3.1 Moving source fit

The yield of each isotope was evaluated, using a moving source fit, where a single intermediate velocity (NN) source with a smeared source velocity around half the beam velocity was used, coupled with the AMD simulations as a reference. A detailed procedure of the multiplicity extraction was given in Ref.[31]. For neutrons and LCPs, three moving source fit was applied, i.e., projectile-like source, NN source and target-like source. In the following analysis the multiplicity of the NN source is used for all particles as their multiplicity.

As direct outputs of the moving source fit, the source size A_s=(N_s+Z_s) and asymmetry parameter, δ_NN=(N_s–Z_s)/A_s, are evaluated from the extracted multiplicity as N_s=ΣM_iN_i and Z_s=ΣM_iZ_i, where the summation is taken over all measured particles, including neutrons, LCPs and IMF with Z up to 18. N_i and Z_i are neutron and proton number of the particle i. In Fig.1, the extracted A_s and δ_NN are plotted as a function of Z_max, the upper limit of the summation. A_s and δ_NN values at Z_max=2, for example, indicate the source size and asymmetry when only light particles are taken into account. The emitting source size of the NN component varies from 45 to 80 for the small (Ni) to the large mass (Th) target. This indicates that all NN source particles originate from a rather small source, about a half of the projectile for the Ni targets to a slightly large size of the projectile for the heavy targets. The observation of a rather small size of the emitting source is consistent with previous observations, which indicates that, as a function of temperature, experimental observables such as the isoscaling parameter, symmetry energy coefficient and pairing coefficient, reflect similar reaction mechanisms , even though the target sizes are quite different in the reactions studied^[30,31,32].

Fig.1Full-screen View

(Color online) Source size (top) and asymmetry δ_NN (bottom) of the NN source for reactions with the ⁶⁴Ni beam. Different targets are shown by different colors as indicated. Horizontal bars on the bottom right indicates the calculated δ_NN values from the source of 1 to 1 mixing of projectile and target nucleons and from the whole given reaction system, indicated by 1:1 and sys, respectively.

Within errors, the evaluated asymmetry parameter, δ_NN, is consistent with the values calculated assuming either from 1 to 1 mixing of the projectile and target nucleons or mixture of all nucleons. However the error bars, mainly originating from the moving source parameterization, are too large to determine the specific experimental asymmetry value for a given reaction system. Therefore in this work the calculated values of the 1 to 1 mixing source are used as δ_NN for the following analysis.

3.2 The ratio (μ_n–μ_p)/T

The ratio of the difference in nucleon chemical potential relative to the temperature, (μ_n–μ_p)/T, can be experimentally determined from the isotopic yield ratios, using the Modified Fisher Model^[35,36,37]. In the Modified Fisher model, the fragment yield of A nucleons with I=N–Z, Y(A, I), is given by

where C is a constant. The A^–τ term originates from the entropy of the fragment and the last two terms are from the entropy contributions for the mixing of two substances in the Fisher Droplet Model^[38]. W(A,I) is the free energy of the cluster at temperature T. In our application W(A,I) is given by the following generalized Weiszȁcker-Beth semi-classical mass formula^[39,40].

When the yield ratio of two isotopes between I=I and I=I+2 with same A is taken, $R\left(I+2,I,A\right)=Y\left(A,I+2\right)/Y\left(A,I\right)$ is given by

where $S_{\text{mix}}\left(I,A\right)=N\ln \left(N/A\right)+Z\ln \left(Z/A\right)$. From these relations, one can get

where $\Delta (I+IA)=S_{\text{mix}}(I+A)S_{\text{mix}}(IA)$, a_c and a_sym are the Coulomb and symmetry coefficient in W(I,A) and δ(N,Z) represent the pairing term.

