On the elliptical flow and asymmetry of the colliding nuclei

A study of elliptical flow is presented with respect to the asymmetry of colliding nuclei using the reactions of 24Cr50 +44 Ru102, 16S32 +50 Sn120 and 8O16 +54 Xe136 at incident energies between 50 and 250 MeV/nucleon within the framework of isospin-dependent quantum molecular dynamics model. For the present analysis, total mass of the colliding pairs is kept fixed and asymmetry is varied as ? = 0.2, 0.3, 0.7. The elliptical flow shows a transition from in-plane to out-of-plane in the mid rapidity region with incident energy. The transition energy is found to increase with the asymmetry for lighter fragments.


I. INTRODUCTION
Large efforts are going on to understand the nuclear matter at the extreme conditions of temperature and density and also to explore the role of symmetry energy under these extreme conditions [1]. One of the most sought after phenomena in this direction is the collective flow and its various forms [2][3][4][5]. For the last few years, collective flow has been used as a powerful useful tool to explore the nuclear equation of state (EOS) as well as in-medium nucleon-nucleon cross-sections [1,6]. Collective flow is a motion characterized by the space-momentum correlations of dynamic origin. Following two different signatures of collective flow have been predicted: a) bounce-off of compressed matter in the reaction plane [7] and b) squeeze-out of the participant matter out of the reaction plane [8].
Such observables together represent the anisotropic part of the transverse flow that appears in the non-central heavy-ion collisions only. The highly stopped and compressed nuclear matter around the mid-rapidity region is seen directly in the squeeze out [9], also known as elliptical flow. The elliptical flow has been proven to be one of the most fruitful probes to study the dynamics of heavy-ion collisions. The elliptical flow describes the eccentricity of an ellipse-like distribution. Quantitatively, it is the difference between the minor and major axis. The orientation of the major axis is confined to the azimuthal angle φ or φ + π 2 for an ellipse-like distribution. The major axis lies within the reaction plane for φ; whereas φ + π 2 indicates that the orientation of the ellipse is perpendicular to the reaction plane (i.e., squeeze-out flow). The parameter of the elliptical flow is quantified by the second order Fourier coefficient < v 2 > = (< p 2 x −p 2 y p 2 x +p 2 y >, p x and p y being the x and y components of the momentum respectively), from the azimuthal distribution of detected particles at mid-rapidity [10]: where φ is the azimuthal angle between the transverse momenta of the particles and reaction plane. The positive values of elliptical flow reflect an in-plane emission, whereas out-off plane emission is reflected by the negative values. The parameters < Cos2φ > of elliptical flow depend on the complex interplay between the expansion, rotation and shadowing of the spectators, apart from the incident energy. It is worth mentioning that both the mean field 2 and binary nucleon-nucleon collision parts play an important role at intermediate energies.
The mean field potential plays a dominant role at low incident energies, which is gradually taken over by the two body collision part at higher incident energies. Therefore, a detailed study on the excitation function of elliptical flow in this energy range can provide useful information about the nucleon-nucleon interactions and origin of the isospin effects in heavyion collisions.
It is worth mentioning that the outcome of a reaction depends also on the asymmetry of the reaction. Unfortunately, very few investigations focus on this aspect. The asymmetry of a reaction can be defined by the asymmetry parameter η = (A T − A P )/(A T + A P ); [11] where A T and A P are the masses of the target and projectile, respectively. The η = 0 corresponds to the symmetric reactions, whereas non-zero values of η define different asymmetries of a reaction. As noted by FOPI group [12][13][14][15][16][17], the reaction dynamics in a symmetric reaction (η = 0) can be quite different compared to an asymmetric reaction (η = 0). This is valid both at low and intermediate energies. This difference emerges due to the different deposition of the excitation energy ('in form of compressional and thermal energies) in symmetric and asymmetric reactions. Though the systematic role of asymmetry has been explored in multifragmentation, no such study yet exists in the literature for elliptical flow [18]. We plan to address this in present paper.
We plan to understand how the elliptical flow is affected by the asymmetry of a reaction.
This study is performed within the isospin-dependent quantum molecular dynamics (IQMD) model discussed in section II. Our results are presented in section III. Finally, we summarize the results in section IV.

