⇦Chinese Physics Letters, 2017, Vol. 34, No. 9, Article code 092101 Collective Flows of $^{16}$O+$^{16}$O Collisions with $\alpha$-Clustering Configurations * Chen-Chen Guo(郭琛琛)1, Wan-Bing He(何万兵)2, Yu-Gang Ma(马余刚)1,3,4** Affiliations 1Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800 2Institute of Modern Physics, Fudan University, Shanghai 200433 3University of Chinese Academy of Sciences, Beijing 100049 4School of Physical Science and Technology, ShanghaiTech University, Shanghai 200031 Received 9 June 2017 *Supported by the National Natural Science Foundation of China under Grant Nos 11421505, 11220101005, 11305239 and 11605270, the Major State Basic Research Development Program of China under Grant No 2014CB845401, the Key Research Program of Frontier Sciences of the Chinese Academy of Sciences under Grant No QYZDJ-SSW-SLH002, and the China Postdoctoral Science Foundation under Grant No 2016M591730.
**Corresponding author. Email: ygma@sinap.ac.cn Citation Text: Guo C C, He W B and Ma Y G 2017 Chin. Phys. Lett. 34 092101 Abstract The main purpose of the present work is to discuss whether or not the collective flows in heavy-ion collision at the Fermi energy can be taken as a tool to investigate the cluster configuration in light nuclei. In practice, within an extended quantum molecular dynamics model, four $\alpha$-clustering (linear chain, kite, square and tetrahedron) configurations of $^{16}$O are employed in the initialization, $^{16}$O+$^{16}$O around the Fermi energy (40–60 MeV/nucleon) with impact parameter 1–3 fm are simulated, and the directed and elliptic flows are analyzed. It is found that collective flows are influenced by the different $\alpha$-clustering configurations, and the directed flow of free protons is more sensitive to the initial cluster configuration than the elliptic flow. Nuclear reaction at the Fermi energy can be taken as a useful way to study cluster configuration in light nuclei. DOI:10.1088/0256-307X/34/9/092101 PACS:21.65.Cd, 21.65.Mn, 25.70.-z © 2017 Chinese Physics Society Article Text With the rapid development in both theoretical and experimental methods, $\alpha$ clustering structure in light nuclei has attracted much attention in recent decades.[1-3] There are various theoretical models to study the $\alpha$ cluster in light nuclei, e.g., the ab initio method,[4,5] the fermion molecular dynamics model (FMD),[6,7] the antisymmetric molecular dynamics model (AMD),[8-11] the extended quantum molecular dynamics model (EQMD),[12-16] the $\alpha$-cluster model,[17] the algebraic cluster model, and [18] the covariant density functional theory.[19] Regarding the $\alpha$-clustering configuration in $^{16}$O, the chain configuration of 4$\alpha$ clusters in $^{16}$O was investigated using a Skyrme cranked Hartree–Fock method[20] as well as the Brink wave functions,[21] while a tetrahedral configuration of $\alpha$ clusters was discussed in Refs. [22,23]. In the experimental point of view, however, the whole picture of $\alpha$-clustering configuration in nucleus has still not emerged so far although some experimental signs have been indicated for information of $\alpha$ clusters in nuclei.[2,22,24] It is necessary to find more observables to explore cluster configuration in nuclei. Recently, it was proposed to use collective flows in relativistic nuclear collisions to probe $\alpha$-clustering in light nuclei, which offers a new idea for investigating the cluster configuration.[25-27] Collective flow, which has been studied over a wide range of beam energies and reaction systems, is one of the most important observables to probe the nuclear equation of state, the in-medium nucleon–nucleon cross section, the quark–gluon plasma, the viscosity and so on.[28-41] Thus it is very interesting to evaluate whether it can be used to probe the cluster configuration in light nuclei at the Fermi energies. In this work, within an extended quantum molecular dynamics (EQMD) model, the influence of cluster configuration on collective flows of protons produced in $^{16}$O+$^{16}$O collisions at the Fermi energy is investigated. The EQMD model is based on the quantum molecular dynamics (QMD) model, which is an $N$-body approach considering several improvements, and the approaches can be used to simulate nuclear reaction and meson-induced reaction at both very low energies and relativistic energies.