Modification of surface energy in nuclear multifragmentation

Within the statistical multifragmentation model we study modifications of the surface and symmetry energy of primary fragments in the freeze-out volume. The ALADIN experimental data on multifragmentation obtained in reactions induced by high-energy projectiles with different neutron richness are analyzed. We have extracted the isospin dependence of the surface energy coefficient at different degrees of fragmentation. We conclude that the surface energy of hot fragments produced in multifragmentation reactions differs from the values extracted for isolated nuclei at low excitation. At high fragment multiplicity, it becomes nearly independent of the neutron content of the fragments.

Figure 1

Influence of the symmetry energy coefficient γ on yields of hot fragments in the freeze-out volume (bottom panels) and on their $N/Z$ ratios (top panels) in SMM calculations for Au sources at excitation energies of 3 and 5 MeV/nucleon (left and right panels, respectively).

Figure 2

Mass distributions of hot fragments with atomic numbers $Z=6,12$, and 26, produced in the multifragmentation of Au sources at excitation energies of 3, 5, and 10 MeV/nucleon, for different symmetry energy coefficients γ.

Figure 3

Influence of the surface energy coefficient $B_0$ on charge yields of hot fragments in the freeze-out volume, as obtained from SMM calculations for Au sources at excitation energies of 3, 5, and 10 MeV/nucleon.

Figure 4

SMM calculations of τ parameters for hot fragments from ${}^{238}\mathrm{U}$, ${}^{197}\mathrm{Au}$, and ${}^{129}\mathrm{Xe}$ sources as a function of their excitation energies. The power-law fitting was performed for fragments with $Z=5\text{−}15$.

Figure 5

SMM calculations of characteristics of hot fragments from Au sources for different surface energy coefficients $B_0=14,18,22$ MeV as a function of the excitation energy $E^{*}$. Top two panels: effective temperature $T_{\mathrm{eff}}$ and freeze-out temperature (caloric curves); middle panel: reduced mass number of the maximum fragment, $A_{\max }/A_0$; bottom panel: τ parameter.

Figure 6

Minimum value ${\tau }_{\min }$ of the τ parameter for hot fragments and the corresponding effective temperature $T_{\min }$, extracted from SMM calculations for sources with different neutron richness ${}^{238}\mathrm{U}$, ${}^{197}\mathrm{Au}$, and ${}^{129}\mathrm{Xe}$, as a function of the surface energy coefficient $B_0$.

Figure 7

Power-law parameters τ obtained from fitting the fragment charge distributions obtained in ALADIN experiments in the range $5\hspace{1em}⩽Z\hspace{1em}⩽\hspace{1em}15$ as a function of the reduced bound charge $Z_{\mathrm{bound}}/Z_0$ for the fragmentation of ${}^{129}\mathrm{Xe}$, ${}^{197}\mathrm{Au}$, and ${}^{238}\mathrm{U}$ projectiles of 600 MeV/nucleon. Top panel: three systems with different projectiles as indicated; bottom panel: results for collisions of ${}^{197}\mathrm{Au}$ with Be, C, Al, Cu, In, and Au targets and of ${}^{238}\mathrm{U}$ with Cu, In, Au, and U targets.

Figure 8

Ensemble of hot thermal sources represented in a scatter plot of reduced mass number $A_s/A_0$ versus excitation energy $E^{*}/A_s$ for the fragmentation of ${}^{197}\mathrm{Au}$ projectiles, as used in the SMM calculations. The intensity of the individual sources is proportional to the area of the squares.

Figure 9

Mean IMF multiplicities versus $Z_{\mathrm{bound}}/Z_0$, the bound charge divided by the projectile charge, predicted by the SMM ensemble calculations for three different projectiles. The meaning of the symbols is explained in the text.

Figure 10

SMM ensemble calculations of τ parameters for cold fragments versus $Z_{\mathrm{bound}}/Z_0$ for ${}^{238}\mathrm{U}$, ${}^{197}\mathrm{Au}$, and ${}^{129}\mathrm{Xe}$ projectiles. The surface energy coefficient is $B_0=18$ MeV for all sources.

Figure 11

SMM ensemble calculations of τ parameters for cold fragments versus $Z_{\mathrm{bound}}/Z_0$ for ${}^{238}\mathrm{U}$, ${}^{197}\mathrm{Au}$, and ${}^{129}\mathrm{Xe}$ projectiles. The Myers-Swiatecki mass formula was used for the hot primary fragments.

Figure 12

Comparison of the τ parameters obtained from the SMM ensemble calculations as a function of $B_0$ (dashed lines and open symbols) with the experimental results (solid horizontal lines and full symbols) for U, Au, and Xe projectiles in different $Z_{\mathrm{bound}}/Z_0$ bins.

Figure 13

The extracted $B_0$ for ensemble sources with different $N/Z$ ratios at the onset ($Z_{\mathrm{bound}}/Z_0=0.85$ bin) and at the region of full multifragmentation ($Z_{\mathrm{bound}}/Z_0=0.65$ and 0.55 bins). The width of the shaded band represents limits given by our method for the lowest bin. The $B_0$ obtained from Cameron (C) and Myers-Swiatecki (MS) mass formulas for cold nuclei are shown by solid lines for illustration.