Power law behavior of the isotope yield distributions in the multifragmentation regime of heavy ion reactions

Isotope yield distributions in the multifragmentation regime were studied with high-quality isotope identification, focusing on the intermediate mass fragments (IMFs) produced in semiviolent collisions. The yields were analyzed within the framework of a modified Fisher model. Using the ratio of the mass-dependent symmetry energy coefficient relative to the temperature, $a_{\mathrm{sym}}/T$, extracted in previous work and that of the pairing term, $a_{\mathrm{p}}/T$, extracted from this work, and assuming that both reflect secondary decay processes, the experimentally observed isotope yields were corrected for these effects. For a given $I=N-Z$ value, the corrected yields of isotopes relative to the yield of ${}^{12}\mathrm{C}$ show a power law distribution $Y(N,Z)/Y({}^{12}\mathrm{C})~A^{-\tau }$ in the mass range $1⩽A⩽30$, and the distributions are almost identical for the different reactions studied. The observed power law distributions change systematically when $I$ of the isotopes changes and the extracted $\tau$ value decreases from $3.9$ to $1.0$ as $I$ increases from $-1$ to $3$. These observations are well reproduced by a simple deexcitation model, with which the power law distribution of the primary isotopes is determined to be ${\tau }^{\mathrm{prim}}=2.4\pm 0.2$, suggesting that the disassembling system at the time of the fragment formation is indeed at, or very near, the critical point.

Figure 1

Simulated impact parameter distributions for violent (downward triangles), semiviolent (upward triangles), semiperipheral (squares), and peripheral (dots) collisions. Stars indicate the events in which at least one IMF $(Z⩾3)$ is emitted at ${20}^{°}\pm 5^{°}$. The summed distribution for a given class is normalized to 1.

Figure 2

Experimental ${}^{16}\mathrm{O}$ energy spectra (closed circles) are compared with the AMD $+$ GEMINI result (open circles) for ${}^{64}\mathrm{Zn}+{}^{112}\mathrm{Sn}$ at 40 $A$ MeV. The spectra for the AMD $+$ GEMINI result is obtained for the semiviolent collisions. Detection angles are given in the figure and the absolute $Y$ scale corresponds to the bottom spectra and the spectra are multiplied by a factor of 10 from the bottom to the top. The curves are the result of the moving-source fit, in which the parameters are determined from the experimental spectra at $17.5^{°}$ and $22.5^{°}$. The source velocity of $V_{\mathrm{s}}=0.62V_{\mathrm{p}}$ and $\Delta V_{\mathrm{s}}=0.11V_{\mathrm{p}}$ are used.

Figure 3

Experimental multiplicity distributions vs. isotope mass $A$ for the ${}^{64}\mathrm{Ni}$ projectiles with different targets. Targets are indicated on the right for each distribution. Each datum represents a summed multiplicity over $Z$ for a given $A$. Solid lines are the resits of power law fits for $1⩽A⩽30$ and dotted lines are for $10<A⩽30$. The extracted $\tau$ values are also given as ${\tau }_a$ and ${\tau }_b$, respectively. The absolute multiplicity value is given for the distribution at the bottom reaction and the multiplicity is multiplied by a factor of 100 from the bottom to the top.

Figure 4

(Upper) $Y/Y({}^{12}\mathrm{C})$ as a function of $A$ for even-even isotopes with $I=0$. (Middle) Same as the upper, but for odd-odd isotopes with $I=0$. (Bottom) The pairing energy term corrected yields for isotopes with $I=0$. The lines are the results fit by $A^{-\tau }$. The extracted $\tau$ values are shown in each figure.

Figure 5

Symmetry-term-corrected yields of $I=3$ for different reaction systems. Different symbols present different reactions. Dotted lines are connected between data points for the smallest and largest $Z/A$ values, respectively.

Figure 6

(Left) Corrected experimental isotope distributions for $I=-1$ to 3 from top to bottom for all 13 reactions. The experimental yields are corrected by $\exp [-(E_{\mathrm{sym}}-\delta )/T+\mathit{kI}]$, where $k=k_1[(Z/A){}_{\mathrm{sys}}-0.5]$. The pairing-term correction $\delta$ is made only for $I=0$ and 2. Solid lines are the results by the $A^{-\tau }$ fit for $1⩽A⩽30$. The extracted $\tau$ values are given in each figure. (Right) Corrected calculated isotope distributions for ${\tau }^{\mathrm{prim}}=2.3$. The same corrections as on the left are made. The extracted $\tau$ values are also given in each figure.

Figure 7

Extracted $\tau$ values from the experiment and the final products of the simulations. The experimental results are shown by dots and, for simulations, different symbols represent results from the simulations with the different ${\tau }^{\mathrm{prim}}$ values, which are indicated in the figure.