Correlation and isospin dynamics of participant-spectator matter in neutron-rich colliding nuclei at 50 MeV/nucleon
The sensitivities of isospin asymmetry and collision geometry dependences of participant (overlapping region)-spectator (quasi-projectile and quasi-target region) matter towards the symmetry energy using the Isospin Quantum Molecular Dynamical model are explored. Particularly, the difference of number of nucleons in overlapping zone to quasi projectile-target matter is found to be quite sensitive towards the symmetry energy at semi-peripheral geometries compared to the individual yield. It gives us a clue of this quantity to act as a measure for isospin migration. Further, the yield of neutrons (charge of the second largest fragment) are provided as a tool for overlapping region (quasi projectile-target) matter to check the sensitivity of above mentioned observable towards the symmetry energy experimentally.
pacs:21.65.Ef, 25.70.Pq, 25.70.-z
The primary goal of isospin physics is to obtain a better understanding of properties of asymmetric nuclear matter or in other words isospin dependence of Nuclear Equation of State (NEOS). Over three decades, the NEOS of symmetric nuclear matter is well understood by the study of giant dipole resonances, collective flow as well as multifragmentation etc. (1); (2); (3); (4); (5). The compressibility , which describes so called stiffness of the symmetric nuclear matter has been determined to 235 14 MeV. The NEOS of isospin asymmetric nuclear matter is recently underway, particularly, the density dependence of symmetry energy, which is critically important also in many astrophysical processes, such as, structural and dynamical evolution of neutron star, the critical density for the direct cooling process etc.(6).
Considerable progress has been made in determining the sub- and supra-saturation density behavior of the symmetry energy (7); (8); (9); (10); (11); (12); (13); (14); (15); (16). The later part is still an unanswered question in spite of recent findings in term of neutron-proton elliptic flow ratio and difference (13); (14). However, The former one is understood to some extent (7); (8); (9); (10); (11), although, more efforts are needed for precise measurements. According to Symmetry Energy project (SEP) started from Michigan State University (MSU) at National Superconducting Cyclotron laboratory (NSCL) in 2011 (includes European as well as Asian countries), the sub-saturation density dependence of symmetry energy still needs more study to verify the previously defined observables such as double ratio, isospin diffusion and then extend the study to define new observables to parametrize it (symmetry energy) more accurately, which has still long range in the literature from soft to linear one (7); (9); (10). Here we will try to search for new observable whose importance in intermediate energy heavy ion collisions is discussed below.
Recently, the role of isospin degree of freedom has been investigated using collective flow and its balance energy(at which flow disappears) (16). The collective flow is proven as an indicator for the symmetry energy. At balance energy, the attractive interactions due to mean field are balanced by the repulsive interactions due to the nucleon-nucleon (NN) collisions. This counterbalancing is reflected in quantities like participant-spectator matter (17). Recently, the participant-spectator matter at balance energy is found to be quite insensitive towards the mass and of the colliding system and hence can act as a barometer for the study of vanishing flow (17); (18). The inter-dependence of collective flow with symmetry energy and collective flow with participant-spectator matter gives us a clue to check the sensitivity of participant-spectator matter towards the symmetry energy in asymmetric colliding nuclei.
Secondly, the elliptic flow is also shaped by the interplay of collisions and mean field (19). In addition, the elliptic flow pattern of the participant matter is affected by the presence of cold spectator matter (20). Specially, the spectator can inhibit the collective transverse expansion of the decompressing participant matter and effectively shadow particle emission directed towards the reaction plane. This study is indicating that elliptical flow is influenced strongly by participant-spectator matter distributions. As we have discussed earlier, the elliptic flow ratio and difference from neutrons and protons was used as an indicator for the symmetry energy (13); (14). It is the another insight/clue to check the sensitivity of participant-spectator matter towards the symmetry energy.
Apart from collective transverse and elliptic flow, the participant-spectator matter also plays an important role in understanding multifragmentation as well as nuclear stopping. In recent years, the correlation between nuclear stopping and light charged particles (LCPs) is investigated by using Quantum Molecular Dynamics (QMD) and Isospin-QMD (IQMD) (21). The relation is established between nuclear stopping, directed flow and elliptic flow (22). Further, the participant matter is declared as an indicator to nuclear stopping (17). All the correlation studies indicate some indirect correlation of participant-spectator matter with different kind of fragments. So, it also becomes our prime duty to correlate the participant-spectator matter with different kind of fragments and then provide specific type of fragments as an experimental measure for participant-spectator matter.
In the literature, mostly participant-spectator terminology is used, which is most suitable at higher incident energies, where the dynamics is almost fully accounted by NN collisions. In the energy region of present manuscript, the reaction proceeds through the interplay of both NN collisions and mean field. We prefer to use quasi-projectile (QP) or quasi-target (QT) for spectator matter and overlap region (OR) for the participant matter.
