Experimental determination of the symmetry energy of a low density nuclear gas

Experimental analyses of moderate-temperature nuclear gases produced in the violent collisions of 35 MeV/nucleon ${}^{64}\mathrm{Zn}$ projectiles with ${}^{92}\mathrm{Mo}$ and ${}^{197}\mathrm{Au}$ target nuclei reveal a large degree of α particle clustering at low densities. For these gases, temperature- and density-dependent symmetry energy coefficients have been derived from isoscaling analyses of the yields of nuclei with $A⩽4$. At densities of 0.01 to 0.05 times the ground-state density of symmetric nuclear matter, the temperature- and density-dependent symmetry energies range from 9.03 to 13.6 MeV. This is much larger than those obtained in mean-field calculations and reflects the clusterization of low-density nuclear matter. The results are in quite reasonable agreement with calculated values obtained with a recently proposed virial equation of state calculation.

Figure 1

Double isotope ratio temperatures, $T_{\mathrm{HHe}}$, as a function of surface velocity for 35 MeV/nucleon ${}^{64}\mathrm{Zn}$$+{}^{92}\mathrm{Mo}$ (open squares) and ${}^{64}\mathrm{Zn}$$+{}^{197}\mathrm{Au}$ (open diamonds). Also shown is the average of the two (solid triangles). Uncertainties are estimated to be $⩽30\%$

Figure 2

Alpha mass fractions, $X_A$, as a function of surface velocity for 35 MeV/nucleon ${}^{64}\mathrm{Zn}$$+{}^{92}\mathrm{Mo}$ (squares) and ${}^{64}\mathrm{Zn}$$+{}^{197}\mathrm{Au}$ (diamonds). Also shown by the dashed line is the average of the two. Solid triangles represent derived values used for comparison to VEOS results. See text.

Figure 3

Isoscaling parameters as a function of surface velocity. The values for the parameters α (open circles), β (open squares), and the parameter ratio $\alpha /\beta$ (solid triangles). Estimated errors are $\pm 5\%$

Figure 4

The quantity $[(Z_1/A_1){}^2\text{−}(Z_2/A_2){}^2]{}^2$ as a function of surface velocity. Experimental determinations are represented by solid circles. Results of AMD-V calculations are represented by open squares.

Figure 5

Estimated nuclear densities as a function of temperature. Values derived from the experimental yield ratios using the Alberga formulas are indicated by solid squares. Theoretical results from Ref. [21] are shown by solid triangles. Values derived from the theoretical calculations of Ref. [32] are shown for confining volumes of 5 V${}_0$ (open squares) and 10 V${}_0$ (open diamonds).

Figure 6

$S_{\mathrm{sym}}$ vs $v_{\mathrm{surf}}$. Experimental data are represented by closed squares and calculated values are shown by open circles.

Figure 7

Derived symmetry energy coefficients as a function of baryon density. Solid diamonds indicate results using densities of column 4 in Table . Solid line indicates the variation predicted by the Gogny interaction. The dotted line represents the function $31.6(\rho /{\rho }_0){}^{1.05}$ [36].