IQMD Model

The QMD model is essentially a quantum extension of the classical molecular dynamics approach, which is widely applied in chemistry and astrophysics; it is designed to describe fragment formation [54-58]. This n-body approach uses a microscopic framework that treats the dynamics of colliding nuclei directly and simulates heavy-ion collisions on an event-by-event basis [54]. The descriptions of mean positions and momenta are purely classical, and particles are considered to be distinguishable in the QMD model [55]. In this work, we use an improved version of the QMD model that incorporates isospin-dependent interactions and the Pauli exclusion principle. In this model, each nucleon is treated as a Gaussian wave packet in a coherent state [54, 55, 59]: ${\varphi }_i(\overrightarrow{r}\mathrm{,}t)=\frac{1}{{(2\pi L)}^{3/4}}\text{e}xp[-\frac{{\left(\overrightarrow{r}-{\overrightarrow{r}}_i(t)\right)}^2}{4L}+\frac{i{\overrightarrow{p}}_i(t)\cdot \overrightarrow{r}}{\hslash }]\mathrm{,}$ (1) where ${\overrightarrow{r}}_i$ and ${\overrightarrow{p}}_i$ represent the position and momentum of the i-th nucleon, respectively. L is the square of the Gaussian wave packet width, which is set to 2.16 fm². The following interaction terms are included in the IQMD model: $V_{\text{tot}}=V_{\text{sky}}+V_{\text{yuk}}+V_{\text{sym}}+V_{\text{MDI}}+V_{\text{Coul}};$ (2) they correspond to Skyrme, Yukawa, symmetry, momentum-dependent, and Coulomb interactions, respectively. The detailed forms of these interaction terms are given in Refs. [54, 55]. The Skyrme potential component of the EOS, which is associated with the Skyrme interaction, is given by $U_{\text{sky}}=\alpha \frac{\rho }{{\rho }_0}+\beta {\left(\frac{\rho }{{\rho }_0}\right)}^{\gamma }\mathrm{.}$ (3) In this work, we set α = -129 MeV, β = 59 MeV, and γ = 2.09, which is called the hard EOS, and α = -390 MeV, β = 320 MeV, and γ = 1.14, which is called the soft EOS.

Formulas

The following probes are calculated on the basis of information on the phase space and fragments: the Fisher parameter τ_eff, fragment multiplicity, information entropy (H), and density moments. The effective Fisher parameter τ_eff can be obtained from the charge distribution by power-law fitting as follows: $\begin{array}{l}\text{d}N/\text{d}Z\sim Z^{-{\tau }_{\text{e}ff}}\mathrm{.}\hfill \end{array}$ (4) Here the charge distribution is taken at 300 fm/c when the system has reached the kinetic freeze-out stage. These charge distributions can be fitted well in the range 3 ≤ Z ≤ 7, as shown in Fig. 1. The charge range is selected to avoid the lightest fragments, with Z = 1 and 2, which could be affected by the evaporation and decay process, as well as heavier residual fragments.

Fig. 1Full-screen View

Charge distributions of the systems in freeze-out stage (t = 300 fm/c) at beam energies of E of (a) 20 MeV/nucleon, (b) 50 MeV/nucleon, (c) 80 MeV/nucleon, and (d) 130 MeV/nucleon. The red lines represent the Fisher power-law fits in the charge range 3 ≤ Z ≤ 7 for the hard EOS.

The information entropy H of all multiplicity events, which was presented for the first time by Ma [44] and introduced into the study of LGPTs of nuclei, is given by $\begin{array}{l}H=-{\displaystyle \sum _i}{f}_i\text{ln}{f}_i\mathrm{,}\hfill \end{array}$ (5) where fi, which is calculated for the event space, is the normalized event probability that i particles are produced, and ${\displaystyle {\sum }_i}{f}_i=1$. The sum is taken over the entire distribution of fi [44].

The density moments can be calculated from the phase space data computed in an IQMD simulation. First, from the phase space data computed by the IQMD simulation, the nuclear matter densities can be calculated at each coordinate space point and at every time as follows: $\begin{array}{l}\rho (\overrightarrow{r}\mathrm{,}t)={\displaystyle \sum _{i=1}^{A_{\text{T}}+A_{\text{P}}}}\frac{1}{{(2\pi L)}^{3/2}}\text{e}xp[\frac{-{(\overrightarrow{r}-{\overrightarrow{r}}_i)}^2}{2L}]\mathrm{,}\hfill \end{array}$ (6) where the summation is taken over all nucleons. Then, the numerical value of the nuclear matter density can be used to calculate the density moments via the formula described in Ref. [49-51], that is, $\begin{array}{l}\langle {\rho }^N\rangle \equiv \frac{1}{A}{\displaystyle \int }\rho {(\overrightarrow{r})}^N\rho (\overrightarrow{r}){\text{d}}^{3}r\mathrm{,}\hfill \end{array}$ (7) where A = ${\displaystyle \int }\rho (\overrightarrow{r}){\text{d}}^{3}r$. The normalized density moments are given by $\langle {\rho }^N\rangle /{\langle \rho \rangle }^N$, which is thus unity for N = 1 [49-51]. These quantities have observational relevance because of their intimate relationship with the relative production yield of fragments.

Time evolution of properties

Figure 2 shows the time evolution of the IMF multiplicity (N_IMF) and information entropy (H) at various incident energies. In Fig. 2 (a), the IMF multiplicity increases, reaches a maximum, and then decreases with increasing time. Here IMFs are defined as fragments with charge number (Z) greater than or equal to 3 and smaller than the charge number of the source. As the energy increases, the IMF multiplicity changes more rapidly. Thus, at higher energies, the collision system generally has a higher excitation energy and breaks more rapidly into numerous fragments.