[42] In the QMD-type model, each nucleon is represented by a Gaussian wave packet. In the initialization of projectile and target in QMD-type models, the centers of the Gaussian wave packet of nucleons are randomly chosen in coordinate space between 0 and the radius of projectile or target as well as momentum space between 0 and the Fermi momentum at the local density with consideration of several constraints, such as a proper binding energy, density and momentum distributions. However, the initialized nuclei are not always at their ground state (energy-minimum state), thus some unexpected nucleons are emitted during the collision process even at zero temperature. To solve those problems, an extended version of the QMD model was developed by Maruyama et al., named as EQMD.[43] In the EQMD model, unlike the standard QMD model, the width of Gaussian wave packet of nucleon is a complex and has the form, i.e., ${\nu _i} \equiv \frac{1}{{{\lambda _i}}}+i{\delta _i}$, where ${{\lambda _i}}$ and ${\delta _i}$ are its real and imaginary parts which are also time-dependent, respectively. The Gaussian wave packet is $$\begin{align} {\phi _{i}}({\boldsymbol r};t)=\,&\Big(\frac{\nu _{i}+{\nu _{i}}^*}{2\pi}\Big)^{3/4}\exp\Big[-\frac{\nu _{i}}{2}{({\boldsymbol r}-{{\boldsymbol r}_{i}(t)})^2}\\ &+\frac{i}{\hbar}{\boldsymbol r} \cdot {\boldsymbol p}_{i}(t)\Big],~~ \tag {1} \end{align} $$ where ${\boldsymbol r}_{i}(t)$ and ${\boldsymbol p}_{i}(t)$ are the centers of wave packet of nucleon $i$ in the coordinate and momentum space, respectively. The total wave function of an $N$-body system is assumed as the direct product of the Gaussian wave packet, i.e., ${\it \Psi} =\prod\limits_{i} {{\phi _{i}}} ({\boldsymbol r};t)$. The Hamiltonian of the system is given as $$\begin{alignat}{1} \!\!\!\!\!\!H=\,&\Big\langle {\it \Psi}\Big|\sum\limits_i - \frac{{{\hbar ^2}}}{{2{m}}}\nabla _i^2 - {{\hat T}_{\rm c.m.}}+{{\hat H}_{\rm int}}\Big|{\it \Psi}\Big\rangle \\ =\,&\sum\limits_i \Big[\frac{{{\boldsymbol p}_i(t)^2}}{{2m}}+\frac{{3{\hbar ^2}(1+\lambda _i^2\delta _i^2)}}{{4m{\lambda _i}}}\Big]-{T_{\rm c.m.}}+ {H_{\rm int}},~~ \tag {2} \end{alignat} $$ where ${T_{\rm c.m.}}$ denotes the spurious zero-point center-of mass kinetic energy caused by the zero-point oscillation of the center-of-mass wave function of the clusters. The subtraction of the spurious zero-point center-of mass kinetic energy is necessary for reproducing binding energy of nuclei and cannot be ignored when one studies nuclear structure and reaction at low energies; more details about ${T_{\rm c.m.}}$ can be found in Ref. [44]. Here ${H_{\rm int}}$ is the potential energy term, which contains several parts as follows: $$\begin{alignat}{1} \!\!\!\!\!\!\!\!H_{\rm int}=H_{\rm Skyrme}+H_{\rm Coulomb}+ H_{\rm Symmetry}+H_{\rm Pauli},~~ \tag {3} \end{alignat} $$ where $H_{\rm Skyrme}$, $H_{\rm Coulomb}$, $H_{\rm Symmetry}$ and $H_{\rm Pauli}$ are the Skyrme, Coulomb, symmetry and Pauli potential terms, respectively. More details can be found in Ref. [43]. Recently, the EQMD model was also extended to treat photonuclear reactions.[14,45,46] In our previous studies, different $\alpha$-clustering structures and their effects on dipole resonance and photonuclear reactions of $^{16}$O and $^{12}$C have been investigated.[12-14] Here, to study the influence of initial $\alpha$-cluster configurations on the collective flow, four configurations, namely linear chain, kite, square and tetrahedron of 4-$\alpha$ structure for $^{16}$O are employed at the initialization, and random orientations are used to obtain the initial state for $^{16}$O+$^{16}$O. More than 600000 events for each configuration are simulated, to reduce statistical error for observables. Figure 1 shows the directed flow parameter $v_1=\langle\frac{p_x}{p_t}\rangle$ and the elliptic flow parameter $v_2=\langle\frac{p_x^2-p_y^2}{p_t^2}\rangle$ of free protons as a function of the longitudinal rapidity $y_z=\frac{1}{2}\ln\frac{E+p_z}{E-p_z}$, where $p_t=\sqrt{p_x^2+p_y^2}$ is the transverse momentum of emitted particles, $p_z$ is the $z$-component of momentum, and $E$ is the total energy in the center-of-mass system. As usual, the $x$-axis is defined to be along the impact parameter vector, the $z$-axis is along the beam direction, and the $y$-axis is perpendicular to the reaction plane ($xz$-plane). First, in Fig. 1(a) we clearly see that the directed flow parameter $v_1$ decreases with increasing rapidity, called negative flow, which means that particles are more likely to undergo a rotational-like motion rather than a bounce-off motion. The value of the elliptic flow parameter $v_2$ at mid-rapidity ($y_z=0$) are sightly larger than zero, as seen in Fig. 1(b), which implies a preferential in-plane emission rather than an out-of-plane emission pattern. Indeed, the observed negative directed flow and in-plane elliptic flow around the Fermi energy have been well established in both experimental and theoretical studies.[28,32,47,48] This results from the domination of the attractive part of the nucleon–nucleon interaction over the repulsive part on the nucleon–nucleon scattering. Secondly, $v_1$ of protons produced from the tetrahedron (more compact shape) configuration is much negative than that from the chain (less compact shape) configuration, because of a stronger interaction originating from a higher density in collisions with the former configuration (as will be seen in Fig. 2). In addition, we can see that the differences in $v_1$ and $v_2$ between different cluster configurations become noticeable around the projectile/target rapidity region. From Fig. 1(b), we find that the calculated $v_2$ at mid-rapidity from the tetrahedron configuration is sightly smaller than that from the chain configuration, while the situation is reverse around the target and projectile rapidities. This phenomenon may result from two factors: (i) relatively more violent two-body scatterings in the tetrahedron configuration determined by the initial geometry lead to smaller values of $v_2$ (towards out-of-plane emission) at mid-rapidity; (ii) weaker interactions existing in the quasi-projectile and quasi-target region in the chain configuration weaken the rotational-like motion of nucleons. In general, we note that the directed flow of free protons is more sensitive to the initial cluster configuration than the elliptic flow.
cpl-34-9-092101-fig1.png
Fig. 1. The directed flow parameter $v_1$ (a) and elliptic flow parameter $v_2$ (b) for free protons as a function of the longitudinal rapidity $y_z$ for $^{16}$O+$^{16}$O collisions at 40 MeV/nucleon and $b=3$ fm.
To better understand the influence of initial configuration on the reaction dynamic process, time evolution of the nucleon density in the reaction plane for $^{16}$O+$^{16}$O collisions at 40 MeV/nucleon is shown in Fig. 2 as an example. At initial stage (here $t=100$ fm/c, because the projectile and target nuclei are put far away), see Figs. 2(a), 2(d), 2(g) and 2(j), the nucleon density contours for those four different configurations have almost spherical shapes. This is because the initial configurations are randomly orientated for all the events, and 1000 events are accumulated to draw these contour plots. Nevertheless, the differences among them can be seen. It is understandable that the contour of nucleon density for the chain configuration is the largest, while for the tetrahedron configuration it is the smallest. The difference in the density distribution at the initial stage will certainly affect the density evolution afterwards. As can be seen in the last two columns of Fig. 2, at $t=160$ fm/c, the nucleon density in the center of the compressed region slightly increases from the chain to the tetrahedron configuration, and at $t=200$ fm/c, the nucleon density in the center of the quasi-projectile and quasi-target region also increases from the chain to the tetrahedron configuration. Different densities achieved during the reaction would lead to different pressures, therefore one can see that the final observables such as collective flows are influenced by the initial configuration.
