector components of the density dependent M3Y effective interaction is found to be in close agreement with the volume symmetry energy coefficient extracted from the measured atomic masses.

Tracing the evolution of the symmetry energy of hot nuclear fragments from the compound nucleus towards multifragmentation

The evolution of the symmetry energy coefficient of the binding energy of hot fragments with increasing excitation is explored in multifragmentation processes following heavy-ion collisions below the Fermi energy. In this work, high-resolution mass spectrometric data on isotopic distributions of projectile-like fragments are systematically compared to calculations involving the statistical multifragmentation model (SMM). Within the SMM picture, the present study suggests a gradual decrease of the symmetry energy coefficient of the hot primary fragments from 25 MeV at the compound nucleus regime towards 15 MeV in the multifragmentation regime. The isotopic distributions of the hot primary fragments are found to be very wide and extend towards the neutron drip line. These findings are expected to have important implications in the modeling of the composition and the evolution of hot and dense astrophysical environments, such as those of core-collapse supernova.

Isoscaling in Statistical Sequential Decay Model

A sequential decay model is used to study isoscaling, i.e. thefactorization of the isotope ratios from sources of different isospinsand sizes over a broad range of excitation energies, into fugacity termsof proton and neutron number,R21(N,Z) = Y2(N,Z)/Y1(N,Z) = Ce(αN+βZ). It isfound that the isoscaling parameters α and β have astrong dependence on the isospin difference of equilibrated sourceand excitation energy, no significant influence of the source sizeon α and β has been observed. It is found that α andβ decrease with the excitation energy and are linearfunctions of 1/T and Δ(Z/A)2 or Δ(N/A)2 of thesources. Symmetry energy coefficient Csym is constrained fromthe relationship of α and source Δ(Z/A)2, β andsource Δ(N/A)2.