Fragment emission and critical behavior in light and heavy charged systems


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Samiksha Sood, Rohit Kumar, Arun Sharma, Sakshi Gautam and Rajeev K. Puri. Fragment emission and critical behavior in light and heavy charged systems[J]. Chinese Physics C.
Samiksha Sood, Rohit Kumar, Arun Sharma, Sakshi Gautam and Rajeev K. Puri. Fragment emission and critical behavior in light and heavy charged systems[J]. Chinese Physics C. shu
Received: 2020-06-03
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Fragment emission and critical behavior in light and heavy charged systems

    Corresponding author: Rohit Kumar,
    Corresponding author: Rajeev K. Puri,
  • 1. Department of Physics, Panjab University, Chandigarh - 160014, India
  • 2. Department of Physics, G. D. College, Billawar, Jammu - 185204, India

Abstract: We study the emission of fragments in central collisions of light and heavily charged systems of 40Ar+45Sc and 84Kr+197Au, respectively using Quantum Molecular Dynamics (QMD) model as primary model. The fragments are identified using energy based clusterization algorithm i.e., Simulated Annealing Clusterization Algorithm (SACA). The charge distributions of intermediate mass fragments [3≤ $ Z_{f} $ ≤12] are fitted with power-law ( $ \propto Z_{f} ^{-\tau} $ ) and exponential fits ( $ \propto {\rm{e}} ^{-\lambda {Z_{f}}} $ ) to extract the parameters τ and $ \lambda $ whose minimum value is also sometimes linked with the onset of fragmentation or critical point for a liquid-gas phase transition. Other parameters such as normalized second moment $ <S_2> $ , $ <\gamma_2> $ , average size of the second largest cluster $ <Z_{\rm max2}> $ , phase separation parameter ( $ S_p $ ), bimodal parameter (P), information entropy (H) and Zipf's law are also analyzed to find the exact energy of the onset of fragmentation. Our detailed analysis predicts an energy point to exist between 20-23.1 MeV/nucleon which is very close to experimentally observed value of 23.9 MeV/nucleon for 40Ar+45Sc reaction. We also found that the critical energy deduced using Zipf's law is higher than predicted from other critical exponents. Also, no minimum is found in τ values for the highly charged system of 84Kr+197Au in agreement with experimental findings and various theoretical calculations. We observe that QMD + SACA model calculations are in agreement with the experimental observations. This agreement supports our results regarding the energy point of the liquid-gas phase transition or the onset of fragmentation.


    1.   Introduction
    • During last three decades, extensive study of nuclear multifragmentation has been carried out due to its possible link with liquid-gas phase transition in nuclear matter [1-25]. This is considered due to similarity between the nuclear force and the Van der Waal's interaction i.e., a long range attractive part and short range repulsive core. The first effort in this regard was done by Fermi-lab-Purdue experiment on the reactions of p+Kr and p+Xe systems. The mass yield of fragments upto mass 30 was fitted with power law dependence ( $ Y(A_f)\propto A_{f}^{-\tau} $ ) [19]. This observation was consistent with the prediction of the Fisher's droplet model and stimulated a large number of studies on the topic. The incident energy at which liquid-gas phase transition takes place is often labeled as "onset of multifragmentation" or "critical energy point" [4-9, 14-21, 23].

      Generally, one fits the charge (or mass) yields of the fragments with power law $ \propto Z_{f} ^{-\tau} $ (or $ A_{f} ^{-\tau} $ ) and extracts the value of τ [4-7, 9, 11, 14-17]. The minima in the τ value, when plotted as a function of incident energy is assumed to indicate the onset of multifragmentation or the critical energy point [4-6, 9-11, 14-17]. To mention a few studies in this regard, Ogilvie et al. [9] performed the experiment with ALADIN forward spectrometer at GSI, Germany and reported the existence of minima in the values of τ for Au induced reactions with C, Al and Cu targets at an incident energy of 600 MeV/nucleon. Later on, Li et al. [15] performed an experiment using Michigan State University (MSU) 4 $ \pi $ -array to study the central reactions of 40Ar+45Sc. They reported a minimum in the power law exponent τ at 23.9 MeV/nucleon. However, the corresponding Percolation model calculations predicted minimum value around 28 MeV/nucleon [15]. William et al. [3] on the other hand, reported the absence of minima in the value of τ for the highly charged colliding system of 84Kr+197Au at incident energies between 35 MeV/nucleon and 400 MeV/nucleon. The dominance of the Coulomb forces was thought to hinder the occurrence of such minima. Gupta and Pan have also reported similar results for 197Au+197Au reactions [26].

      In addition to minima in the values of τ, other characteristic signals have also been advocated for the study of the onset of multifragmentation or liquid-gas phase transition [5, 21-23, 27-31]. Among these are, the fluctuation in the size of the largest fragment [27], the size of the second largest fragment [28], the asymmetry in the size of the first two large fragments [29], the parameters based on the moments of the charge distribution i.e., $ S_2 $ , $ \gamma_2 $ [21, 22], bimodality of the order parameter [30, 31], the multiplicity derivatives [23] as well as information entropy (H) [5] and the Zipf's law [5]. With these advocated signals many studies have been performed and a great progress on the theoretical and experimental front has been made on the topic of liquid-gas phase transition [5-8, 11, 18, 20, 23, 32]. See Refs. [1, 2, 33] for the reviews on the topic.