cpl-34-9-092101-fig2.png
Fig. 2. Contour plots of the average nucleon density in the reaction plane of $^{16}$O+$^{16}$O collisions at 40 MeV/nucleon. Simulations with the chain [panels (a), (b), and (c)], kite [panels (d), (e), and (f)], square [panels (g), (h), and (i)], and tetrahedron [panels (j), (k), and (l)] configurations at three different times (100 fm/c, 160 fm/c, and 200 fm/c) are shown. In the insets of (a), (d), (g) and (j), sketches for the four different $\alpha$-clustering configurations are plotted. Here 1000 events were collected to make the statistical error small enough.
cpl-34-9-092101-fig3.png
Fig. 3. The directed flow parameter $v_1$ (upper panels) and elliptic flow parameter $v_2$ (lower panels) of free protons in $^{16}$O+$^{16}$O collisions at 40 MeV/nucleon with impact parameters of 1, 2 and 3 fm, as a function of the longitudinal rapidity $y_z$.
To exhibit more systematically the influence of cluster configuration on the collective flows, $^{16}$O+$^{16}$O reaction with different impact parameters and beam energies are focused on. The results of the directed and elliptic flows of free protons are shown in Figs. 3 and 4. As we can see from upper panels of Fig. 3, the difference between different cluster configurations is more evident for the larger impact parameters. We have checked that the difference in the nucleon density contour in the reaction plane increases with the impact parameter, which results in a larger sensitivity of the directed flow to the cluster configuration. For more central collision, the collision number is hardly influenced by cluster configuration due to random orientations of the initial projectile and target, while for more peripheral collision, the collision number is strongly dependent on the initial geometry, thus the flow is sensitive to the cluster configuration. The directed flow at different beam energies is shown in Fig. 4. The sensitivity of the directed flow to the cluster configuration decreases with increasing the beam energy, e.g., at 60 MeV/nucleon, we can see almost the same directed flow with different cluster configurations. In Figs. 3 and 4, we find that the elliptic flow is less sensitive to different cluster configurations than the directed flow. Beam energy below 40 MeV/nucleon is not considered, because nuclei are more likely to undergo fusion then the collective flows of nucleons become quite weak. The energy dependence of collective flow in HICs has been widely studied both experimentally and theoretically (see, e.g., Refs. [47,48]). With increasing the beam energy, the multiple scattering process is dominant in comparison with the contribution coming from the effect of the clustering configuration.
cpl-34-9-092101-fig4.png
Fig. 4. The same as Fig. 3, but for $^{16}$O+$^{16}$O collisions at beam energies of 40, 50 and 60 MeV/nucleon with impact parameter of $b=3$ fm.
In summary, $^{16}$O+$^{16}$O reactions with the linear chain, kite, square, and tetrahedron 4-$\alpha$ configurations around the Fermi energy have been simulated by the EQMD model and their directed and elliptic flows of free protons are focused on. It is found that the directed flow of protons produced from the tetrahedron configuration is more negative than that from the chain configuration, while in-plane elliptic flow from the tetrahedron configuration is smaller than that from the chain configuration. This can be understood from the difference in the mean-field potential and two-body scattering caused by the initial geometry. Moreover, we also found that the directed flow of protons shows a larger sensitivity to the initial cluster configuration than the elliptic flow. In addition, through the simulation of $^{16}$O+$^{16}$O reaction at different impact parameters and beam energies, it