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Influence of nuclear structure in relativistic heavy-ion collisions

Many probes are proposed to determine the quark-gluon plasma and explore its properties in ultra-relativistic heavy-ion collisions. Some of them are related to initial states of the collisions, such as collective flow, Hanbury-Brown-Twiss (HBT) correlation, chiral magnetic effects and so on. The initial states can come from geometry overlap of the colliding nuclei, fluctuations or nuclear structure with the intrinsic geometry asymmetry. The initial geometry asymmetry can transfer to the final momentum distribution in the aspect of hydrodynamics during the evolution of the fireball. Different from traditional methods for nuclear structure study, the ultra-relativistic heavy-ion collisions could provide a potential platform to investigate nuclear structures with the help of the final-state observables after the fireball expansion. This chapter first presents a brief introduction of the initial states in relativistic heavy-ion collisions, and then delivers a mini-review for the nuclear structure effects on experimental observables in the relativistic energy domain.


A brief introduction to the relativistic heavy-ion collisions and the initial state
Relativistic heavy-ion collisions aim at investigating a new state of matter, quarkgluon plasma (QGP) which was predicted by quantum chromodynamics (QCD) [1] and is considered to be produced at the early stage of central nucleus-nucleus collisions in experiments [2,3]. The collective motions in partonic level were reported by the RHIC-STAR collaboration [4]. Recently, a mount of experimental results are obtained to investigate the relationship between the initial states of the collision system and the properties of QGP, such as collective flow [5,6], Hanbury-Brown-Twiss (HBT) correlation [7], fluctuation [8] and so on. In theoretical side, the initial geometry fluctuations are studied by various models, such as a multi-phase transport (AMPT) model [9], hydrodynamics [10] etc. Some methods are proposed to perform collective flow and geometry eccentricity analysis related to initial fluctuations [11].
The probes, such as collective flow, HBT correlation or fluctuations, have been extensively investigated and a new state of matter, so called QGP, was declared to be created in ultra-relativistic heavy-ion collisions, such as in Au + Au or Pb + Pb collisions, as introduced above. Some similar phenomena are also observed in small systems (p + Al, p + Au, d + Au, 3 He + Au, p + Pb and p + p) with high multiplicity events [6,12] as in large systems (Au + Au and Pb + Pb). It is an open question how to understand transformation coefficient from initial geometry distribution or fluctuation to momentum distribution at final stage in hydrodynamic mechanism [10,13] and if the matter created in different-size systems undergoes the similar dynamical process and has similar viscosity [14]. There are already a lot of theoretical works contributing to physics explanation and analysis method in this subject [10,15]. Some detailed introduction and discussion can be found in some recent review articles [16]. Suggestions to perform small collision system experiments could provide great opportunities to uncover the initial geometry effect in different-size collision systems.
The collective flow and the related observables play the key roles in experimental and theoretical investigations, and great progress has been made at relativistic energies [17]. Systematic measurements of collective flow at center of mass energy from GeV to TeV in nucleus-nucleus collisions have been conducted at RHIC and LHC energies [2,3] and the number of constituent quark (NCQ) scaling law of elliptic flow [4] becomes one of the significant probes to demonstrate the partonic level collectivity of the super hot-dense matter. It arises more efforts to understand the transformation of asymmetry from initial geometry space into final momentum space. Specifically the system size dependence of collective flow has been analyzed in experiments [6,12], and the initial state fluctuations and shear viscosity effect are taken into account in theoretical works [14]. The viscous relativistic hydrodynamics models [14] suggest a relationship between the collective flow v n and the initial geometry eccentricity ε n , ln(v n /ε n ) ≈ −n 2 η s (T ) N ch −1/3 , here η s (T ) represents the shear viscosity over entropy density, N ch is the number of charge particles. The intuitive picture of the initial state fluctuations is shown in figure 1 [11], where the par-ticipants contribute to triangularity or higher order geometrical eccentricities, and the triangular flow and higher order flow harmonics are discussed and measured in literature such as Refs. [11,18].
Similarly, the number of nucleon (NN) scaling of elliptic flow was first proposed in Ref. [19] by using the quantum molecular dynamics model simulation for lower energy heavy-ion collisions. Even though this conclusion was drawn from low energy heavy-ion collisions but it also works at relativistic heavy-ion collisions [20]. Actually this NN scaling of elliptic flow was confirmed by the STAR Collaboration for light nuclei d, t, 3   The harmonic coefficients of collective flow can be calculated from the Fourier series of particle's azimuthal distribution [22], where E is the energy, p T is transverse momentum, y is rapidity, and φ is azimuthal angle of the particle. Ψ RP denotes the reaction plane angle. And the Fourier coefficients v n (n = 1, 2, 3, ...) are collective flows to characterize different orders of azimuthal anisotropies with the form, where the bracket means statistical averaging over particles and events. The true reaction plane angle Ψ RP is always estimated by event plane angle [22] or by participant plane angle [11]. The harmonic flow can be calculated with respect to participant plane angle or event plane angle, called as participant plane (PP-) method and event plane (EP-) method, respectively. Some methods avoiding to reconstruct the reaction plane are developed, such as Q-cumulant (QC-) method [9,23] and two-particle correlation (2PC-) method with rapidity gap [24]. In fact, two-particle correlation method has been already developed earlier in low-intermediate energy heavy-ion collision for flow analysis, eg. see Refs. [25]. The collective flow is driven from the initial anisotropy in geometry space. To investigate transformation from initial geometry to final momentum space, the initial geometry eccentricity coefficients ε n are calculated from the participants via [9,11,13], E n ≡ ε n e inΦ n ≡ − r n part e inφ part r n part , where, r part = x 2 part + y 2 part and φ Part are coordinate position and azimuthal angle of initial participants in the collision zone in the recentered coordinates system ( x part = y part = 0). Φ n is the initial participant plane and ε n = |E n | 2 1/2 . The bracket means the average over the transverse position of all participants event by event. Note that for the definition of eccentricity coefficients ε n , r 2 part weight is alternative and it was discussed in Refs. [11,13].