      In literature, the studies do exist where many-body dynamical models were used to investigate the liquid-gas phase-transition in nuclear matter e.g., Ma et al. studied the reactions of 40Ar+27Al between 25 and 150 MeV/nucleon and found a minima in τ at 65 MeV/nucleon using Quantum Molecular Dynamics (QMD) model [4]. Belkacem et al. have analyzed the experimental data of MULTICS-MINIBALL collaboration for the reactions of 197Au+197Au at 35 MeV/nucleon and also presented the theoretical calculations with Classical Molecular Dynamics (CMD) model [18]. They obtained critical behavior in both theory and experiment. Ma et al. using Neutron Ion Multi-detector for Reaction Oriented Dynamics (NIMROD) experimental set up studied 40Ar on 27Al,48Ti and 58Ni reactions [6-8] and also put forward the calculations with the CMD model [6, 7]. In their studies, they have used most of the above mentioned characteristic signals and found critical behavior for the systems. Two of us and collaborators, however reported no minimum in the power law exponent or clear onset of fragmentation in the reaction of40Ar+45Sc using Isospin dependent Quantum Molecular Dynamics (IQMD) model [10]. Later on, minimum was obtained when Coulomb potential was neglected [11].

      The common point in all the above mentioned studies with dynamical models is that all have used Minimum Spanning Tree (MST) algorithm to clusterize the phase space. Recently, we used various extensions of MST method (that includes momentum and binding cuts) and found that there is no effect of these extensions [32] on the extraction of minimum in the exponent τ for reaction of 40Ar+45Sc. The absolute values were, however, far from the experimental data. To resolve the general problem of MST algorithm, Puri and Aichelin, based on the method of Dorso et al. [34] had proposed Simulated Annealing Clusterization Algorithm (SACA) [35, 36]. This algorithm was found to resolve the failure that was reported with MST method for weakly excited systems [36, 37]. Recently, two of us confronted the QMD+SACA model to the experimental data near Fermi-energy domain and were successful in the effort [38]. The aim of the present study is atleast two-fold: 1) to confront the QMD+SACA model with the experimental data at and away from the Fermi-energy domain for the lighter i.e., 40Ar+45Sc and heavily charged systems 84Kr +197Au and 2) to see, if various different signals of the liquid-gas phase transition are put together in the framework of QMD+SACA model, they show consistent picture or which signals are best suited for the purpose. The present calculations also discuss the utilization of the Zipf's law, which was introduced as possible signal of the liquid-gas phase transition, but reported to show signatures at much higher excitation energies as shown in Refs. [20, 39]. Our study will include lighter systems of 40Ar+45Sc and heavier systems of 84Kr +197Au. Note that the main difference here is not only the size of the system but also the large Coulomb forces in the later compared to former. The study will show if the involvement of these large Coulomb forces influence the compatibility of the QMD+SACA model.

      In section 2, we will briefly discuss both primary and secondary models. The detailed analysis of theoretical calculations is given in section 3. Finally, we summarize our findings in section 4.

    2.   Methodology

      2.1.   Quantum molecular dynamics model

    • The quantum molecular dynamics [40] model is a many-body dynamical model that simulates heavy-ion collisions on an event-by-event basis. In this model, the time evolution of a reaction is studied via following individual nucleons. The trajectories of nucleons during the reaction are determined by mean field and nucleon-nucleon collisions. The model describes the reaction from the well separated target and projectile followed upto freeze-out (final) stages, where the nuclear matter is fragmented. In this model, each nucleon of two colliding nuclei is represented by a Gaussian wave packet in the position and momentum space as [40]:

      $ \psi_{i}({{{r}}},{{{p}}}_{i}(t), {{{r}}}_{i}(t)) = \dfrac{1}{(2\pi L)^{\frac{3}{4}}}\; {\rm e}^{ [ \frac{i}{\hbar} { {{p}}}_{i}(t)\cdot{{{r}}}-\frac{({{{r}}}-{{{r}}}_i(t))^2}{4L}]}. $


      To built a nucleus, the coordinates are assigned to nucleons inside a sphere of radius R $ = 1.142 A^{1/3} $ (where A is the mass number of the nucleus) in accordance to liquid drop model. The nucleons are also assigned Fermi-momentum values between 0 and $ P_{\rm F} $ . Here, $ P_{\rm F}(r_i) $ = $ \sqrt{2mU(r_i)} $ with U( $ r_i $ ) being the local potential energy of the $ i^{\rm th} $ nucleon. If the momentum chosen leads to an overlap in phase-space with any previously defined nucleon that configuration is rejected. Now to keep the QMD formulation as close as possible to the classical transport theory, instead of the wave functions the Wigner densities are used. The Wigner densities corresponds to the phase-space densities in classical mechanics. The Wigner representation of the $ A_T + A_P $ nucleon system is given by

      $ f({{r}}, {{p}}, t) = \displaystyle\sum_i \dfrac{1}{(\pi \hbar)^3} {\rm e}^{-[{{r}}-{{r}}_i(t)]^2/2L-[{{p}}-{{p}}_i(t)]^2 2L/\hbar^2}. $


      The Wigner representation of the gaussian wave packets obeys the uncertainty principle and the one body densities in coordinate and momentum space are given by