is shown that the directed flow from reaction with a larger impact parameter and a smaller beam energy (around 40 MeV/nucleon) is suggested for future measurement to probe the cluster configuration in $^{16}$O. From an experimental point of view, a precise measurement of the collective flow is relatively easier for a larger colliding system due to the better determination of the reaction plane. Thus it would probably be advisable to investigate the cluster configuration with nuclear reaction, such as $^{16}$O($^{12}$C)+$^{124}$Sn ($^{197}$Au). In a future study, such reactions will be simulated within the same microscopic transport model. Finally, it should be mentioned that here we did not touch the study of high-order flows, such as triangular flow ($v_3$) and 4th flow ($v_4$), which have been extensively studied in relativistic heavy ion collision.[26,35] In principle, $v_3$ and $v_4$ will be significant for nuclear reaction with triangle 3-$\alpha$ clustering nuclei ($^{12}$C) and 4-$\alpha$ clustering nucleus ($^{16}$O), respectively. Anyway, such high-order flows leave us a topic to be addressed in the near future. We thank Professor Qingfeng Li for helpful discussions and acknowledge support by the computing server C3S2 in Huzhou University. References Nuclear clusters and nuclear moleculesThe clustered nucleus—cluster structures in stable and unstable nucleiProgress of theoretical and experimental studies on α cluster structures in light nucleiAb Initio Calculation of the Hoyle StateStructure and Rotations of the Hoyle StateFermionic molecular dynamicsStructure of the Hoyle State in C 12 Antisymmetrized molecular dynamics and its applications to cluster phenomenaTemperature of intermediate mass fragments in simulated 40Ca + 40Ca reactions around the Fermi energies by AMD modelA Model Comparison Study of Fragment Production in 140 A MeV 58,64 Ni+ 9 Be ReactionsGiant Dipole Resonance as a Fingerprint of α Clustering Configurations in C 12 and O 16 Dipole oscillation modes in light α -clustering nucleiPhotonuclear reaction as a probe for α -clustering nuclei in the quasi-deuteron regionAlpha-clustering effects on 16O( $\gamma$ γ ,np)14N in the quasi-deuteron regionDynamical α -cluster model of O 16 Anatomy of molecular structures in 20 NeLinear Chain Structure of Four- α Clusters in O 16 One-Dimensional α Condensation of α -Linear-Chain States in C 12 and O 16 Evidence for Tetrahedral Symmetry in O 16 Ab Initio Calculation of the Spectrum and Structure of O 16 Cluster decay of the high-lying excited states in 14 CSignatures of α Clustering in Light Nuclei from Relativistic Nuclear CollisionsNuclear cluster structure effect on elliptic and triangular flows in heavy-ion collisions α clusters and collective flow in ultrarelativistic carbon–heavy-nucleus collisionsCOLLECTIVE FLOW IN HEAVY-ION COLLISIONSCollective flow signals the quark–gluon plasmaRotational behavior in intermediate energy heavy ion collisionsCollective motion of reverse-reaction system in the intermediate-energy domain via the quantum-molecular-dynamics approachScaling of anisotropic flow and momentum-space densities for light particles in intermediate energy heavy ion collisionsElliptic flow of ϕ mesons and strange quark collectivityInitial fluctuation effect on harmonic flows in high-energy heavy-ion collisionsMomentum-Dependent Symmetry Potential from Nuclear Collective Flows in Heavy Ion Collisions at Intermediate EnergiesEffects of the HMT on Nucleon Collective Flows within BUU Transport ModelCorrelation between elliptic flow and shear viscosity in intermediate-energy heavy-ion collisionsCollective flow and hydrodynamics in large and small systems at the LHCElliptic Flow Splitting between Particles and their Antiparticles in Au+Au Collisions from a Multiphase