Experimental reconstruction of primary hot isotopes and characteristic properties of the fragmenting source in heavy-ion reactions near the Fermi energy

The characteristic properties of the hot nuclear matter existing at the time of fragment formation in multifragmentation events produced in the reaction ${}^{64}\mathrm{Zn}+{}^{112}\mathrm{Sn}$ at 40 MeV/nucleon are studied. A kinematical focusing method is employed to determine the multiplicities of evaporated light particles, associated with isotopically identified intermediate-mass fragments. From these data the primary isotopic yield distributions are reconstructed using a Monte Carlo method. The reconstructed yield distributions are in good agreement with the primary isotope distributions obtained from antisymmetrized molecular dynamics transport model simulations. Utilizing the reconstructed yields and power distribution, characteristic properties of the emitting source are examined. The primary mass distribution exhibits a power-law distribution with the critical exponent $A^{-2.3}$ for $A\ge 15$ isotopes but significantly deviate from that for lighter isotopes. Based on the modified Fisher model, the ratios of the Coulomb and symmetry energy coefficients relative to the temperature, $a_c/T$ and $a_{\mathrm{sym}}/T$, are extracted as a function of A. The extracted $a_{\mathrm{sym}}/T$ values are compared with results of the antisymmetrized molecular dynamics simulations using Gogny interactions with different density dependencies of the symmetry energy term. The calculated $a_{\mathrm{sym}}/T$ values show a close relation to the symmetry energy at the density at the time of fragment formation. From this relation the density of the fragmenting source is determined to be $\rho /{\rho }_0=0.63\pm 0.03$. Using this density, the symmetry energy coefficient and the temperature of fragmenting source are determined in a self-consistent manner as $a_{\mathrm{sym}}=24.7\pm 3.4$ MeV and $T=4.9\pm 0.2$ MeV.

Figure 1

Comparisons of total neutron multiplicity obtained in this work (circles), from neutron ball measurement (squares), and from an AMD + gemini calculation (stars). The figure is taken from Ref. [55].

Figure 2

Simulated impact parameter distributions for violent (downward triangles), semiviolent (upward triangles), semiperipheral (squares), and peripheral (circles) collisions. Stars indicate events in which at least one IMF $(Z\ge 3)$ is emitted at ${20}^{\circ }\pm 5^{\circ }$. The summed distribution for a given class is normalized to 1. The figure is taken from Ref. [58].

Figure 3

Experimental ${}^{16}\mathrm{O}$ energy spectra (filled circles) are compared with the AMD + gemini result (open circles) for ${}^{64}\mathrm{Zn}+{}^{112}\mathrm{Sn}$ at 40 MeV/nucleon. The spectrum for the AMD + gemini result was obtained for semiviolent collisions. Detection angles are given, the absolute Y scale corresponds to the bottom spectrum, and spectra are multiplied by a factor of 10 from bottom to the top. Curves are the result of a moving-source fit, in which the parameters were determined from the experimental spectra at ${17.5}^{\circ }$ and ${22.5}^{\circ }$. Source velocities of $V_s=0.62V_p$ and $\Delta V_s=0.11V_p$ are used. The figure is taken from Ref. [58].

Figure 4

Typical cold-isotope distributions are compared with the results of AMD + gemini simulations for three interactions discussed in the paper. Results for $Z=6$ (left column), 8 (middle column), and 11 (right column). Results of AMD with g0 (top row), g0AS (middle row), and g0ASS (bottom row) are plotted with the experimental data. Experimental data are taken from the IV source component from the moving-source fit and represented by filled circles. The same experimental data are used in each column. Multiplicity distributions from AMD simulations are calculated in two ways. Circles represent the results of the IV source component from the moving-source fit. Triangles are those calculated from the approximated method (see details in text).

Figure 5

Average reconstructed excitation energy vs gemini input excitation energy.

Figure 6

Isotope distribution in a two-dimensional plot of Z vs N (a) for experimental and (b) for reconstructed fragments. The dashed line is the $\beta$ stability line. The Z axis is the multiplicity given on an absolute logarithmic scale.

Figure 7

Isotopic multiplicity distributions of experimental cold fragments (filled circles), reconstructed hot fragments (filled squares), and AMD primary hot fragments as a function of fragment mass number $A$ for a given charge Z, which is indicated. AMD results are from the g0AS interaction.

Figure 8

Isotopic multiplicity distributions of (a) $Z=8$ and (b) $Z=12$ for reconstructed hot fragments (filled squares) as well as AMD primary hot fragments with g0 (open circles), g0AS (open squares), and g0ASS (open triangles) as a function of the fragment mass number $A$ for a given charge Z, which is indicated.

Figure 9

Calculated Ne isotope distributions when the $\sigma$ values of the LP multiplicity distribution are changed from for $0.75\sigma$ to $1.25\sigma$.

Figure 10

Absolute multiplicities of reconstructed hot isotopes are plotted by filled circles as a function of A together with those of the AMD primary isotopes of the IV source (open circles). Those of the IV + PLF sources (filled triangles) are plotted only for $A>30$, where one can see a clear deviation from those of the IV source.

Figure 11

$ln\left[R\right(1,-1,A\left)\right]$ values are plotted as a function of the fragment mass number A for experimental (filled circles) and reconstructed (filled squares) hot fragments as well as primary fragments of the AMD simulation (open triangles). Curves were obtained by free parameter search of $\Delta \mu /T$ and $a_c/T$ in Eq. (5). The extracted parameter from the reconstructed data is $a_c/T=0.14\pm 0.04$, using $\Delta \mu /T=0.67$. Values used for experimental and AMD primary data are from Ref. [57], and values of $(\Delta \mu /T,a_c/T)=(0.71,0.35)$ for experimental and $(\Delta \mu /T,a_c/T)=(0.40,0.18)$ for AMD primary data are used.

Figure 12

Calculated values of $a_{\mathrm{sym}}/T$, are plotted as a function of $A$ from experimental cold isotopes (filled circles) and reconstructed hot isotopes (filled squares). The solid curve is from Ref. [57]. The dashed curve is the average value of those from the reconstructed ones. AMD results are shown by open symbols: g0 (squares), g0AS (triangles), and g0ASS (circles). The average $a_{\mathrm{sym}}/T$ values for those from reconstructed and AMD simulations are listed in the fifth column in Table 1.

Figure 13

(a) Ratios of the calculated $a_{\mathrm{sym}}/T$ values for g0/g0AS (open triangles), g0/g0ASS (open squares), and g0/reconstructed experimental yield (filled circles). Dotted lines are the average values for AMD simulations. The values are listed in the second column in Table 1. (b) Symmetry energy coefficient vs density for g0, g0AS, and g0ASS used in AMD simulations. The hatched vertical area indicates the density of the fragmenting system extracted from the ratio of the symmetry energy coefficient. Two hatching patterns are used for the density values given in the third column in Table 1. (c) Ratio of symmetry energy coefficients used in (b), g0/g0AS and g0/g0ASS, as a function of the density. Dotted horizontal lines indicate ratio values extracted from the $a_{\mathrm{sym}}/T$ values in (a).