      $ \rho ({{r}},t) = \int f({{r}}, {{p}}, t) {\rm d}^3 {{p}} =\displaystyle\sum_i \frac{1}{(2 \pi L)^{3/2}} {\rm e}^{-[{{r}}-{{r}}_i(t)]^2/2L}, $


      $ g({{p}}, t) = \int f({{r}}, {{p}}, t) {\rm d}^3 {{r}}. $


      Each nucleus generated by above procedure is checked for its ground state properties such as binding energy, nuclear density etc. If it fulfils these conditions than a nucleus is said to be successfully initialized. The successfully initialized target and projectile are boosted towards each other. During the reaction, the centroid of each Gaussian propagates using classical equations of motion:

      $ \begin{array}{l} \dot{{{{r}}}}_i = \dfrac{\partial H}{\partial{{{p}}}_i}, \;\;\;\;\; \dot{{{{p}}}}_i = -\dfrac{\partial H}{\partial{{{r}}}_i}, \end{array} $


      where H represents the Hamiltonian and is given by

      $ \begin{aligned}[b] \langle H\rangle = \langle T\rangle+\langle V \rangle =& \sum_{i}\frac{p^{2}_{i}}{2m_{i}} + \sum\limits_{i}\sum\limits_{j>i}\int f_{i}({{r}}_{i},{{p}}_{i},t) {V_{ij}} ({{r}}_{i},{{r}}_{j}) \\ & \times f_{j}({{r}}_{j},{{p}}_{j},t) {\rm d}{{r}}_{i} {\rm d}{{r}}_{j} {\rm d}{{p}}_{i} {\rm d}{{p}}_{j}. \end{aligned} $


      Here, $ f_i $ and $ f_j $ are the Wigner distribution function of the i $ ^{\rm th} $ and j $ ^{\rm th} $ nucleon, respectively, and V $ _{ij} $ ( ${{r}} _{i} $ , ${{r}} _{j} $ ) is the baryonic potential that has essential components like Skyrme, Yukawa and Coulomb potentials and reads as:

      $ \begin{array}{l} V_{ij} ({{r}}_{i}, {{r}}_{j}) = V_{ij}^{\rm Skyrme} + V_{ij}^{\rm Yukawa} + V_{ij}^{\rm Coulomb}, \end{array} $


      $ \begin{aligned}[b] =& t_1 \delta({{r}}_{i}-{{r}}_{j})+ t_2 \delta({{r}}_{i}-{{r}}_{j}) \rho^{\gamma-1} \left(\frac{{{r}}_{i}+{{r}}_{j}}{2}\right) \\ & + t_{3}\frac{\exp(-|({{r}}_{i}-{{{r}}}_{j})|/\mu)}{(|({{{r}}}_{i}-{{{r}}}_{j})|/\mu)} + \frac{Z_{i}Z_{j}e^{2}}{|{{{r}}}_{i}-{{{r}}}_{j}|}. \end{aligned} $


      Here, ${{t}} _1 $ and ${{t}} _2 $ depends on the values of $ \alpha $ and $ \beta $ (described below), ${{t}} _{3} $ and $ \mu $ are -6.66 MeV and 1.5 fm, respectively, and $ {{Z}}_{i} $ and $ {{Z}}_{j} $ denote the charges of the $ i^{\rm th} $ and $ j^{\rm th} $ baryon. In the limit of nuclear matter, the (two- and three-body) Skyrme interactions are parameterized as:

      $ \begin{array}{l} V^{\rm Sky} = \alpha\left(\frac{\rho}{\rho}_o\right)+ \beta \left(\frac{\rho}{\rho}_o\right)^{\gamma}. \end{array} $


      The parameters $ \alpha $ and $ \beta $ are adjusted so that the average binding energy shows minimum at the normal nuclear matter density ( $ {\rho}_o $ ) and should be equal to -15.76 MeV, whereas $ \gamma $ decides the compressibility. During the course of propagation, if the two nucleons come closer than $ \surd{\sigma_{NN}/\pi} $ , where $ \sigma_{NN} $ is the nucleon-nucleon cross-section, suffer collision. Here used $ \sigma_{NN} $ is energy dependent nucleon-nucleon cross-section [40]. Before every collision nucleons are checked for the Pauli blocking i.e., if the phase space where nucleons will go after scattering is already filled that collision is neglected, otherwise, allowed. Within the framework of QMD model, for simplicity to calculate Pauli blocking, corresponding to each nucleon a sphere in coordinate and momentum space is considered. This way, the same Pauli blocking ratio can be achieved as one obtains for each Gaussian, for which calculations take much longer times. Here, we calculate the fractions P $ _1 $ and P $ _2 $ of the already filled final phase space where scattered partners will occupy. The collision is then blocked with a probability;

      $ \begin{array}{l} P_{\rm block} = 1-[1-{\rm min}(P_1, 1)][1-{\rm min}(P_2, 2)], \end{array} $


      and, the corresponding collision is allowed with the probability (1- $ P_{\rm block} $ ). If the collision is blocked, the same momenta that the nucleons have before the collision are assigned. The information of the reaction, in the form of phase space of the nucleons, is stored at various time steps during the propagation. For the nuclear matter incompressibility K = 200 MeV, the values of the parameters i.e., $ \alpha $ , $ \beta $ and $ \gamma $ are -356 MeV, 303 MeV and 1.17. With this set i.e., soft equation of state coupled with the energy-dependent nucleon-nucleon cross-section various experimental observations have been explained in previous studies [37, 38, 41-46].