From the above introduction, it can be seen that the collective flow and the related observables are sensitive to the initial state in collisions, which may come from the overlap of the collision zone between the collided nuclei, the initial fluctuation as well as the intrinsic geometry of the nuclei. Next the chapter mainly focuses on how the intrinsic geometry (nuclear structure) affects the observables at the final state in the relativistic heavy-ion collisions.
A brief introduction to the nuclear structure There are a lot of theoretical works (or models) to describe nuclear structure, such as Woods-Saxon model, Shell-model, effective field theory or Ab initio calculations. Here the Woods-Saxon model is only introduced and a short review is made for the exotic nuclear structures, specifically for α-clustering nuclei as well as neutron (proton) skin nuclei in theories and experiments. This is due to that the main purpose of this chapter is to investigate how the exotic nuclear structures affect on final observables in relativistic heavy-ion collisions, or on the other hand how the final observables can be used to distinguish the exotic nuclear structures.
The Woods-Saxon potential was introduced by R. D. Woods and D. S. Saxon [26] in 1954 to describe nucleon-nucleus scattering, and written as, here r is radial distance of nucleon and V 0 , R, a are different parameters. The ground state of nuclei can be obtained by solving the Schrödinger equation by using this potential [27].
In relativistic heavy-ion collisions, the nucleon density distribution in the initial collided nuclei can be usually parameterized by a three-parameter Fermi (3pF) distribution [28], namely where ρ 0 is the nucleon density in the center of the nucleus, the three parameters R, a and ω are related to the nuclear radius, the (surface) skin depth and deviations from a spherical shape. If ω sets to zero, it becomes the so-called two-parameter Fermi (2pF) distribution [29][30][31], To describe non-spherical nuclei, the extended formula can be found in Refs. [32,33] by introducing spherical harmonics and the spatial distribution of deformed nuclei in the rest frame could be written in the Woods-Saxon form, where ρ 0 is the normal nuclear density, R 0 and a represent the "radius" of the nucleus and the surface diffuseness parameter, respectively, and β 2 was the deformation of the nucleus. The Woods-Saxon distribution of nucleons, introduced above, always gives the same radius of proton and neutron distribution in the nucleus, denoted as R p and R n , respectively. However, if the density profiles are the same for protons and neutrons in neutron-rich nucleus, the radius of neutron distribution should be larger than that of proton's, i.e. R n > R p , which is called the neutron skin. From the measurements at lower energies, the neutron skin thickness which is defined as the difference between the root mean square radii for the neutron and proton distributions, is presented and the mechanism is investigated in Refs. [31]. To configure a nucleus with neutron skin in a model, one can refer to literature [30], and here a method based on the two-parameter Fermi (2pF) distribution in equation (6) is introduced. For a nucleus with the neutron skin, the nuclear density is written by ρ A (r) = ρ p,A (r) + ρ n,A (r) and different choice of parameters for protons and neutrons results in the neutron skin effect.
In nature, cluster structure is an ordinary phenomenon that can be observed everywhere, such as cloud in the sky, water clusters into ice in the river and atom forms crystal. In general, it can be considered the constituents to form a stable structure through interaction each other even though some of the potential among constituents has not been found very well. Keep in mind that the nuclear structure effects are taken into account in heavy-ion collisions, or on the other hand, it will be discussed for the investigation of the exotic nuclear structures from the final observables, then a brief introduction to α-clustered nuclei will be given for theoretical works and experimental progress. The history of clustered nuclei hypothesis which was first proposed by Gamow [35] can be traced back decades ago and the light 4n nuclei is considered to be composed of α-particles, which is the so-called α cluster model. And then in the last decades, there are amount of works contributing to this exciting subject of exotic nuclear structure [34]. Some recent reviews can be found in references [34]. Fifty years ago, the famous threshold diagram 2 was put forwarded, as shown in figure, which pointed out that cluster structures were mainly found close to cluster decay thresholds [34]. In this prediction, for example the ground state of 8 Be would be considered as a bound state of 2αs. Later on this diagram was extended to N = Z nuclei which can be composed of αs with valence neutrons in Ref. [34], as shown in figure 3.
