Density dependence of the symmetry energy and the nuclear equation of state: A dynamical and statistical model perspective

The density dependence of the symmetry energy in the equation of state of isospin asymmetric nuclear matter is of significant importance for studying the structure of systems as diverse as the neutron-rich nuclei and the neutron stars. A number of reactions using the dynamical and the statistical models of multifragmentation, and the experimental isoscaling observable, are studied to extract information on the density dependence of the symmetry energy. It is observed that the dynamical and the statistical model calculations give consistent results assuming the sequential decay effect in dynamical model to be small. A comparison with several other independent studies is also made to obtain important constraints on the form of the density dependence of the symmetry energy. The comparison rules out an extremely “stiff” and “soft” forms of the density dependence of the symmetry energy with important implications for astrophysical and nuclear physics studies.

Figure 1

Symmetry energy as a function of density predicted by microscopic ab initio calculations. The left panel shows the low-density region, while the right panel displays the high-density range. The figure is taken from Ref. [6].

Figure 2

Relative yield distribution of the fragments for the lithium (left) and carbon (right) isotopes in ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ (stars and solid lines), ${}^{58}\mathrm{Fe}+{}^{58}\mathrm{Ni}$ (circles and dashed lines), ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Fe}$ (squares and dashed lines), and ${}^{58}\mathrm{Fe}+{}^{58}\mathrm{Fe}$ (triangles and dotted lines) reactions at various beam energies.

Figure 3

Relative yield distribution of the fragments for lithium (left), berillium (center), and carbon (right) isotopes in ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ (stars and solid lines), ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Ni}$ (circles and dashed lines), and ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Fe}$ (squares and dotted lines) reactions at various beam energies.

Figure 4

Experimental isotopic yield ratios of the fragments as a function of neutron number $N$, for various beam energies. The left column correspond to ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Fe}$ and ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ pair of reactions. The right column correspond to ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Ni}$ and ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ pair of reactions. The different symbols correspond to $Z=3$ (circles), $Z=4$ (open stars), $Z=5$ (triangles), $Z=6$ (squares), and $Z=7$ (filled stars) elements. The lines are the exponential fits to the data as explained in the text.

Figure 5

Experimental isotope yield ratios (top) and isotone yield ratios (bottom) from ${}^{58}\mathrm{Fe}+{}^{58}\mathrm{Fe}$ and ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ reactions as a function of $N$ and $Z$ for 30 MeV$/$nucleon beam energy. The solid lines are fit to the data as discussed in the text.

Figure 6

Experimental isoscaling parameter α, as a function of the beam energy. The solid circles are for the ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Fe}$ and ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ reactions. The open triangles are for ${}^{58}\mathrm{Fe}+{}^{58}\mathrm{Fe}$ and ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ reactions. The solid stars are for ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Ni}$ and ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ reactions. The open squares are for ${}^{58}\mathrm{Fe}+{}^{58}\mathrm{Ni}$ and ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ reactions.

Figure 7

AMD calculations of the fragment asymmetry $(Z/A){}^2$, at $t=300$ fm/$c$ for the Gogny (solid line and solid squares) and Gogny-AS (dotted line and hollow squares) interactions at 35 MeV/nucleon. The calculations are taken from Ref. [31] for the systems shown by the square symbols. The lines are linear fit to the square symbols. The other symbols are the interpolated values for the systems studied in this work.

Figure 8

Isoscaling parameter α, as a function of the difference in fragment asymmetry for 35 MeV/nucleon. The solid and the dotted lines are the AMD calculations for the Gogny and Gogny-AS interactions, respectively [31]. The solid and the hollow squares, stars, triangles, and circles are from the present work as described in the text. The other symbols corresponds to data taken from [50] (asterisks) and [36] (crosses, diamonds, inverted triangles).

Figure 9

Different forms of the density dependence of the nuclear symmetry energy used in the dynamical analysis of the present measurements on isoscaling data and the isospin diffusion measurements of NSCL-MSU [51]. The curves are as described in the text.

Figure 10

Isoscaling parameter α, temperature $T$, symmetry energy $C_{\mathrm{sym}}$, and density as a function of excitation energy for the Fe + Fe and Ni + Ni (inverted triangles), and Fe + Ni and Ni + Ni (solid circles) reactions at 30, 40, and 47 MeV/nucleon. (a) Experimental isoscaling parameter as a function of excitation energy. The solid and the dashed curves are the SMM calculations as discussed in the text. (b) Temperature as a function of excitation energy. The solid stars correspond to the measured values and are taken from Ref. [56]. The solid and the dashed curves correspond to the Fermi-gas relation. The dotted curve corresponds to the one obtained from Eq. (4). (c) Symmetry energy as a function of excitation energy. (d) Density as a function of excitation energy. The solid stars correspond to those from Ref. [70]. The open triangles are those from Ref. [71]. The solid curve is from Ref. [58].

Figure 11

Symmetry energy as a function of density for the Fe + Fe and Ni + Ni pair of reaction (inverted triangles), and Fe + Ni and Ni + Ni pair of reactions (solid circles) for the 30, 40, and 47 MeV/nucleon. The solid curve is the dependence obtained form the dynamical model analysis as explained in the text.

Figure 12

Comparison between the results on the density dependence of the symmetry energy obtained from various different studies. The various curves and the symbols are described in the text.

Figure 13

SMM calculated isoscaling parameter α as a function of symmetry energy for various excitation energies. The open circles joined by dotted lines correspond to the primary fragments and the open stars joined by solid lines to the secondary fragments. The left column shows the calculation for ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Ni}$ and ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ pair, and the right column for the ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Fe}$ and ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ pair.

Figure 14

Same as in Fig. 13, but with the modified secondary deexcitation with evolving symmetry energy.