    • 2.2.   Simulated annealing clusterization algorithm

    • It is well known now that as soon as the nucleons come out of the compression phase the clusterization algorithms, also known as secondary models, are evoked to obtain the fragments. Here we use simulated annealing clusterization algorithm (SACA) [35, 36]. This method utilizes the concept of energy minimization via simulated annealing technique to obtain the most bound fragment configuration. To exclude the formation of loosely bound clusters at intermediate stages of the algorithm a binding energy cut is implemented on ${{k}} ^{\rm th} $ cluster:

      $ \begin{aligned}[b] \zeta_k =& {\sum\limits_{i = 1}^{A_f}}\left[\sqrt{\left({{p}}_{i}-{{p}}^{\rm cm}_{A_f}\right)^2 + m^2_i} - m_i \right.\\&\left.+ \frac{1}{2}{\sum\limits_{j\neq i}^{A_f}}{V_{ij}({{r}}_i,{{r}}_j)} \right] < E_{\rm bind}\times A_f, \end{aligned} $


      with E $ _{\rm bind} $ = - 4.0 MeV, if A $ _{f} $ $ \geqslant $ 3 and E $ _{\rm bind} $ = 0 MeV, otherwise. In the above equation, A $ _{f} $ and $ {{p}}_{A_f}^{\rm cm} $ represent the total number of nucleons in a fragment and center-of-mass momentum of that fragment, respectively. It is worth mentioning that the potentials used in the above equation are same as are used in the QMD model to remain self consistent. It has been shown previously that the exact value of the binding energies does not affect the most bound structures of the fragments [41]. In the absence of such condition, fragments are still bound close to their true ground state binding energies as per QMD model. Within the SACA method, the total binding energy of the clusters $ \zeta_{\{C_k\}} $ for cluster set $ \{C_k\} $ is calculated at each step:

      $ \begin{array}{l} \zeta_{\{C_k\}} = \displaystyle\sum\limits_k \zeta_k. \end{array} $


      After the initial configuration is obtained, usually using the MST method, at the first stage, exchange of individual nucleons is allowed to find the possible best configuration. The new cluster configuration is always accepted if the new cluster configuration is more stable i.e., the sum of the binding energies of the clusters is greater than the previous configuration, otherwise, an exponential probability is assigned to the new configuration. In this way a large number of configurations are build. At the later stage, fragment exchange procedure is also applied so as to achieve the global most bound configuration. The iterations are terminated if the exchange of nucleons do not alter the sum of the binding energies of the clusters. After millions of iterations that cluster configuration is accepted which is most stable e.g., if we start from the ground state of the nuclei i.e., 20Ne, 40Ca, 93Nb, 208Pb, generated by QMD model and breaks them into number of fragments, the SACA finds the same nuclei with ground state binding energy after nearly 8000, 12000, 62000 and 280000 iterations, respectively [36]. The reader is referred to Ref. [36] for the detailed procedure followed in SACA.

      In earlier studies, the SACA method was found exceptionally successful in reproducing experimental data for wide entrance channels [35-38, 41, 47]. On one hand this method explains experimental observations such as multiplicity of fragments and size of the largest cluster, on the other hand it is also able to explain physics of event-by-event based observables such as multiplicity probability and probability distribution of the first three largest clusters. This algorithm not only identifies the realistic fragment structures but also realizes the fragment structure as early as $ \sim $ 60-90 fm/c where matter is still dense and hot. This time is much shorter than the one needed by the standard MST method to provide the final fragment structures ( $ \sim $ 300 fm/c). In other words, SACA also enables one to understand the reaction dynamics at the violent stages of a reaction. For more details about the SACA method, reader is referred to Refs. [35, 36].

    3.   Results and discussion
    • For the present study, several thousand events of the reactions of 40Ar+45Sc and 84Kr+197Au were generated, at different beam energies (ranging between 15 MeV/nucleon and 400 MeV/nucleon). The reactions are simulated for reduced impact parameter, $ \hat{b} \leqslant $ 0.25, this choice is guided by the previous studies [3, 15]. Here, soft equation of state supplemented by Cugnon parametrization of the nucleon-nucleon (NN) cross-section is used to simulate the above reactions [40]. It is worth mentioning that above choice of the equation of state and NN cross-section has been very successful in explaining various experimental results [37, 38, 41-46].

    • 3.1.   Analysis of charge distribution and multiplicity values of 40Ar+45Sc reaction

    • In Fig. 1, we display the charge yields (see crossed squares) calculated using the QMD model coupled with SACA method for the central collisions ( $ \hat{b} \leqslant $ 0.25) of 40Ar+45Sc at different incident energies between 15 and 115 MeV/nucleon. The choice of the centrality and incident energy range is guided by experimental measurements reported in Ref. [15]. Along with calculated yields, we also show the available experimental data (see stars) [15]. It was observed that as one increases the incident energy of the projectile the excitation energy of the composite system increases, therefore, the yield of the heavier (lighter) fragments decreases (increases). Hence, the slope of the charge yield becomes steeper with incident energy signifying the violence of the binary collisions. We see that the results of QMD+SACA model shows the same behavior and are consistent with experimental results for intermediate mass fragments (IMFs), $ 3 \leqslant Z_f \leqslant 12 $ , and calculations reported in earlier studies [3, 4, 9-11, 13-15, 38, 47]. Very encouragingly, the QMD + SACA can reproduce the measured charge yields at all incident energies very closely, except at few points at 15 MeV/nucleon. This discrepancy at 15 MeV/nucleon is due to slightly less appropriate Pauli blocking at this low incident energy [40]. In earlier studies, the charge distribution (over 3≤ $ Z_{f} $ ≤12) is often fitted with power law $ \propto $ $ Z_f^{-\tau} $ to investigate the critical energy point of the possible liquid-gas phase transition. Hence, we also fit the calculated charge yields [3≤ $ Z_{f} $ ≤12] with power law $ \propto $ $ Z_f^{-\tau} $ .