II. THE MODEL
The isospin-dependent quantum molecular dynamics (IQMD) [17][18][19]  In this model, baryons are represented by Gaussian-shaped density distributions (1) where L is the Gaussian width which is taken to be 2.16 f m 2 . In ref [19], this Gaussian width is found to be dependent on the size of the system. Nucleons are initialized in a sphere with radius R = 1. This helps in achieving the correct distribution of protons and neutrons within the nucleus.
The hadrons propagate using Hamilton equations of motion: with < H >=< T > + < V > is the Hamiltonian, which is written as: The baryon-baryon potential V ij , in the above relation, reads as  [20]. Note that the Gaussian shaped behavior is integrated over the entire rapidity range. One also sees that elliptical flow of nucleons/LCP's is positive over entire P t range. One also notices that for larger asymmetries (e.g. η = 0. elliptical flow decreases indicating that not enough particles occupy such higher P t values. One also notices lower peaks of Gaussian for heavier fragments. The reason for this shift is that the emitted free nucleons can feel the role of mean-field directly, while the LCP's have a weaker sensitivity [21]. Interestingly, < v 2 > depends significantly on the symmetry of the reaction also. Further, we see that neutron-rich system ( 8 O 16 + 54 Xe 136 with N/Z = 1.4) exhibits a weaker squeeze-out flow compared to other reactions. These findings are in agreement with the one reported by Zhang et al., [22]. Moreover, the N/Z effect is more pronounced at E = 50 MeV/nucleon. This also indicates that in addition to the mass asymmetry, the mean field effects such as isospin effects are also responsible for different elliptical flow.
In Fig. 2 where fragments were found to exhibit similar trends.
Since the change of the asymmetry of a reaction can affect the elliptical flow via symmetry energy, we display in Fig. 3, the transverse momentum dependence of elliptical flow for LCP's in the mid-rapidity region for the reaction of 24 Cr 50 + 44 Ru 102 with and without symmetry energy The effect of symmetry energy is clearly visible. This is in agreement with the findings of Chen et al., [23], where it was concluded that the production of LCP's act as a probe for symmetry energy.
In Fig. 4, we display the variation of excitation function < v 2 > for LCP's as a function of incident energy for entire rapidity region and for mid rapidity region −0.1 ≤ Y red ≤ 0.1 only.
The general behavior of excitation functions for various asymmetries is quite similar. The microscopic behavior, however, depends on the asymmetry of the reaction. Interestingly, no transition in the elliptical flow occurs when entire rapidity region is considered. In contrast, a transition from the preferential in-plane (rotational like) emission (v 2 > 0) to out-of-plane (squeeze-out) emission (v 2 < 0) occurs at mid rapidity zone. This happens due to the fact that the contribution of spectator matter increases with rapidity region, leading to less squeeze-out of the particles in entire rapidity region. On the other hand, the contribution of the participant zone dominates the reaction in midrapidity region leading to the transition from in-plane to out-of-plane. This happens because the mean field which contributes to the formation of a rotating compound system becomes less important and collective expansion process (based on the nucleon-nucleon binary scattering) starts to be predominant [24]. The competition between the mean-field and nucleon-nucleon collisions should strongly depend on the effective interactions, which leads to the divergence of the transition energies calculated by varying the asymmetry of a reaction. In other words, participant zone is primarily responsible for the transition from in-plane to out-of-plane.
The energy at which this transition is observed is dubbed as the transition energy E trans .
That is why, LCP's, which originate from the participant zone, show a clear and systematic transition with the beam energy as well as with the asymmetry of a reaction. One should note that transition energies increases with the asymmetry of a reaction.

IV. CONCLUSION
In the present study, elliptical flow is studied for different asymmetries leading to same