Reexamination of a novel determination of density, temperature, and symmetry energy based on a modified Fisher model

In this article, a mistake in the formulation of the modified Fisher model (MFM) derived in the pioneering works of the Purdue group is addressed and corrected by reversing the sign of the mixing entropy term in the original formulation. The errors in the results of the previous MFM-related studies, such as isotopic yield distribution, isobaric yield ratios, isoscaling, $m$ scaling, self-consistent determination of density, symmetry energy, and temperature, and density and temperature determination related to the intermediate mass fragment (IMF) freezeout, are quantitatively analyzed. It is found that the errors originating from the mistake in sign of the mixing entropy term are generally small and even have no effect in some cases.

Figure 1

Mixing entropy per nucleon as a function of $m$. The quantum results from Eq. (19) are shown by different symbols for $T$ = 1 MeV (dots), 3 MeV (solid squares), 5 MeV (solid triangles), 7 MeV (solid inverted triangles), and 30 MeV (open circles) at density $\rho ={\rho }_0/6$. The solid line corresponds to the mixing entropy per nucleon calculated using the classical formula, Eq. (12).

Figure 2

Isotopic yields as a function of $I$. Dots: isotopic yield of all available measured fragments in $p+Xe$ system at the incident momentum between 80 and 350 GeV/c. Triangles: fitting results from the original MFM formulation of the Purdue group. Squares: fitting results from the corrected MFM formulation, Eq. (20).

Figure 3

Numerical values of ${\mathrm{\Delta }}_{Err}$ calculated using Eq. (24) as a function of $A$ with $I=-5\text{to}4$ from the bottom to the top.

Figure 4

Numerical values of $\mathrm{\Delta }(a_{\mathrm{sym}}/T)$ calculated using Eq. (26) as a function of $A$ with $I=1–9$.

Figure 5

Comparisons between the present and original results together with available published results. The line is from the fitting of the available data using Eq. (28). Data are taken from Khoa et al. (2005) [41], Kowalski et al. (2007) [42], Wada et al. (2012) [43], Roca-Maza et al. (2013) [44], Shetty et al. (2004) [45], Shetty et al. (2007) [46], Trippa et al. (2008) [47], and Tsang et al. (2009) [48].