      Figure 1.  (colored online) The charge distributions obtained in the central reactions of 40Ar+45Sc at beam energies between 15 and 115 MeV/nucleon. The crossed squares show the calculated results of QMD + SACA method whereas, stars represent the experimental data [15]. The lines correspond to power law and exponential fits of fragment charge distributions for IMFs [3≤ $ Z_{f} $ ≤12] using QMD + SACA model.

      Figure 2.  (color online) (left) The extracted values of power law parameter τ, (obtained from the power law fits $ \propto $ $ Z_{f} $ $ ^{-\tau} $ of IMFs as shown in Fig. 1.) plotted as a function of incident energy. The solid lines correspond to fourth order polynomial fits over extracted τ values obtained using QMD + SACA model predicted τ values. Dashed vertical line represents the point of onset of multifragmentation with QMD+SACA model. (inset) We displayed the τ values in the incident energy range of 120 to 200 MeV/nucleon. (right) The values of multiplicities of IMFs ( $ <N_{\rm IMFs}> $ ) and charged particle multiplicity ( $ <N_{\rm c}> $ ) is shown as a function of incident energy of projectile. Data points are taken from Refs. [14, 15] and symbols have same meaning as in Fig. 1.

      Figure 3.  (color online) The extracted values of parameter $ \lambda $ , obtained using exponential fits $ \propto $ e $ ^{-\lambda Z_{f}} $ of IMFs, normalized second moment $ <S_{2}> $ , variance $ <\gamma_{2}> $ , average charge of the second largest fragment $ <Z_{\rm max2}> $ , phase separation parameter $ S_p $ and bimodality parameter P are plotted as a function of incident energy. The dotted line shows the exact energy point of onset of fragmentation.

      Figure 4.  (color online) The values of $ <Z> $ versus rank, n, in the decreasing order of of the charge of fragments for different incident energies for 40Ar+45Sc reactions. The lines denote the power law fit, $ <Z_n>\propto n^{-\xi}. $

      The above mentioned extracted values of τ are plotted in Fig. 2(a). We displayed the experimental values of τ obtained using the power law fit to charge yield of IMFs with stars and our theoretical predictions with squares. From the figure, we see that the value of τ increases with beam energy (>20 MeV/nucleon) reflecting the steepening of the slope of the charge distribution with incident energy. Experimentally, it was observed that the value ofτ changes from $ \tau \simeq $ 1.2 at 25 MeV/nucleon to $ \tau \simeq $ 4.72 at 115 MeV/nucleon. The corresponding excitation energy for this was observed to be varying from 8 to 29 MeV/nucleon. The minimum value which is considered as signal of criticality was obtained at 23.9 MeV/nucleon with $ \tau \simeq $ 1.21 via fitting the τ values with fourth order polynomial fit. We also fitted the τ values obtained using the QMD+SACA model with the fourth order polynomial fit and observed a minima in the extracted $ " $ τ" value at 20.1 MeV/nucleon with $ \tau \simeq 1.39 $ . Also, it is important to mention here that one usually expects the τ = 2.2 at the critical point but here values are much smaller than this. This deviation is due to the exclusion of lighter particles ( $ Z_f \leqslant 2 $ ) from the fitting. It was shown in Ref. [49] that τ values depends crucially on the choice of the mass/charge range which is fitted to obtain the τ values. We see that our present prediction about the minimum value for the light charged system of 40Ar+45Sc is in a close agreement with the measured one (23.9 MeV/nucleon) [15]. One also looks for $ \chi^{2} $ values, which should be minimum at critical point and large at other values. To check the goodness of fit, we also calculated $ \chi^{2} $ for the power law fit [26]. We obtained minimum value at 20.1 MeV/nucleon i.e., $ \chi^{2} \sim $ 1.1 which rises to very large values at other incident energies. This shows that best fit is obtained at critical energy point which further supports our results. Note that, this is the closest value ever reported in the literature so far by any theoretical model. On the other hand, QMD + MST predicted the minimum value to occur at 18.03 MeV/nucleon and values of τ were nowhere close to experimental values [12, 32]. Also, the Percolation model calculations predicted the energy of critical point at 28 MeV/nucleon with τ = 1.5, the values are much higher than the experimentally observed values [15]. Thus, the QMD + SACA calculations not only reproduced the measured charge yields but also explained the behavior of power law factor "τ" over the entire energy range nicely. We have also shown τ values for the QMD+SACA above incident energy of 115 MeV/nucleon upto 200 MeV/nucleon, where experimental values are not available, to see the trends (see inset of Fig. 2(a)). We find that the τ value keep on increasing with rise in incident energy. The bottleneck in the present calculations is that the SACA method recognizes the stable fragments structures as early as $ \sim $ 60-90 fm/c. Thus it can provide the vital information of the hot and dense nuclear matter.

