Symmetry energy for fragmentation in dynamical nuclear collisions

We extract values for the free symmetry energy as a function of the fragment size (the proton number $Z$) from antisymmetrized molecular dynamics calculations of calcium collisions. Simple statistical physics describe well the distribution of hot nuclei at breakup, provided the surface symmetry term in the free energy is much smaller at high excitation than in ground state nuclei. This result may reflect the condition of low density and finite temperature when these systems disassemble.

Figure 1

(Color online) Binding energies (not per nucleon) of nuclei calculated by AMD with the Gogny force. The quantities $-\mathrm{BE}(N,Z)+8(N+Z)\mathrm{MeV}$ are shown by filled and open squares for even-$Z$ and odd-$Z$ nuclei, respectively. Lines connect isotopes.

Figure 2

(Color online) The values of $K(N,Z)$ for $3⩽Z⩽18$ are shown by symbols for the abscissa of $N+Z$. The values are obtained by combining the results of ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$, ${}^{48}\mathrm{Ca}+{}^{48}\mathrm{Ca}$, ${}^{60}\mathrm{Ca}+{}^{60}\mathrm{Ca}$, and ${}^{46}\mathrm{Fe}+{}^{46}\mathrm{Fe}$ simulations. The error bars show the statistical uncertainty due to the finite number of events. The curve for each $Z$ was obtained by fitting $K(N,Z)$ using Eq. (7).

Figure 3

(Color online) The solid points are the extracted values of the coefficients $\zeta \left(Z\right)$ of Eq. (7) using the combined fragment yields of four systems shown in Fig. 2. The thick solid curve, the thin sold curve, and the dashed line show functions $\zeta \left(Z\right)\propto 1-k{\left(2Z\right)}^{-1∕3}$ normalized at $Z=10$ for $k=1.14$, 0.5, and 0, respectively.

Figure 4

(Color online) The solid points show $\zeta \left(Z\right)T$ when $T=3.4\mathrm{MeV}$ is assumed. The dot-dashed horizontal line shows the volume symmetry energy $c_{\mathrm{v}}=30.9\mathrm{MeV}$ for the ground state nuclei calculated with AMD. The solid line shows the symmetry energy $c(A=2Z)=c_{\mathrm{v}}+c_{\mathrm{s}}{\left(2Z\right)}^{-1∕3}$ for the ground state nuclei ($c_{\mathrm{v}}=30.9\mathrm{MeV}$ and $c_{\mathrm{s}}=-35.2\mathrm{MeV}$).