1 Introduction

The characterization of collective flow has proven to be one of the more powerful probes of the dynamics in heavy in collisions. Elliptic flow is an excellent collective flow observable which has been studied extensively at Bevalac and SIS as well as AGS, SPS, and RHIC^[1]. In intermediate energy, microscopic transport model calculations have stressed the importance of elliptic flow for extracting the equation of state (EOS) of nuclear matter.

The elliptic flow is mathematically defined as the second coefficient of the Fourier expansion of particle azimuthal distribution with respect to the reaction plane. The origin of the elliptic flow is the initial geometry anisotropy of the collision system which originates from the uneven distribution of the density in the early time of the collisions. Subsequent dynamical evolution of the system transforms the anisotropy from the coordinate space to the momentum space, which leads to the collective motion observed in the final state. Based on this physical picture, it can be imagined that the elliptic flow should be very sensitive to the initial state.

Recently, extensive studies of the initial fluctuation effect on anisotropic flow in relativistic nuclear collisions have been made. The initial fluctuation can be illustrated as the event-by-event fluctuation in the shape of the overlap region created in initial collisions. The initial eccentricity (ε₂), which is a parameter used for quantifying the initial spatial anisotropy, is affected by the initial spatial anisotropy.

In high energy collisions, it has already been predicted by model calculations that the anisotropic flow considering the initial fluctuation is larger^[2]. Han et al. shows that the ratio of the elliptic flow to initial eccentricity (v₂/ε₂) is sensitive to the initial fluctuation^[2]. The value of v₂/ε₂ stands for the conversion from the initial geometry anisotropy to the final momentum anisotropy. Many studies prove that the hydrodynamic calculations on flow are in good agreement with the experimental data, when the initial fluctuation is taken into account^{[3,4,5,6,7,8,9,10]}. Furthermore, the initial fluctuation makes higher odd-order harmonic flow nontrivial. Previous studies have shown that the triangle flow is non-negligible, after the initial fluctuation is taken into account^[11,12].

However, the initial fluctuation effect on flow has only been investigated in high energy collisions so far. In the intermediate energy domain, its effect is not addressed yet.

In this article, the initial fluctuation effect on the collective flow will be demonstrated in the intermediate energy collisions. A transport model, named as an Isospin Quantum Molecular Dynamics (IQMD), which allows the generation of events with event-by-event fluctuating initial conditions, is used. We will also compare the flow results using two different methods. Finally, the centrality dependence of the eccentricity ε₂ and the ratio of ν₂/ε₂ will be presented.

2 Analysis method

In the collisions, the particle azimuthal distribution relative to the reaction plane is not isotropic and is usually expanded in the Fourier series^[13]:

where the v_n=<cos[n(φ-ψ_RP)]> coefficients are normally referred to n-th collective flow or anisotropic flow^[14], and the angle bracket means an average over all particles in all events. ψ_RP is the reaction plane angle. The reaction plane is spanned by the vector of the impact parameter and the beam direction. It can not be measured experimentally, but can be estimated in several ways.

2.1 Initial fluctuation effect on the reaction plane

The original reaction plane is XZ plane in the model where x_RP represents for X-axis, however, the initial fluctuation has effects on the reaction plane as illustrated in Fig.1.

The effect of the initial fluctuation on the reaction plane is that the initial fluctuation makes the participant plane (x_PP) deviate from the reaction plane (x_RP)^[15]. The participant plane angle is defined as:

${\psi }_n\left\{PP\right\}=\frac{1}{n}[arctan\frac{<r^2\sin (n\varphi )>}{<r^2\cos (n\varphi )>}+\pi ]$ (2)

where the r and φ are the coordinate position and azimuthal angle of each particle and $<\cdot \cdot \cdot >$ is the participant density weighted in the initial state.

The n-th order eccentricity calculated with respect to this participant plane is defined as:

${\epsilon }_n=\frac{\sqrt{<r^2\cos (n\varphi ){>}^2+<r^2\sin (n\varphi ){>}^2}}{<r^2>}$ (3)

When n=2, the above two variables ν₂ and ε₂, correspond to elliptic flow and eccentricity, respectively. In high energy heavy-ion collisions, it is found that the eccentricity taking the initial fluctuation into account is different from the eccentricity without the consideration of initial fluctuation^[16].

Fig.1Full-screen View

Reaction plane and participant plane coordinate systems.

3 Results and discussion

Figure 2 shows the y₀ and u_t0 dependencies of the elliptic flow measured with respect to the reaction plane ({RP}), participant plane ({PP}) and event plane ({EP}), where y₀=y/y_p is the scaled rapidity and u_t0=u_t/u_p is the scaled transverse 4-velocity. Please note that in the v₂ calculation, we have applied the detector geometry cut of the FOPI detector so that a quantitative comparison can be made^[24].