      In Fig. 2 (b-c), we also displayed the multiplicities of intermediate mass fragments (IMFs) ( $ <{{N}}_{\rm IMFs}> $ ) and charge particle multiplicities ( $< {N}_{\rm c}> $ ) for 40Ar + 45Sc system as a function of incident energy. For these observables the experimental observations are only available for incident energies $ \geqslant $ 35 MeV/nucleon [14]. We see that the theoretical calculations are consistent for $ < {{N}}_{\rm IMF}> $ which are decrease in the multiplicity value as the incident energy is increased. But $< {{ N}}_{\rm c} >$ is predicted to be quite large compared to experimental values. This larger values of $< {{N}}_{\rm c}> $ are due to excess lighter particles. The results of $ <{{N}}_{\rm c}> $ are qualitatively reproduced by the QMD+SACA model.

      We also found that in some studies, the exponential fitting $ \propto $ e $ ^{-\lambda} $ $ ^{Z_{f}} $ is used instead of power law fit [7, 8]. The critical exponent $ \lambda $ here is also considered to show minima near critical point. To check this, we also fitted the above calculated yields (see Fig. 1, straight lines) with exponential fit of the form $ \propto $ e $ ^{-\lambda} $ $ ^{Z_{f}} $ and plotted the extracted $ \lambda $ values in Fig. 3 (a). Very interestingly, once again QMD + SACA method found a minimum in the extracted $ \lambda $ value. The exact minimum is extracted by fitting the $ \lambda $ values with fourth order polynomial fit. The minimum value, obtained at 23.1 MeV/nucleon, is very close to the one obtained using power law fit (= 20.1 MeV/nucleon) and experimental value (= 23.9 MeV/nucleon). It is first time that such close agreement between model and experimental results is obtained. It is worth mentioning this fitting is very poor and $ \chi^{2} $ in this case is very-very large (except at higher incident energies $ \geqslant $ 45 MeV/nucleon where it reduces) and this fitting is used only for the case study considering earlier studies [7, 8]. We have also shown some values of $ \lambda $ above 115 MeV/nucleon upto 200 MeV/nucleon to see trends (see inset of Fig. 3 (a)). We see same trends i.e., gradually increasing trends for $ \lambda $ with rise in incident energy.

    • 3.2.   Maximal fluctuations in 40Ar+45Sc reaction

    • At the minimum of critical exponent 'τ' the fluctuations are maximum pointing towards its possibility of being point of onset of multifragmentation or critical point of liquid-gas phase transition. Campi [21] was first to exploit this feature and introduced powerful methods to characterize the critical behavior in fragmentation. These methods are based on the conditional moments of the asymptotic cluster charge distributions. Generally, the $ k^{\rm th} $ moment of charge of cluster is defined as:

      $ \begin{array}{l} M_k = {\displaystyle\sum\limits_{{Z_{f}}\neq{Z_{max}}}} {Z_{f}^{k}}\; {n}{(Z_{f})}, \end{array} $


      where $ Z_{\rm max} $ is the charge of the largest fragment and $ n(Z_{f}) $ is defined as the multiplicity of fragment having charge $ Z_{f} $ in an event. The moments are calculated event-by-event and then averaging is done on the number of events. The normalized second moment ( $ S_{2} $ ) is defined as [21, 22]:

      $ S_2 = \frac{{\sum_{{{Z_{f}}\neq{Z_{max}}}} Z_{f}^2 \; n(Z_f)}}{{\sum_{{Z_{f}}\neq{Z_{max}}}} Z_{f}\; n(Z_{f})}, $


      and $ \gamma_{2} $ is constructed as:

      $ \gamma_{2} = \frac {M_{2}M_{0}}{M_{1}^{2}}. $


      These parameters $ <S_{2}> $ , $ <\gamma_{2}> $ and $ <Z_{\rm max2}> $ (averaged over large number of events) should exhibit a peak at incident energy where minimum in the exponent parameter τ is obtained [4, 19-22, 28].

      In Fig. 3 (b-d), we display the average values of $ <S_{2}> $ , $ <\gamma_{2}> $ and $ <Z_{\rm max2}> $ as a function of incident energy. We see that all these parameters passes through maximal value over the incident energy. Interestingly, all parameters $ <S_{2}> $ , $ <\gamma_{2}> $ , and $ <Z_{\rm max2}> $ predict a maximum at 20 MeV/nucleon, which is again very close to the earlier predicted value using τ/ $ \lambda $ i.e., 20.1 (23.1) MeV/nucleon. We also used other critical parameters i.e., phase separation parameter and bimodal parameter which we discuss below.

    • 3.3.   Phase separation parameter and bimodality in 40Ar+45Sc reaction

    • The phase separation parameter ( $ S_p = <Z_{\rm max2}>/ $ $ <Z_{\rm max}> $ ) which was introduced in the percolation model calculations suggested that it reaches a value close to 0.5 near critical point [49]. Ma et al. also utilized this parameter and found consistent results for the extraction of critical energy point in their experimental study [8]. We use this parameter and the results of QMD+SACA model is displayed in Fig. 3 (e). If we see our calculations the value of $ S_p $ exhibits a linear behavior with two different slopes above and below incident energy 20 MeV/nucleon. We see $ S_p $ = 0.52 at 20 MeV/nucleon. Therefore, the present calculations of QMD+SACA calculations indeed show characteristic signal of this parameter.

