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New results for the strength of the symmetry energy are presented which illustrate the complementary aspects encountered in reactions probing nuclear densities below and above saturation. A systematic study of isotopic effects in spectator fragmentation was performed at the ALADIN spectrometer with 124Sn primary and 107Sn and 124La secondary beams of 600 MeV/nucleon incident energy. The analysis within the Statistical Fragmentation Model shows that the symmetry-term coefficient entering the liquid-drop description of the emerging fragments decreases significantly as the multiplicity of fragments and light particles from the disintegration of the produced spectator systems increases. Higher densities were probed in the FOPI/LAND study of nucleon and light-particle flows in central and mid-peripheral collisions of 197Au+197Au nuclei at 400 MeV/nucleon incident energy. From the comparison of the measured neutron and hydrogen squeeze-out ratios with predictions of the UrQMD model a moderately soft symmetry term with a density dependence of the potential term proportional to (rho/rho_0)^{gamma} with gamma = 0.9 +- 0.3 is favored.
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The relative isobaric yields of fragments produced in a series of heavy ion induced multifragmentation reactions have been analyzed in the framework of a Modified Fisher Model, primarily to determine the ratio of the symmetry energy coefficient to the temperature, $a_a/T$, as a function of fragment mass A. The extracted values increase from 5 to ~16 as A increases from 9 to 37. These values have been compared to the results of calculations using the Antisymmetrized Molecular Dynamics (AMD) model together with the statistical decay code Gemini. The calculated ratios are in good agreement with those extracted from the experiment. In contrast, the ratios determined from fitting the primary fragment distributions from the AMD model calculation are ~ 4 and show little variation with A. This observation indicates that the value of the symmetry energy coefficient derived from final fragment observables may be significantly different than the actual value at the time of fragment formation. The experimentally observed pairing effect is also studied within the same simulations. The Coulomb coefficient is also discussed.
The nuclear symmetry energy is a fundamental quantity important for studying the structure of systems as diverse as the atomic nucleus and the neutron star. Considerable efforts are being made to experimentally extract the symmetry energy and its dependence on nuclear density and temperature. In this article, we review experimental studies carried out up-to-date and their current status.
Intermediate-energy heavy-ion collisions can produce a spin polarization of the projectile-like species. Spin polarization has been observed for both nucleon removal and nucleon pickup processes. Qualitative agreement with measured spin polarization as a function of the momentum of the projectile-like fragment is found in a kinematical model that considers conservation of linear and angular momentum and assumes peripheral interactions between the fast projectile and target. Better quantitative agreement was reached by including more realistic angular distributions, de-orientation caused by gamma-ray emission, and by correcting for the out-of-plane acceptance. The newly introduced corrections were found to apply to both nucleon removal and nucleon pickup processes.
Within an isospin and momentum dependent transport model, the dynamics of isospin particles (nucleons and light clusters) in Fermi-energy heavy-ion collisions are investigated for constraining the isospin splitting of nucleon effective mass and the symmetry energy at subsaturation densities. The mass splitting of $m^{*}_{n}>m^{*}_{p}$ and $m^{*}_{n}<m^{*}_{p}$ in nuclear matter and the different stiffness of symmetry energy are used in the model. The single and double neutron to proton ratios of free nucleons and light particles are thoroughly investigated in the isotopic nuclear reactions of $^{112}$Sn+$^{112}$Sn and $^{124}$Sn+$^{124}$Sn at the incident energies of 50 and 120 MeV/nucleon, respectively. It is found that the both effective mass splitting and symmetry energy impact the kinetic energy spectra of the single ratios, in particular at the high energy tail (larger than 20 MeV). Specific constraints are obtained from the double ratio spectra, which are evaluated from the ratios of isospin observables produced in $^{124}$Sn+$^{124}$Sn over $^{112}$Sn+$^{112}$Sn collisions. A mass splitting of $m^{*}_{n}<m^{*}_{p}$ is constrained from the available data at the energy of 120 MeV/nucleon. A soft symmetry energy with the stiffness of $gamma_{s}=$0.5 is close to the experimental double ratio spectra at both energies.
A three parameter scaling relationship between isotopic distributions for elements with Z$leq 8$ has been observed that allows a simple description of the dependence of such distributions on the overall isospin of the system. This scaling law (termed iso-scaling) applies for a variety of reaction mechanisms that are dominated by phase space, including evaporation, multifragmentation and deeply inelastic scattering. The origins of this scaling behavior for the various reaction mechanisms are explained. For multifragmentation processes, the systematics is influenced by the density dependence of the asymmetry term of the equation of state.
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