When we focus on the yields of isobars with I=–1 and 1, one can get

This equation was initially applied to the ⁶⁴Zn+¹¹²Sn reaction. ln[R(1,–1,A)] values were calculated from the multiplicities of isotopes in the atomic number range of 3≤Z≤18. (μ_n–μ_p)/T=0.71 and a_c/T=0.35 were extracted^[31]. Assuming the same temperature and density at the time of the fragment formation for different reaction systems, which results in the same Coulomb energy contribution in Eq.(4), (μ_n–μ_p)/T is evaluated as:

Here $\langle \ln \left[R(1,-1,A)\right]\rangle$ and $\langle 2a_c(Z-1)/A^{1/3}/T\rangle$ are the averaged experimental value over a certain range of Z of isotopes. Isotopes with $3\le Z\le 12$ are used, in order to minimize the errors between the reaction systems, since the statistics of higher Z isotopes becomes low for some of the reaction systems in the experiment. The resultant $({\mu }_n-{\mu }_p)/T$ is shown in Fig.2 as a function of the asymmetric parameter ${\delta }_{NN}=(N-Z)/A$ of the emitting source. The values are well scaled by δ_NN. $({\mu }_n-{\mu }_p)/T=11.1{\delta }_{NN}\pm 0.21$ is obtained as shown in the figure.

Fig.2Full-screen View

(Color online) (μ_n–μ_p)/T vs δ_NN from all 13 reactions. Red, black and blue represent for ⁶⁴Ni, ⁷⁰Zn and ⁶⁴Zn beams, respectively. Results for different targets are also indicated by different symbols as shown. Data are fit by a linear function and the parameters are shown in the figure.

3.3 Temperature

In order to determine the temperature at the time of the

fragment formation, the double isotope ratio thermometer between different IMF isotopes was used. The double isotope ratio temperature is generally given by

where R is the double yield ratio of four isotopes, B is a binding energy parameter and a relates to their spins^[20]. There are many choices of the double ratio thermometers. Here we examined the following thermometers^[25],

Fig.3Full-screen View

(Color online) Primary temperature vs emitting source density. Symbols are from the double ratio temperatures and typical errors are given only for the results from T_dthα. The others have a similar errors in each point. Solid line is the result from (μ_n ̶ μ_p)/T fit.

The calculated values from the different thermometers represent the temperatures of the final products, which may be significantly modified by the secondary statistical decay in the cooling process. The secondary decay effects are carefully examined, using the QSM calculation as described in Ref.[25]. The resultant primary temperatures extracted are fitted by a polynomial function of the secondary temperature for a given density. In Fig.3 the calculated primary temperature for these different thermometers are shown as a function of the density of the emitting source. The secondary temperatures used are those averaged over all 13 reactions. For all cases the primary temperature increases when the source density increases.

Fig.4Full-screen View

(Color online) (bottom) Experimental and calculated (μ_n ̶ μ_p)/T values as a function of δ_NN. T=5.2 MeV is used in the figure. The different symbols represent the values for different given densities as indicated. (top) Chi-square values as a function of the density.

For a given temperature and density in Fig.3, the differential chemical potential, $({\mu }_n-{\mu }_p)/T$, is calculated, using QSM and compared with the experimental values in Fig.2. An example is shown in Fig.4 for the case T=5.2 MeV. For a given density and δ_NN, the differential chemical potential, $({\mu }_n-{\mu }_p)/T$, is calculated. The density is varied from 0.05ρ₀ to 0.7ρ₀ in this case. For a given density, $({\mu }_n-{\mu }_p)/T$ increases linearly as δ_NN increases, as shown in the bottom of Fig.4. From the chi-square fit, ρ=0.3ρ₀ is extracted for the case. T<4.5 MeV and T>6.0 MeV, no solution is found to get the minimum chi-square value below 0.4. T and ρ/ρ₀ values at the minimum chi square are plotted by a solid curve in Fig.3. From the results of $({\mu }_n-{\mu }_p)/T$ analysis, the possible temperature range of 4.5≤T≤6.0 MeV was evaluated, but no specific density range was able to be determined in these methods.

3.4 Symmetry energy

The energy per nucleon of an asymmetric nuclear matter is approximately given by

On the other hand the differential chemical potential is given by

From these equations, one can get $({\mu }_n-{\mu }_p)=4E_{\text{sym}}(\rho )\delta$^[33]. In Fig.2, the experimental data are fit by a linear function and $({\mu }_n-{\mu }_p)/T=(11.1\pm 1.4)\delta -0.2$is obtained. Using $({\mu }_n-{\mu }_p)=4E_{\text{sym}}(\rho )\delta$ from Eq.(14) and 4.5≤T≤6.5 MeV, E_sym coefficient is evaluated as E_sym=14.6±3.5 MeV.