Transport ModelAzimuthal Asymmetry of Pion-Meson Emission around the Projectile and Target Sides in Au+Au Collision at 1 A GeV“Quantum” molecular dynamics—a dynamical microscopic n-body approach to investigate fragment formation and the nuclear equation of state in heavy ion collisionsExtension of quantum molecular dynamics and its application to heavy-ion collisionsAntisymmetrized molecular dynamics for heavy ion collisionsA Photonuclear Reaction Model Based on IQMD in Intermediate-Energy RegionSystematics of stopping and flow in Au+ Au collisionsNonequilibrium dynamics in heavy-ion collisions at low energies available at the GSI Schwerionen Synchrotron
[1] Oertzen W, Freer M and Kanada-Eno Y 2006 Phys. Rep. 432 43
[2] Freer M 2007 Rep. Prog. Phys. 70 2149
[3] Cao X G and Ma Y G 2015 Chin. Sci. Bull. 60 1557
[4] Epelbaum E, Krebs H, Lee D and Meissner U G 2011 Phys. Rev. Lett. 106 192501
[5] Epelbaum E, Krebs H, Lahde T A, Lee D and Meissner U G 2012 Phys. Rev. Lett. 109 252501
[6] Feldmeier H 1990 Nucl. Phys. A 515 147
[7] Chernykh M, Feldmeier H, Neff T, Neumann-Cosel P and Richter A 2007 Phys. Rev. Lett. 98 032501
[8] Kanada-En'yo Y, Kimura M and Ono A 2012 Prog. Theor. Exp. Phys. 2012 01A202
[9] Kanada-En'yo Y et al 2015 Nucl. Sci. Tech. 26 S20501
[10] Ma C W et al 2016 Nucl. Sci. Tech. 27 111
[11] Ma C W et al 2015 Chin. Phys. Lett. 32 072501
[12] He W B, Ma Y G, Cao X G, Cai X Z and Zhang G Q 2014 Phys. Rev. Lett. 113 032506
[13] He W B, Ma Y G and Zhang Q G 2016 Phys. Rev. C 94 014301
[14] Huang B S, Ma Y G and He W B 2017 Phys. Rev. C 95 034606
Huang B S, Ma Y G and He W B 2017 Eur. Phys. J. A 53 119
[15]He W B, Ma Y G, Cao X G et al 2014 Nucl. Tech. 37 100511 (in Chinese)
[16]Wang S S, Cao X G, Zhang T L et al 2015 Nucl. Phys. Rev. 32 24
[17] Halcrow C J, King C and Manton N S 2017 Phys. Rev. C 95 031303
[18]Bijker R 2010 AIP Conf. Proc. 1323 28
[19] Zhou E F, Yao J M, Li Z P, Meng J and Ring P 2016 Phys. Lett. B 753 227
[20] Ichikawa T, Maruhn J A, Itagaki N and Ohkubo S 2011 Phys. Rev. Lett. 107 112501
[21] Suhara T, Funaki Y, Zhou B, Horiuchi H and Tohsaki A 2014 Phys. Rev. Lett. 112 062501
[22] Bijker R and Iachello F 2014 Phys. Rev. Lett. 112 152501
[23] Epelbaum E et al 2014 Phys. Rev. Lett. 112 102501
[24] Tian Z Y, Ye Y L, Li Z H et al 2016 Chin. Phys. C 40 111001
[25] Broniowski W and Arriola W R 2014 Phys. Rev. Lett. 112 112501
[26] Zhang S, Ma Y G, Chen J H et al 2017 Phys. Rev. C 95 064904
[27] Bozek P, Broniowski W, Ruiz A E and Rybczynski M 2014 Phys. Rev. C 90 064902
[28] Reisdorf W and Ritter H G 1997 Annu. Rev. Nucl. Part. Sci. 47 663
[29] Stoecker H 2005 Nucl. Phys. A 750 121
[30]Kumar S and Ma Y G 2013 Nucl. Sci. Tech. 24 050509
[31]Ko C M, Chen L W, Vincenzo G et al 2013 Nucl. Sci. Tech. 24 050525
[32] Ma Y G et al 1993 Phys. Rev. C 48 R1492
Ma Y G et al 1995 Phys. Rev. C 51 1029
[33] Yan T Z, Ma Y G, Cai X Z et al 2006 Phys. Lett. B 638 50
[34] Chen J H, Ma Y G, Ma G L et al 2006 Phys. Rev. C 74 064902
[35] Han L X, Ma G L, Ma Y G et al 2011 Phys. Rev. C 84 064907
[36] Xie W J and Feng Z Q 2015 Chin. Phys. Lett. 32 122502
[37] Zhang F 2016 Chin. Phys. Lett. 33 012501
[38] Zhou C L, Ma Y G, Fang D Q et al 2014 Phys. Rev. C 90 057601
[39] Song H C, Zhou Y and Gajdoov K 2017 Nucl. Sci. Tech. 28 99
[40] Xu Z Y et al 2017 Chin. Phys. Lett. 34 062501
[41] Wang T T et al 2015 Chin. Phys. Lett. 32 062501
[42] Aichelin J 1991 Phys. Rep. 202 233
[43] Maruyama T, Niita K and Iwamoto A 1996 Phys. Rev. C 53 297
[44] Ono A and Horiuchi H 2004 Prog. Part. Nucl. Phys. 53 501
[45] Huang B S and Ma Y G 2017 Chin. Phys. Lett. 34 072401
[46]Wu H L et al 2015 Nucl. Sci. Tech. 26 050103
[47] Andronic A, Lukasik J, Reisdorf W et al 2006 Eur. Phys. J. A 30 31
[48] Li Q F, Shen C W, Guo C C et al 2011 Phys. Rev. C 83 044617