In the present work, IQMD model is used, which is discussed in detail in our recent publications (11); (23), originally developed by Hartnack and collaborators (24). The model is modified by the authors for the density dependence of symmetry energy, having form:
with the parameters of = 25 MeV and = 35.2 MeV. When we set and 1.5, respectively, it corresponds to the soft and stiff symmetry energy (11).
Ii Results and Discussion
The present OR and QP+QT matter demonstration is based on the fireball model (25) as reported in Ref.(17). All nucleons having experienced at least one collision are supposed to originate from OR matter (labeled as ). The remaining matter is called QP,QT matter (labeled as ). The are the total number of nucleons in the reaction system. This concept gives similar results as has been demonstrated in the fireball model of Gosset et al. (25) and further verified by QMD in last couple of years to till date (17); (18). There is another way to define the OR, QP and QT matter on the basis of different rapidity cuts (17); (8). However, both these definitions have been reported to give the same results (17); (8). In the present study, we shall use the first definition to construct the OR and QP+QT matter. During the expansion stage, this definition will lead to the production of two matters with different densities. The OR must have relatively low density compared to QP,QT matter. This density gradient must increase, when one goes towards the peripheral collisions. Due to the density gradient, the transfer of particles from high to low density region is found to relate with the phenomenon known as isospin migration (26).
Mathematically, isospin migration can be understood as: , with the mass coefficients, which are directly given by the variation of chemical potentials with respect to density and asymmetry. For more detail see Ref. (26). From the above equation, it is clear that isospin migration depends on the slope of the symmetry energy, or the symmetry pressure. When one moves towards the semi-peripheral collisions, QP or QT of about normal density are in contact with the OR where the density is quite low than saturation density. In this region of density, a stiff symmetry energy has a smaller value but a larger slope in comparison with a soft symmetry energy. In brief, this definition can help us to find the observable for isospin migration.
ii.1 Theoretical probe for isospin migration
In the present study, thousand of events are simulated using static soft equation of state and energy dependent NN cross-section for the isotopes of Sn, namely Sn + Sn, Sn + Sn and Sn + Sn at incident energy 50 MeV/nucleon along the whole collision geometry.
Fig. 1 displays the impact parameter (b) and isospin asymmetry of the system () dependences of OR and QP+QT including the neutrons and protons contribution. With the increase of impact parameter, the number of nucleons in OR (QP+QT) matter decreases (increases). The similar behavior is followed by the impact parameter dependence of neutrons and protons. In contrast, there exists more sensitivity for the number difference between the neutrons and protons near the central (peripheral) geometries from OR (QP+QT) matter, which is due to the dominance of more NN collisions (very less collisions) near the respective collision geometries.
In semi-peripheral collisions, the isospin asymmetry dependence of OR and QP+QT matter along with neutrons and protons contribution have dramatic behavior. With the increase in isospin asymmetry, the number of nucleons in OR (QP+QT) matter increases (decreases). Furthermore, the neutrons (protons) contribution from OR as well as QP+QT matter increases (decreases). Interestingly, the sensitivity is stronger for QP+QT matter at higher isospin asymmetry. This is due to the fact that (1) With isospin asymmetry, there emerges a sharp increase in neutron content inside QP+QT about 4-5 times compared to the decrease of QP+QT matter (see the red circles vs black circles in right bottom panel); (2) As the proton content is constant for all the isotopes of Sn and while the QP+QT matter is decreasing, it will eventually lead to the decrease in protons content in the QP+QT matter.
Lastly, the effect of symmetry energy is relatively weak on the impact parameter and isospin asymmetry dependences on the nucleons originated from OR and QP+QT matter. This difference is true as the error bars are less than the size of the symbols. The soft (stiff) symmetry energy contributes more for OR (QP+QT) matter. This is due to the gradient in densities of two matters, which can be explained by slope of symmetry energy rather than magnitude. The slope (magnitude) of symmetry energy changes its behavior below (at) the saturation density for the soft and stiff symmetry energy. Due to the large slope (less magnitude) for the stiff symmetry energy even below the saturation density, the QP+QT is more neutron-rich with the stiff symmetry energy, which is true with the soft symmetry energy for OR matter as the density at freeze-out time in comparison to QP+QT is very low.
From the above discussions, it is clear that number of nucleons contribution from OR and QP+QT matter is a good candidate to explore the isospin physics, but not a potential candidate for symmetry energy (due to weak dependence). However, the density gradient of symmetry energy gives us some preliminary clues of isospin migration between the nucleons of OR and QP+QT matter.
In order to reveal the effect of symmetry energy in term of isospin migration, in Fig. 2, we display the difference of nucleon number between OR and QP+QT () as a function of kinetic energy of nucleons at different impact parameter and for two neutron-rich systems. In central collisions, no effect of symmetry energy is observed. However, with the increase of impact parameter, sensitivity of symmetry energy towards the observable seems increasing. This indicates that the effect of symmetry energy on the observable under study depends weakly on the isospin asymmetry, but strongly on the density gradient. As discussed earlier, the density gradient increases strongly (weakly) with the impact parameter (isospin asymmetry). The effect of symmetry energy on the observable () mainly originates from the density difference and not due to the isospin asymmetry. We further know, the density difference is a direct measure of the isospin migration. The findings are also supporting the results of the Ref. (26).
