Analysis of fragment yield ratios in the nuclear phase transition

The critical phenomenon of the liquid-gas phase transition has been investigated in the reactions ${}^{78,86}\mathrm{Kr}$$+$${}^{58,64}\mathrm{Ni}$ at beam energy of 35 MeV/nucleon using the Landau free energy approach with isospin asymmetry as an order parameter. Fits to the free energy of fragments showed three minima, suggesting that the system is in the regime of a first-order phase transition. The relation m $=$ −∂F/∂H, which defines the order parameter and its conjugate field $H$, has been experimentally verified from the linear dependence of the mirror nuclei yield ratio data on the isospin asymmetry of the source. The slope parameter, which is a measure of the distance from a critical temperature, showed a systematic decrease with increasing excitation energy of the source. Within the framework of the Landau free energy approach, isoscaling provided similar results as obtained from the analysis of mirror nuclei yield ratio data. In the present work, it is shown that the external field is primarily related to the minimum of the free energy, which implies a modification of the source concentration Δ used in isospin studies.

Figure 1

(a) Plot of fragment yields arising from the fragmentation of the quasiprojectiles with isospin asymmetry of 0.169 as a function of their isospin asymmetry m. (b) Plot of fragment free energies, calculated by the procedure discussed in Ref. [11], as a function of their isospin asymmetry $m$. Solid line (Fit 1) is a fit to data with the Landau equation (1). Fit 2 is a fit to the data using Eq. (1) with only first and last terms. Fit 3 is a fit to the data using $H$/$T$ and parameter $a$ obtained from the analysis of mirror nuclei yield ratio data.

Figure 2

(a) Plot of $0.5\ln (Y_2/Y_1)$ for $A$ $=$ 3 and 7 as a function of isospin asymmetry ($m_s$) of the quasiprojectile source for the reactions ${}^{78,86}\mathrm{Kr}$$+$${}^{86,64}\mathrm{Ni}$ at beam energy of 35 MeV/nucleon. Circles, squares, up triangles, and down triangles correspond to ${}^{78}\mathrm{Kr}$$+$${}^{58}\mathrm{Ni}$, ${}^{78}\mathrm{Kr}$$+$${}^{64}\mathrm{Ni}$, ${}^{86}\mathrm{Kr}$$+$${}^{58}\mathrm{Ni}$, and ${}^{86}\mathrm{Kr}$$+$${}^{64}\mathrm{Ni}$ reactions, respectively. Filled and open symbols correspond to $A$ $=$ 3 and $A$ $=$ 7, respectively. The subscripts “1” and “2” refer to the neutron-deficient and neutron-rich members of the mirror nuclei pair. (b) Plot of $0.5\ln (Y_2/Y_1)$, averaged over different reaction systems for $A$ $=$ 3 and 7 as a function of $m_s$. Solid and dashed lines are linear fits to the data for $A$ $=$ 3 and 7, respectively.

Figure 3

Plot of the position of the central minima in free energy (${\varepsilon }_0$) as a function of isospin asymmetry of the source ($m_s$). Average isospin asymmetries of the fragments $\langle m_f\rangle$ calculated with and without neutrons and protons are also shown in the figure for comparison.

Figure 4

Plot of $0.5\ln (Y_2/Y_1)$ as a function of isospin asymmetry of the quasiprojectile source ($m_s$) for $A$ $=$ 3. Filled and open symbols correspond to the excitation energies of 4 and 7 MeV/nucleon. The data of different reaction systems have been combined before taking the ratio of yields.

Figure 5

Slope (a′) values, obtained by fitting the plots of mirror nuclei yield ratios as a function of isospin asymmetry of the source for $A$ $=$ 3 (filled circles) and 7 (open circles), as a function of excitation energy of the quasiprojectile. Squares were obtained by fitting isoscaling plots, similar to that in Fig. 6, with a gate on excitation energy.

Figure 6

Plot of $1/A\ln (Y_{\mathit{ms}2}/Y_{\mathit{ms}1})$ as a function of $m$($m_s$${}_2$−$m_s$${}_1$). $Y_{\mathit{ms}1}$ and $Y_{\mathit{ms}2}$ are, respectively, the yields of a fragment from sources with isospin asymmetry of $m_s$${}_1$ and $m_s$${}_2$. $m$ is the isospin asymmetry of the fragment. $Y_{\mathit{ms}}$ was normalized with respect to the total number of events in the bin corresponding to $m_s$. Yield ratios were calculated for various possible combinations of $m_s$ bins such that $m_s$${}_2>m_s$${}_1$.