Symmetry energy and the isoscaling properties of the fragments produced in ${}^{40}\mathrm{Ar}$, ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Fe}$, ${}^{58}\mathrm{Ni}$ reactions at 25, 33, 45, and 53 MeV/nucleon

The symmetry energy and the isoscaling properties of the fragments produced in the multifragmentation of ${}^{40}\mathrm{Ar}$, ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Fe}$, ${}^{58}\mathrm{Ni}$ reactions at 25, 33, 45, and 53 MeV/nucleon were investigated within the framework of statistical multifragmentation model. The isoscaling parameters α, from the primary (hot) and secondary (cold) fragment yield distributions, were studied as a function of excitation energy, isospin (neutron-to-proton asymmetry), and fragment symmetry energy. It is observed that the isoscaling parameter α decreases with increasing excitation energy and decreasing symmetry energy. The parameter α is also observed to increase with increasing difference in the isospin of the fragmenting system. The sequential decay of the primary fragments into secondary fragments, when studied as a function of excitation energy and isospin of the fragmenting system, show very little influence on the isoscaling parameter. The symmetry energy, however, has a strong influence on the isospin properties of the hot fragments. The experimentally observed scaling parameters can be explained by symmetry energy that is significantly lower than that for the ground-state nuclei near saturation density. The results indicate that the properties of hot nuclei at excitation energies, densities, and isospin away from the normal ground-state nuclei could be significantly different.

Figure 1

Schematic diagram of the experimental setup showing the placement of the telescopes inside the scattering chamber of the neutron ball.

Figure 2

Relative isotopic yield distribution for the lithium (left), berillium (center) and carbon elements 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 3

Double isotope ratio as a function of the difference in binding energy for various beam energies. The circle symbols correspond to ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ reaction, square symbols to ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Ni}$ reaction, and triangle symbols to ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Fe}$ reaction. The solid lines are the best fit to the data.

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

Figure 6

Isoscaling parameters α (solid symbols) and β (open symbols) as a function of the beam energy. The solid circles and open squares correspond to ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Fe}$/${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ pair of reactions. The solid stars and open triangles correspond to ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Ni}$/${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ pair of reactions. The lines are the exponential fits to the data. The error bars are of the size of the symbols.

Figure 7

The scaled isotope ratio $S(\beta ,N)$, as a function of the neutron number $N$ for the 25 and 45 MeV/nucleon beam energies. The top panel is for the ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Fe}$/${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ pair, and the bottom panel is for the ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Ni}$/${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ pair of reactions. The symbols correspond to $S(\beta ,N)$ obtained from various elements ($Z$). The lines are the best fits to the data.

Figure 8

Relative free neutron (solid symbols) and proton (open symbols) densities as a function of the difference in $N/Z$ of the systems for the beam energies of 25, 33, and 45 MeV/nucleon.

Figure 9

SMM calculated primary (left) and secondary (right) fragment isotope yield distributions for the carbon element 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 excitation energies. The calculations are for $\gamma =25$ MeV.

Figure 10

Calculated isotopic yield ratios from the primary (left) and the secondary (right) fragment yield distributions for the ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Ni}$ and ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ pairs at various excitation energies. The calculations are for $\gamma =25$ MeV. The different symbols shown correspond to $Z=3$ (circles), $Z=4$ (open stars), $Z=5$ (triangles), $Z=6$ (square), and $Z=7$ (solid stars) elements. The lines are the exponential fits to the ratios.

Figure 11

Comparison of the SMM calculated α (lines) with the experimentally determined α (symbols) as a function of excitation energy for different values of the symmetry energy coefficient γ. The dotted lines correspond to the primary fragments and the solid lines to the secondary fragments. The left column shows the comparison for the ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Ni}$ and ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ pair, and the right column shows the comparison for the ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Fe}$ and ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ pair.

Figure 12

Isotopic yield distribution for the carbon element in ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ reaction at 45 MeV/nucleon. The solid line is the SMM calculation with symmetry energy coefficient $\gamma =25$ MeV, and the dashed line is the calculation with $\gamma =15$ MeV. The solid points correspond to the experimental result.

Figure 13

SMM calculated isoscaling parameter α as a function of symmetry energy coefficient 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 shows the calculation for the ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Fe}$ and ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ pair.

Figure 14

Same as in Fig. 11, but with the modified secondary deexcitation with evolving symmetry energy coefficient.

Figure 15

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

Figure 16

Comparison between the calculated primary (top) and the secondary (bottom) isotopic yield distribution for the carbon element in ${}^{40}\mathrm{Ar}+{}^{58}\mathrm{Fe}$ reaction at $E^{*}=6$ MeV/nucleon. The left panels correspond to the old and the right to the new deexcitation prescription used in the SMM calculation. The solid and the dashed curves correspond to the calculations using two different values of the symmetry energy.