Fig.2Full-screen View

Comparison for v₂{RP} (red circle), v₂{PP} (blue triangle) and v₂{EP} (green diamond) in Au+Au collisions at E=1 GeV/A and centrality 0.25<b₀<0.45 by the IQMD model with a parameter set of Soft and Momentum-dependent equation of state (SM). The black stars are the experimental data from the FOPI Collaboration. The panel (a) plots y₀ dependence where v₂ is integrated over u_t0, but constrained to u_t0>0.4. The panel (b) shows the u_t0 dependence in the indicated y₀ bin.

The rapidity dependence of Fig.2(a) shows a V-shape. The proton shows an in-plane emission (v₂>0) in project-like and target-like regions (larger rapidity value in absolute value) while it shows a squeeze-out emission in the overlapping region (mid-rapidity). From the mid-rapidity to the projectile-like/target-like rapidities, the proton favors the squeeze-out emission to the in-plane emission. This is consistent with the shadowing effect.

The transverse velocity dependence (Fig.2(b)) shows that with the increase of u_t0, the squeeze-out emission is stronger, i.e. the proton with higher velocity can be more easily escaped from overlapping zone.

From Fig.2, we can see that the value of v₂{RP} is a little larger than v₂{PP}. However, there is almost no discrepancy between v₂{EP} and v₂{PP}. That means v₂ becomes weak by the initial fluctuation but is not sensitive to the dynamical evolution. This phenomenon is different from what is known at high energies, where v₂ is both enlarged by the initial fluctuation and the dynamical evolution. In the figure, the FOPI data is also plotted in order to check our model calculations. Even though quantitative agreements of various calculations are not reached, the trend of v₂ as functions of y₀ and u_t0 is consistent with the data.

Fig.3Full-screen View

v₂ of protons in Au+Au collisions for different centrality ranges. In the upper three panels, v₂ is calculated for transverse 4-velocities u_t0>0.4. In the lower three panels, v₂ is calculated for the rapidity |y₀|<0.4.

Fig. 4Full-screen View

ε₂ and v₂/ε₂ as a function of impact parameter for Au+Au collisions at E=1GeV/A.

Figure 3 displays that v₂ versus y₀ and u_t0 in three different centralities. While the shapes look similar in all centralities, the elliptic flow of protons in mid-rapidity (upper panels) or high velocity (lower panels) regions shows stronger squeeze-out effect at some intermediate centralities (0.45<b₀<0.55). With the increase of b₀, the initial fluctuation and dynamical evolution also play an increasing role as seen from the behavior of increasing difference between v₂{PP}, v₂{RP} and v₂{EP}. Again, the similarity between v₂{PP} and v₂{EP} indicates that there is almost no effect by the evolution of dynamics.

The upper panel of Fig.4 shows the impact parameter (b) dependence of eccentricity ε_2. It shows that ε₂ increases with b and the ε₂{PP} is larger than ε₂{EP}. The lower panel shows the ratio v₂/ε₂ as a function of b. We can see the absolute values of the ratio also increase with b and |v₂{PP}/ε₂ | is smaller than |v₂{RP}/ε₂|. The smaller the absolute ratio is, the less efficient the conversion from the initial geometry anisotropy to the final momentum anisotropy is.

4 Conclusion

The elliptic flow of Au+Au collision at 1 GeV/A by three different methods for the determination of reaction plane was studied. And it was found that the initial event-by-event geometry fluctuation had effects on the final-state elliptic flow. In the present calculation, the initial fluctuation weakens the elliptic flow, in other words the initial fluctuation makes the conversion from initial spatial eccentricity to final momentum smaller. In this paper, we studied the dependence of ε₂ on impact parameter with the consideration of the initial fluctuation and compared the ratio v₂/ε₂ with or without the consideration of the initial fluctuation. It shows that the initial fluctuation enhances the eccentricity ε₂ but it decreases the ratio of |v₂/ε₂|. Moreover, we also found that the dynamical evolution has little influence on the elliptic flow in our studied cases. Our simulation gives a similar trend for rapidity and velocity dependencies, but the results are quantitatively smaller than the experimental data for mid-rapidity and high velocity protons, which may be caused by two reasons. One is the variation of physical parameters (such as the ground state densities, interaction ranges) whose precise values are not known; the other is the consequence of the impossibility to build a ground state nucleus with all its detailed structure in a semi-classical molecular dynamics approach^[25].