From the study, we can conclude that the difference of number of nucleons from OR to QP+QT can act as a probe to the slope of symmetry energy versus the density in term of isospin migration at sub-saturation densities.
ii.2 Correlation with fragments and observables for experiments
The second aspect of this paper is to correlate the number of nucleons in OR and QP+QT matter with fragmentation process and then provide some observables to test the above observable as a sensitive probe for symmetry energy.
To this end, we displayed the impact parameter and isospin asymmetry dependences of different kind of fragments in Fig. 3, which is just like in Fig. 1. The multiplicity of different kind of light charged particles goes on decreasing with the increase of impact parameter as well as size of the fragments. This type of the behavior has been already observed many times in the literatures (21). On the other hand, (charge of the second largest fragment normalized by the mass number of projectile) and (charge of the largest fragment normalized by the mass number of projectile) increase with impact parameter. These two L. H.S. panels are behaving exactly the similar just like the L. H.S. panels in Fig. 1, indicating some clues of the correlation for the measure of number of nucleons in OR(QP+QT) using the light charged particles (charge of the heavier fragments).
To further strengthen the correlation, the R.H.S. of Fig. 3 give us some interesting features. Just like the isospin asymmetry dependence of nucleons in OR (QP+QT) matter, the multiplicity of free nucleons (charge of the heavier fragments) is increasing (decreasing). Moreover, with the isospin asymmetry, the increasing (decreasing) trend of neutrons (protons) contribution in OR matter (Fig.1) is similar to the behavior of the multiplicity of neutrons (protons) (Fig.3). In addition, the isospin asymmetry dependence of charges of heavier fragments is also similar to the contribution of protons from the QP+QT (Fig. 1). These are the strong evidences for correlation. The important point to check here is the sensitivity of yield of different kind of fragments and charge of the heavier fragments with nucleons of OR and QP+QT matter, which will be further checked in Fig. 4 by using the power-law fitting method.
From the light charged particles, interesting isospin effects are observed with isospin asymmetry of the system. With the increase in the one neutron in the LCPs, i.e. from HHH and HeHe, the multiplicity changes its trend from decreasing H, He to increasing for H, He, respectively. However, with the increase of one proton from H to He, the multiplicity reveals a sharp change from increasing to decreasing. The neutrons-protons distributions within the LCPs is also satisfying the neutrons-protons distributions from the OR zone. This indicates the strong correlation between LCPs and OR matter, but the point of interest is which type of LCPs are perfect indicator for OR matter which will be discussed in Fig.4.
In Fig. 4, the sensitivity of number of nucleons in OR and QP+QT matter with the multiplicity of LCPs as well as charge of the heavier fragments is checked by fitting the power law of the form . The positive correlation is observed between OR (QP+QT) nucleons and multiplicity of LCPs (). Interestingly, from the power-law slope, it is found that although, the multiplicity decreases with the size of light fragments, the slope parameter or sensitivity increases with the fragment size (shown in Table 1). This type of prediction is also true for the second largest fragment and the largest fragment for the QP+QT nucleons. The slope parameter for neutrons (0.6) with nucleons of OR matter (also shown in Table 1) and for with nucleons of QP+QT matter (0.58) (Fig.4) is almost the same, which reveals the similar sensitivity of the respective matter towards the respective fragments. From here, one can say that if one uses the neutron as a measure for the nucleons of OR matter and as a nucleons of QP+QT matter, then the difference of two parameters at semi-peripheral geometry, just like in Fig. 2, can act as a probe for the isospin migration or slope of symmetry energy. From the Table 1, we also observed that there is a linear correlation between the nucleons of OR matter and He fragments (i.e., having very close to one), which indicates that the He can be taken as a direct measure for the nucleons of OR matter.
In summary, we tried to find an observable to measure density dependence of symmetry energy or its slope by studying the OR and QP+QT matter in intermediate energy heavy-ion collisions. The difference in number of nucleons of OR to QP+QT matter is particularly sensitive towards the slope of symmetry energy at semi-peripheral geometries. This gives us a clue that this observable can act as probe for the isospin migration. Principally, the yield of neutrons and charge of the second largest fragment () could be provided as experimental tools to check the sensitivity of nucleons in OR and QP+QT matter towards the symmetry energy in term of isospin migration. Interestingly, the nucleons of OR matter has a linear correlation with the yield of He ( close to 1).
This work is supported in part by the Chinese Academy of Sciences Fellowships for young international scientists under the Grant No. 2010Y2JB02, and by NSFC of China under contract No.s 11035009, 10979074, the Knowledge Innovation Project of CAS under Grant No. KJCX2-EW-N01.
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