Primary isotope yields and characteristic properties of the fragmenting source in heavy-ion reactions near the Fermi energy

For central collisions of ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$ at 35 MeV/nucleon, the density and temperature of a fragmenting source have been evaluated in a self-consistent manner using the ratio of the symmetry energy coefficient relative to the temperature, $a_{\mathrm{sym}}/T$, extracted from the yields of primary isotopes produced in antisymmetrized molecular dynamics (AMD) simulations. The $a_{\mathrm{sym}}/T$ values are extracted from all isotope yields using an improved method based on the modified Fisher model (MFM). The values of $a_{\mathrm{sym}}/T$ obtained, using different interactions with different density dependencies of the symmetry energy term, are correlated with the values of the symmetry energies at the density of fragment formation. Using this correlation, the fragment formation density is found to be $\rho /{\rho }_0=0.67\pm 0.02$. Using the input symmetry energy value for each interaction temperature values are extracted as a function of isotope mass $A$. The extracted temperature values are compared with those evaluated from the fluctuation thermometer with a radial flow correction.

Figure 1

$ln\left[R\right(I,-I,A\left)\right]/I$ versus $A$ for $I=1$ and $I=3$ from the events with the g0 interaction. The curve is the fit result of Eq. (4) for $k=0$. The extracted values of $\Delta \mu /T_0$ and $a_c/T_0$ are given in the third and fifth columns of Table 1.

Figure 2

(a) Calculated ratio of free energy to $T_0$ for $N=Z$ isotopes from the AMD events with g0 (solid points). Circles represent the fit using Eq. (5). (b) Calculated $\frac{\Delta F(N,Z)}{T_0}$ values for the g0 interaction and quadratic fits using Eq. (6) for $Z=2$ to 18. The same symbols are used for isotopes with a given $Z$. (c) Extracted $a_{\mathrm{sym}}/T_0$ values from (b) for g0 (dots), g0AS (squares) and g0ASS (triangles). Errors are from the quadratic fits. All results are from the first round $(k=0)$.

Figure 3

(a) The ratios of the $a_{\mathrm{sym}}/T_0$ values shown in Fig. 2, dots for g0/g0AS and circles for g0/g0ASS. (b) Symmetry energy coefficient vs density used in the AMD simulations. Solid curve (g0), dashed (g0AS), and dotted (g0ASS). (c) The ratio of the symmetry energy coefficient in (b). The shaded horizontal lines indicate the ratios extracted in (a) and the vertical shaded area shows the density region corresponding to these ratios. Two different shadings are used for the two ratio values. All results are from the first round.

Figure 4

Extracted $T_0$ values as a function of $Z$. Open symbols are for the first round ($k=0$) and closed symbols are for the final round ($k=0.007$). Different symbols represent results with g0 (circles), g0AS (squares), and g0ASS (triangles) interactions.

Figure 5

Same plots as Fig. 2, but for the final round ($k=0.007$). See also the figure caption of Fig. 2.

Figure 6

(a) Temperatures from the Monte Carlo model calculation. The solid curve represents the fluctuation temperature values for a thermally equilibrated source using $T_0=5.5$ MeV. The dashed curve represents those corresponding to the source with additional radial flow with $f_r=0.022$ fm/$c$. No momentum conservation is applied. (b) Temperature values extracted in different methods. Closed symbols (dots, squares, triangles) represent the apparent temperature values from the self-consistent symmetry energy analysis, using the AMD events from the final round with $k=0.007$. Open circles represent the fluctuation temperature values, $T_{Q_{xy}}$. Open squares represent those from the AMD events with no Coulomb potential. Solid and dashed curves are those similar to (a) but under the condition of the momentum conservation.

Figure 7

Radial flow energy vs the available energy. The experimental data are taken from Ref. [47].

Figure 8

Fit results of $a_{\mathrm{sym}}/T$ from Fig. 2 of the first round ($k=0$) with a function, $a_{\mathrm{sym}}/T=a_{\mathrm{sym}}^{\left(V\right)}/T(1-k_{S/V}{A}^{-1/3})$. $a_{\mathrm{sym}}^{\left(V\right)}/T=11.1$ and $k_{S/V}=1.2$ is used.