With an isospin- and momentum-dependent transport model, we find that the degree of isospin diffusion in heavy-ion collisions at intermediate energies is affected by both the stiffness of the nuclear symmetry energy and the momentum dependence of the nucleon potential. Using a momentum dependence derived from the Gogny effective interaction, recent experimental data from NSCL-MSU on isospin diffusion are shown to be consistent with a nuclear symmetry energy given by $E_{\mathrm{s}\mathrm{y}\mathrm{m}}(\rho )\approx 31.6(\rho /{\rho }_0{)}^{1.05}$ at subnormal densities. This leads to a significantly constrained value of about $-550\mathrm{M}\mathrm{e}\mathrm{V}$ for the isospin-dependent part of the isobaric incompressibility of isospin asymmetric nuclear matter.

Expressing explicitly the parameters of the standard Skyrme interaction in terms of the macroscopic properties of asymmetric nuclear matter, we show in the Skyrme-Hartree-Fock approach that unambiguous correlations exist between observables of finite nuclei and nuclear matter properties. We find that existing data on neutron skin thickness $\Delta${}rnp of Sn isotopes give an important constraint on the symmetry energy Esym($\rho${}0) and its density slope L at saturation density $\rho${}0. Combining these constraints with those from recent analyses of isospin diffusion and the double neutron/proton ratio in heavy-ion collisions at intermediate energies leads to a more stringent limit on L approximately independent of Esym($\rho${}0). The implication of these new constraints on the $\Delta${}rnp of Pb208 as well as the core-crust transition density and pressure in neutron stars is discussed.

Within an isospin- and momentum-dependent hadronic transport model, it is shown that the recent FOPI data on the ${\pi }^{-}/{\pi }^{+}$ ratio in central heavy-ion collisions at SIS/GSI energies [Willy Reisdorf et al., Nucl. Phys. A 781, 459 (2007)] provide circumstantial evidence suggesting a rather soft nuclear symmetry energy $E_{\mathrm{sym}}(\rho )$ at $\rho \ge 2{\rho }_0$ compared to the Akmal-Pandharipande-Ravenhall prediction. Some astrophysical implications and the need for further experimental confirmations are discussed.

Correlations between the thickness of the neutron skin in finite nuclei and the nuclear matter symmetry energy are studied in the Skyrme Hartree-Fock model. From the most recent analysis of the isospin diffusion data in heavy-ion collisions based on an isospin- and momentum-dependent transport model with in-medium nucleon-nucleon cross sections, a value of $L=88\pm 25$ MeV for the slope of the nuclear symmetry energy at saturation density is extracted, and this imposes stringent constraints on both the parameters in the Skyrme effective interactions and the neutron skin thickness of heavy nuclei. Predicted thickness of the neutron skin is $0.22\pm 0.04$ fm for ${}^{208}\mathrm{Pb}$, $0.29\pm 0.04$ fm for ${}^{132}\mathrm{Sn}$, and $0.22\pm 0.04$ fm for ${}^{124}\mathrm{Sn}$.