Statistical analysis of experimental multifragmentation events in ${}^{64}\mathrm{Zn}+{}^{112}\mathrm{Sn}$ at 40 MeV/nucleon

A statistical multifragmentation model (SMM) is applied to the experimentally observed multifragmentation events in an intermediate heavy-ion reaction. Using the temperature and symmetry energy extracted from the isobaric yield ratio (IYR) method based on the modified Fisher model (MFM), SMM is applied to the reaction ${}^{64}\mathrm{Zn}+{}^{112}\mathrm{Sn}$ at 40 MeV/nucleon. The experimental isotope distribution and mass distribution of the primary reconstructed fragments are compared without afterburner and they are well reproduced. The extracted temperature $T$ and symmetry energy coefficient $a_{\mathrm{sym}}$ from SMM simulated events, using the IYR method, are also consistent with those from the experiment. These results strongly suggest that in the multifragmentation process there is a freezeout volume, in which the thermal and chemical equilibrium is established before or at the time of the intermediate-mass fragments emission.

Figure 1

(a) Quasitemperature of SMM without (solid circles) and with (open circles) the symmetry entropy as a function of fragment mass A for $\gamma =25$ MeV. (b) The average temperature of SMM without (solid circles) and with (open circles) the symmetry entropy as a function of $\gamma$.

Figure 2

$ln\left[R\right(1,-1,A\left)\right]$ are plotted as a function of the fragment mass number $A$ produced in SMM without (solid circles) and with (open circles) the symmetry entropy for $\gamma =25$ MeV. The solid and dashed lines are the fitting results with Eq. (4) for SMM without and with the symmetry entropy, respectively.

Figure 3

Extracted $a_{\mathrm{sym}}$ values as a function of fragment mass $A$ for system size $A_s=40$ (solid circles), $A_s=60$ (solid squares), $A_s=100$ (open circles), and $A_s=200$ (open squares).

Figure 4

Quasitemperature of SMM with the symmetry entropy for $A_s=40$, 60, 100, and 200 with the fixed $Z_s/A_s=0.45$.

Figure 5

(a) $ln\left[R\right(1,-1,A\left)\right]$ without (solid circles) and with (open circles) the finite-size correction are plotted as a function of the fragment mass number $A$ for SMM with the symmetry entropy. The solid and dashed lines are the fitting results of Eq. (4). (b) Extracted average $a_{\mathrm{sym}}$ values without (solid circles) and with (open circles) the finite-size correction as a function of the fragment mass number A for SMM with the symmetry entropy.

Figure 6

(a) Extracted $a_c$ values as a function of input $\gamma$ before (solid circles) and after (open circles) the finite-size correction for SMM with symmetry entropy. The fitting errors are smaller than the symbol size. The line is from a constant fit. (b) Average $a_{\mathrm{sym}}$ values of the constant fit as a function of input $\gamma$ before (solid circles) and after (open circles) the finite-size correction for SMM with symmetry entropy. The open squares are those after the finite-size correction for SMM without the symmetry entropy. The line is corresponding to the SMM input value of $a_{\mathrm{sym}}=\gamma$. (c) Extracted $k$ values are plotted as a function of $\gamma$ in the case of system size = 100. The $k$ value for $\gamma =0$ MeV is averaged over those of $\gamma >$ 0 MeV. The inset shows the extracted $k$ values as a function of the system size for the case of $\gamma =25$ MeV.

Figure 7

Average temperature from self-consistent analysis (shaded area) from Refs. [36, 37] and that from SMM with $V/V_0=4$, 6, and 10. Open circles and open squares are without and with the symmetry entropy, respectively.

Figure 8

(a) $ln\left[R\right(1,-1,A\left)\right]$ as a function of fragment mass from IMFs generated by SMM for different breakup volumes of $V/V_0=4$, 6, and 10. Lines are the fitting results using Eq. (4) for each breakup volume. (b) Open circles are the extracted $a_c$ values from the fitting in (a) and solid circles are the SMM input values under the Wigner-Seitz approximation.

Figure 9

(a) The experimental mass distribution (solid squares) is compared with that of SMM without (open circles) and with (open squares) the symmetry entropy at $T=5.9$ MeV and the breakup volume of $6V_0$. The mass distribution of AMD from Refs. [36, 37] is also shown by triangles. The distributions of the simulated results are normalized to the reconstructed data at $A=15$. (b) The experimental mass distribution is compared with that of SMM with different breakup volumes at $T=5.9$ MeV. (c) The experimental mass distribution is compared with that of SMM with different temperatures at $V=6V_0$.

Figure 10

Isotope distributions of the experimentally reconstructed primary fragments (solid squares) and those from SMM without (open circles) and with (open squares) the symmetry entropy at $V=6V_0$ are compared for $Z=3$–14. AMD results from Refs. [36, 37] are also shown by open triangles. All results are plotted in an absolute scale.

Figure 11

$a_{\mathrm{sym}}$ as a function of fragments mass A for the reconstructed data (shaded area) and the SMM results without (open circles) and with (open squares) the symmetry entropy at $V=6V_0$ are shown, together with those of AMD from Refs. [36, 37].

Figure 12

Symmetry entropy per nucleon, $S_{\mathrm{sym}}/A$, as a function of $m$ for $T=3$ MeV (solid circles), 6 MeV (solid squares), 9 MeV (solid up-triangles), 15 MeV (solid down-triangles), and 50 MeV (open circles) at density $\rho ={\rho }_0$. Solid line corresponds to the symmetry entropy at the classical limit, and dashed line represents the results using Eq. (A9).