Fig. 2Full-screen View

Time evolution of (a) IMF multiplicity and (b) information entropy at different beam energies for the hard EOS.

Using the distribution {fi} calculated from all events, one can obtain the information entropy (H), as shown in Fig. 2 (b). The pattern of H evolution is slightly different from that of the IMF multiplicity. With increasing H, the system becomes more chaotic. In the compression stage, the information entropy increases rapidly. With increasing time, more fragments are created, and the multiplicity probability distribution fi becomes more diverse. As time increases further, the information entropy decreases slightly because long-term binding of nucleons in clusters is difficult under the IQMD Hamiltonian. Thus, the number of particles produced will increase with time, which will result in a change in the probability distribution fi and finally a slight decrease in H.

Figure 3 shows the time evolution of the normalized density moments ($\langle {\rho }^N\rangle /{\langle \rho \rangle }^N$), which is the major focus of this work, in the central region of [-3,3]³ fm³ at various incident energies for N = 2 and 6. The time evolution of the normalized density moments is similar to that of the IMF multiplicity. It increases with time and then shows near-saturation or a slight decrease. At a given energy, for higher-order density moments, the density moment values are larger. In particular, high-order moments show cleaner structure as a function of beam energy, that is, a clearer broad peak structure at a certain energy, although there are larger fluctuations, as shown in Fig. 3 (b), because obtaining higher-order density moments with the same precision requires better statistics. In addition, higher-order density moments are more sensitive to density fluctuations.

Fig. 3Full-screen View

Time evolution of (a) normalized density moments for the orders N = 2 and (b) N = 6 at different beam energies. Here the normalized density moments are calculated for the central region of [-3,3]³ fm³ for the hard EOS.

Discussion of liquid-gas phase transition

To discuss the LGPT in collisions, we extracted the effective Fisher parameter τ_eff (Fig. 1), maximum IMF multiplicity (Fig. 2 (a)), and maximum information entropy (Fig. 2 (b)) as a function of incident energy, as shown in Fig. 4. Both τ_eff and the maximum information entropy exhibit non-monotonic behavior, and the peak values are approximately 50–80 MeV/nucleon. For the maximum IMF multiplicity, a plateau appears at energies above 70 MeV/nucleon. All of these results seem consistent with each other, and they indicate that the LGPT for this system could occur in this energy region. The hard and soft EOSs are indicated by black lines with circles and blue lines with squares, respectively, in Fig. 4. The turning energies are not sensitive to the EOS for all τ_eff, $N_{\text{IMF}}^{\text{max}}$, and H_max. In addition, only the value of $N_{\text{IMF}}^{\text{max}}$ is higher for the soft EOS than for the hard EOS, as shown in Fig. 4 (b). We also deduced the temperature using the fluctuation in the proton transverse momentum [60], which is defined as ${\sigma }^2=\langle Q_{xy}^2\rangle -{\langle Q_{xy}\rangle }^2=4m^2{T}^2$, where $Q_{xy}=p_x^2-p_y^2$. When temperature is used as a variable instead of incident energy, as shown in Fig. 5, the turning temperature is shifted to slightly lower temperature for all τ_eff, $N_{\text{IMF}}^{\text{max}}$, and H_max for the soft EOS. Thus, the soft EOS can reduce the phase transition temperature of a collision system.

Fig. 4Full-screen View

Extracted values of (a) effective Fisher parameter (τ), (b) maximum values of IMF multiplicities ($N_{\text{IMF}}^{\text{max}}$), and (c) maximum values of information entropy (H_max) as a function of incident energy for the hard and soft EOSs.

Fig. 5Full-screen View

Same as Fig. 4 but as a function of temperature.

Further, we give the maximum normalized density moments for different orders, as shown in Fig. 6. To present all the orders of the normalized density moment in a single figure, these lines are scaled by different factors. In addition, because statistical fluctuations exist, these lines are fitted by polynomial functions. For the order N = 2 [black dotted line in Fig. 6 (a)], the line seems flat. As the order of the density moment increases to N = 6, the maximum values appear around E = 90 MeV/nucleon, as shown in Fig. 6 (b). Thus, the collision system has the maximum density fluctuation around E = 90 MeV/nucleon, indicating that the LGPT could occur here when the system enters spinodal instability [61, 53]. However, if we choose a larger central region of [-5,5]³ fm³, as shown in Fig. 7, the maximum normalized density moments reach maximum values around 70 MeV/nucleon and have a plateau at higher incident energies, like the maximum IMF multiplicity shown in Fig. 4 (b). This energy is also close to the energy given by the effective Fisher parameter τ and the maximum information entropy, as shown in Fig. 4.

Fig. 6Full-screen View

Maximum values of normalized density moments for N = 2, 3, 4, 5, and 6 as a function of incident energy in central region of [-3,3]³ fm³. The lines are the fitted results for the hard EOS.

Fig. 7Full-screen View

Same as Fig. 6, but for central region of [-5,5]³ fm³.

Note that the turning energy varies with the central region. That is, the turning point has regional dependence. In fact, this can be understood from the central densities of the selected regions. We found that the [-3,3]³ fm³ region can reach a higher average density than the [-5,5]³ fm³ region. Thermodynamically, the temperature, pressure, and density are correlated; therefore, the turning point of the energy depends on the density. Actually, as reported in our previous work [62], the temperature extracted from heavy-ion collisions is lower for a smaller central region, where a higher beam energy would be required to reach the same temperature in a smaller central region. Thus, the calculated density moments for a region of a certain size correspond to a certain density and temperature. The [-5,5]³ fm³ is generally a more reasonable option, because our other observables, that is, the fragment distributions and their effective Fisher parameters, the IMF multiplicities, and the information entropy, are those of the entire space.