Constraining the symmetry term in the nuclear equation of state at subsaturation densities and finite temperatures

Methods of extraction of the symmetry energy (or enthalpy) coefficient to temperature ratio from isobaric and isotopic yields of fragments produced in Fermi-energy heavy-ion collisions are discussed. We show that the methods are consistent when the hot fragmenting source is well characterized and its excitation energy and isotopic composition are properly taken into account. The results are independent of the mass number of the detected fragments, which suggests that their fate is decided very early in the reaction.

Figure 1

Isotopic distributions for $Z=4$ (a) and $Z=14$ (b) for a Super Telescope. The $x$ axis is obtained by linearizing $\Delta E-E$ spectra as described in [43]. The Gaussian fits (black, dashed curves) show the overlapping of the mass distributions for different isotopes. The overall Gaussian fit is also shown (red, solid curves) for each $Z$.

Figure 2

Reconstructed quasiprojectile $m_s$ distribution from the ${}^{64}$Zn $+$ ${}^{64}$Zn (squares), ${}^{64}$Ni $+$ ${}^{64}$Ni (diamonds), and ${}^{70}$Zn $+$ ${}^{70}$Zn (triangles) reactions. The total yield of each system was normalized to unity. The $m_s$ distribution of the quasiprojectile obtained from the combined systems is plotted as stars. The total yield of this distribution was normalized to 3.

Figure 3

Isoscaling parameter $\alpha$ extracted for $\Delta \overline{m_s}=0.185$ and $3.5\le E_t^{☆}/A\le 5$ MeV from each reaction and for the combined systems as a function of the fragment mass ($A=2Z$).

Figure 4

Natural logarithm of the isotopic yield ratio as a function of the fragment $N$ [panels (a) and (c)] and natural logarithm of the isotopic yield ratio per fragment mass as a function of the fragment isospin asymmetry $m_f$ [panels (b) and (d)]. Panels (a) and (b) show system-to-system isoscaling and m-scaling from ${}^{64}$Zn $+$ ${}^{64}$Zn and ${}^{70}$Zn $+$ ${}^{70}$Zn reactions. Panels (c) and (d) show isoscaling and m-scaling using different $m_s$ bins of the reconstructed quasiprojectile (see text).

Figure 5

$\frac{C_{\mathrm{sym}}}{T}$, $\alpha$, and $\frac{\Delta H}{T}$ obtained from isobaric yield ratio (top), isoscaling (middle), and m-scaling (bottom) methods, applied to two different reactions (“SysToSys”) and two $m_s$ bins ($\Delta m_s$). The source reconstruction constraints are applied to study their impact on the measured values. The error bars, when not visible, are comparable to the size of the symbols.

Figure 6

Isoscaling parameters $-\beta$ vs $\alpha$ for all the $\Delta \overline{m_s}$ and excitation energy bins. The full line is the best fit to the data, while the dashed line represents $\alpha =-\beta$.

Figure 7

Quantity $-\left[\ln R(3,1)-\ln R(1,-1)\right]/A^2$ vs $1/A^2$ calculated from isobaric yield ratios for an average source isospin asymmetry $\overline{m_s}=0$. The dashed and full lines are the best fits to the data with a first-degree function, including and not including the Coulomb contribution, respectively.

Figure 8

Values of $\Delta$ computed according to the different definitions of Eqs. (8), (9), and (10) vs the fragment charge $Z$ for $\Delta \overline{m_s}=0.185$ and $3.5\le E_t^{☆}/A\le 5$ MeV.

Figure 9

Values of $\frac{C_{\mathrm{sym}}}{T}$ obtained with the three different methods. Each panel corresponds to the noted $\Delta \overline{m_s}$ and the noted window in $E_t^{☆}/A$. Three different choices of $\Delta$ have been used to compute $\frac{C_{\mathrm{sym}}}{T}$ from the isoscaling parameter $\alpha$.