      The other parameter used is bimodal parameter which is based on bimodality i.e., observation of double peaked distribution of order parameter [30, 31]. Here each component is supposed to represent different phase and provides a definition of the order parameter that separates the two peaks. Accordingly, if the nuclear system is in the region of coexistence, the distribution of the probability of the order parameter is bimodal in nature. Borderie et al. [50] at INDRA defines the $ Z_f $ = 12 as the limit between the liquid-gas phase. They defined $ (\sum_{Z_i \geqslant 13} Z_i-\sum_{3\geqslant Z_i \leqslant 12} Z_i/\sum_{Z_i \geqslant 3}Z_i) $ as order parameter. This may be supposed to connect with the density difference between the liquid and gas phase. Ma et al. [8] changed this limit between the two phases for lighter system to $ Z_f $ = 3 ( $ Z_f \leqslant $ 3 as gas and $ Z_f \geqslant $ 4 as liquid) and defined the bimodal parameter as:

      $ P = \frac{\displaystyle\sum_{Z_i \geqslant 4} Z_i-\sum_{1\geqslant Z_i \leqslant 3} Z_i} {\displaystyle\sum_{Z_i \geqslant 1}Z_i}. $


      Where, P = 0 was the point of equal distribution of charge in liquid and gas phases. In Fig. 3 (f), we show the results for the bimodal parameter for 40Ar+45Sc reactions as a function of incident energy. We see a change in slope occurs near 20 MeV/nucleon which is the point of coexistence. Next, we analyzed the Zipf's law and information entropy [5] for characteristic signals of phase-transition.

    • 3.4.   Zipf's law in 40Ar+45Sc reaction

    • Zipf's law is well known in linguistics which is relation between the English words and their frequency used in the literature [51]. Y. G. Ma [5] introduced the Zipf's law as characteristic signal of liquid-gas phase transition in heavy-ion collisions. In heavy-ion collisions, it is utilized as following: first the clusters are arranged in the decreasing order of their charges in an event. Starting from the largest to second largest to third largest and so on. The largest one is assigned rank 1, second largest rank 2, and so on. The obtained values of size of the fragments of n $ ^{\rm th} $ rank i.e., $ <Z_n> $ are fitted with the power law of the form $ <Z_n> \propto n^{-\xi} $ , with $ \xi $ being the order parameter. At critical energy point, the Zipf's law with $ \xi $ = 1 is considered to be followed [5]. In Ref. [52] it was shown that the Zipf's law is just a consequence of power law fit and do not add to any further information. In Refs. [7, 8] the Zipf's law is followed. In Refs. [20, 39], the the Zipf's law is followed away from the critical point. It is interesting to see results of our calculations with QMD+SACA model for Zipf's law.

      In Fig. 4, we display the $ <Z> $ versus rank (n) graph at incident energies ranging between 15 to 200 MeV/nucleon. The values obtained via power law fit ( $ \xi $ ) are also shown at the corresponding incident energies. Note that, we only use the power law fit for n $ \leqslant $ 6. The obtained values of $ \xi $ are plotted in Fig. 5 (a) as a function of incident energy. We see the Zipf's law $ \xi $ = 1 is satisfied at $ \sim $ 35 MeV/nucleon. This energy is far away from the critical energy point predicted by using other order quantities. Our results with QMD+SACA model are same as were reported in Refs. [20, 39]. Therefore, within the QMD+SACA model a consistent picture is not found between other characteristic signals and Zipf's law.

      Figure 5.  (color online) (a) Energy dependence of the $ \xi $ parameter extracted from the power law fit, $ <Z_n>\propto n^{-\xi} $ , (b) values of the information entropy (H) for the 40Ar+45Sc reactions in the incident energy range of 15-200 MeV/nucleon. The Zipf's law is followed when $ \xi = 1 $ , represented by dashed horizontal line [5].

    • 3.5.   Information entropy in 40Ar+45Sc reaction

    • C. E. Shannon introduced the concept of Shannon information entropy that measures the information contained in a message that is sent along a transmission line [53]. Since its introduction, it has been applied to wide variety of problems such as nuclear/particle physics, astrophysics, life sciences, economics, engineering, and many other fields (see Ref. [33] for recent review on this). Cao and Hwa first used this concept in nuclear/particle physics to study the multi-particle production in high energy hadron collisions [54]. They defined information entropy method in whole reaction system and in event space. Y. G. Ma introduced this concept as a characteristic signal to study the liquid-gas phase transition in heavy-ion collisions. The information entropy was constructed as:

      $ H = - \displaystyle\sum\limits_i p_i\; ln (p_i), $


      with $ \displaystyle\sum_i p_i $ = 1, here, $ p_i $ is the normalized probability distribution of the total multiplicities having 'i' particles produced in a event. It should be kept into mind that the emphasis is on event space and not on phase space. The information entropy is found to show peak at the point of maximum fluctuations. The results of QMD+SACA model is shown in Fig. 5 (b). The information entropy (H) shows the peak at 20 MeV/nucleon, reflecting the largest uncertainty of the probabilities at this energy. We also see that this energy of largest value of H is consistent with the observation with other critical parameters (except Zipf's law).

      Combining all the results from Figs. 1-5, our study predicts the critical point (or onset of fragmentation) for 40Ar+45Sc to be in the band of 20-23.1 MeV/nucleon (except Zipf's law) and is in close agreement with experimentally observed value of 23.9 MeV/nucleon.

    • 3.6.   Critical behavior in 84Kr+197Au reaction

    • Next, we extended our study to the heavily charged system of 84Kr+197Au for reduced impact parameter $ \hat{b}\leqslant $ 0.25, where no minimum in the power exponent was observed experimentally [3]. In Fig. 6, we display the charge distributions (crossed squares) [3≤ $ Z_{f} $ ≤12] obtained in the highly charged reaction of 84Kr+197Au at six different incident energies in the range of 35 and 400 MeV/nucleon. The available experimental data [3] for the same reaction is also displayed (see stars). Note that QMD+SACA is able to reproduce the experimental measurements for IMFs in most of the cases.

      Figure 6.  (color online) The charge distributions obtained from the central reactions of 84Kr+197Au at six different beam energies between 35 and 400 MeV/nucleon. Solid lines represent the power law fitting of IMFs [3≤ $ Z_f $ ≤12] obtained using QMD + SACA model.

      Further, we again fit the charge yields at each incident energy with power law fits $ \propto $ $ Z_f^{-\tau} $ and the extracted values of power law factor τ are plotted against incident energy in Fig. 7. From the figure, one notices that the extracted values of τ increase monotonically with incident energy without passing through minimum value. This absence of minimum in τ has been advocated to be due to dominance of long range Coulomb forces in highly charged systems [3, 11, 26]. In the figure, we also display the results of previous calculations (represented by different lines) reported using Statistical Multifragmentation Model (SMM) with and without sequential decay [3]. In the present study, we find that the calculations using QMD+SACA give τ values that are close to the experimentally measured ones as well as to the SMM calculations with sequential decay [3]. In the inset, we also displayed the results of τ down to the incident energy of 15 MeV/nucleon and no minima in the extracted values of τ is seen. We have also analyzed the other critical parameters $ \lambda $ , $ <S_{2}> $ , $ <\gamma_{2}> $ , $ <Z_{\rm max2}> $ , P, $ S_p $ , H and Zipf's law for this highly charged system (results are not shown here) and no characteristic signal of liquid-gas phase transition is observed. Indicating the absence of liquid-gas phase-transition signals due to Coulomb forces in this highly charged system.

      Figure 7.  (color online) Extracted values of the power law factor τ, obtained from the power law fits of IMFs [3≤ $ Z_{f} $ ≤12] for the central reactions of 84Kr+197Au as shown in Fig. 6. Symbols have same meaning as in Fig. 1. Here, different lines represent the Statistical Multifragmentation Model (SMM) calculations with and without sequential decay [3].

      The consistency of the QMD+SACA approach to reproduce experimental data for light and heavily charged systems gives us faith that it can provide much reliable information about the critical point of liquid-gas phase transition or the point of onset of multifragmentation. It is worth mentioning that SACA method does not have free parameters as in other calculations. Moreover, the fragments can be realized as early as $ \sim $ 60-90 fm/c when nuclear matter is still hot and dense. Also, the QMD+SACA model gives consistent results with all the characteristic signals of liquid-gas phase-transition, except the Zipf's law which gives higher value of critical energy for lighter system. Therefore, the present study suggests that Zipf's law is not suitable to predict liquid-gas phase transition [20, 39]. Further, this is the first ever consistent calculation which is in accordance with the onset of fragmentation in lighter system and subsequent absence of such trends in highly charged system. Recently, Lin et al. [20] used SMM model, and studied various critical signals for primary and secondary fragments. It was found that few signals may give different results for primary and secondary fragments. If we see present results, where SACA gives realistic fragments, all signals except Zipf's law shows characteristic signals for lighter system. Therefore, no additional statistical decay codes are indeed required. Considering the consistency of QMD+SACA model one can use this model to guide the experiments to study the possible liquid-gas phase transition in nuclear matter.

    4.   Summary
    • In this article, we investigated the charge yield of fragments and its connection with the liquid-gas phase transition (i.e., the onset of multifragmentation) in light and heavily charged systems of 40Ar+45Sc and 84Kr+197Au, respectively. We have also used various different parameters and found that QMD+SACA calculations are consistent with the experimental measurements. Using different critical parameters, we obtained a critical point of liquid-gas phase transition or the point of onset of multifragmentation around 20-23.1 MeV/nucleon for 40Ar+45Sc system (except for Zipf's law which predicts critical energy $ \sim $ 35 MeV/nucleon) which is close to the experimentally observed value of 23.9 MeV/nucleon. No such critical point of liquid-gas phase transition (or onset of multifragmentation) is observed for 84Kr+197Au in agreement with experimental findings and other theoretical calculations. We feel that the existing model is highly suggestive of the critical behavior in lighter colliding system, although more work needs to be done to establish this more firmly. The present model, we believe is one step forward from the QMD model coupled with the MST or its variants, and is also very helpful for the analysis of experimental data for light and heavily charged systems. We believe that the QMD+SACA model can be very helpful in guiding the future experiments.

Reference (54)



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