From the above introduction, it can be found that there is a chain of clustering nuclei from α, 8 Be, 12 C, 16 O ..., and another chain for Nα+mn (N, m the number of α and neutron) nuclei. Actually the investigation on α-conjugate nuclei, such as 8 Be, 12 C, 16 O, is still an open question for exotic nuclear structures [36]. Effort on investigation of 12 C is very impressive and the obtained precise structural data of 12 C provides a benchmark for verifying first-principles calculations. Fermionic molecular dynamics (FMD), anti-symmetrized molecular dynamics (AMD) and covariant density functional theory [37] support that ground state of 12 C is in a triangle-like structure with three αs. The Brink type THSR-wave function was employed to demonstrate that there is strong two-α correlation in ground state of 12 C [38]. The experimental evidence was reported for triangular symmetry in 12 C at the ground state [39]. The spectra of giant dipole resonance (GDR) was proposed as fingerprint for α-clustering light nuclei by using an Extended Quantum Molecular Dynamics (EQMD) [40]. 16 O is another interesting nucleus with an additional α with respect to 12 C. The four αs in 16 O are considered to be arranged in regular tetrahedral distribution, which corresponds to the ground state in the Ikeda diagram as in figure 2. Chiral nuclear effective field theory [41] and covariant density functional theory [37] support this regular tetrahedral structure in α-clustered 16 O. The algebraic cluster model (ACM) [41,42] was used to produce the rotation-vibration spectrum of 16 O which is comparable to that in experiment and provides a strong evidence for tetrahedral symmetry in 16 O. The EQMD calculations presented that 16 O ground state shows the tetrahedral structure and the corresponding characteristic spectra of GDR is comparable to the experimental result [40].

Influence of α-clustering effects
It is an open question how the initial geometrical asymmetry transfers to final momentum space and how the intrinsic deformation of nuclei plays an important role for the collective motion properties in relativistic heavy-ion collisions. The αclustered light nuclei are arranged in some special geometry structures with the constituent α-clusters, such as 12 C in triangle with 3-αs [37][38][39][40].
Refs. [43,44] proposed an α-clustered Carbon colliding against a heavy-ion, 12 C + Au, to investigate the collective flow in the evolution of the fireball. The large deformation in the initial intrinsic nucleus was transformed into the anisotropy of the final momentum space in the fireball. The configuration of the α-clustered Carbon is described as following: the centers of three clusters were placed in an equilateral triangle of edge length l and the rms radius of the α-cluster is r α . The model parameters l and r α are optimized by fitting the radial density distribution from the so-called BEC model [45] and the VMC calculations [46] as provided at http://www.phy.anl.gov/theory/research/density. Within the Glauber framework [47], the so-called mixed model [48] was used to simulate the initial state of the collisions and the event-by-event (3+1)-dimensional viscous hydrodynamics [49] was employed to simulate the evolution of the created fireball. In those works, the initial fluctuations and the intrinsic geometry from α-clusters were simulated and discussed. Figure 4 from reference [44] simulating with clustered and uniform 12 C showed that the geometry enhances the triangularity at high values of the number of wounded nucleons, N w (corresponding to central collisions), and raises ellipticity at lower values of N w (corresponding to peripheral collisions). With the help of hydrodynamics [13], the ratio of v n {4}/v n {2} was proposed as a probe to distinguish the initial α-cluster structure of 12 C for the relationship to the initial geometry deformations, where v n {k} or ε n {k} means nth-order harmonic flow coefficients or eccentricity coefficients via k-particle cumulant moments as defined in Ref. [44]. Figure 5 showed the simulation results of v n {4}/v n {2} and it indicated that the geometric triangularity increased for collisions with a larger number participants, corresponding to central collisions, which was straightforward to measure in ultra-relativistic heavyion collisions with standard techniques devoted to analysis of harmonic flow [44].
Recently the simulations involving α-clustered 7,9 Be, 12 C and 16 O were presented in ultra-relativistic heavy-ion collisions [50]. The influence of fluctuation and the intrinsic geometry effect were investigated systematically at the SPS collision energy of √ s NN = 17 GeV. In reference [50], the ratios of v 2 {4}/v 2 {2} presented increasing (decreasing) trend with the increasing of the wounded nucleons N w in the collisions with 7,9 Be beams for α-clustered (uniform) configuration and the ratios of v 3 {4}/v 3 {2} gave the similar trend in 16 O + Pb collisions as shown in figure 6. As proposed by Refs. [51,52], the collisions with 16 1% 0.1% Fig. 4 The eccentricity simulated by the GLISSANDO model for the Bose-Einstein condensation (BEC) case at RHIC as a function of the wounded nucleons N w which is corresponding to centralities [44].
in details [50,53] and were considered as a platform to invetigate the effect from the intrinsic geometry of nuclei at RHIC or LHC. The effect from the intrinsic geometry was also studied in a multi-phase transport (AMPT) model [54,55]. AMPT model describes the whole heavy-ion collision processes dynamically as following, the binary collisions among the initial nucleons result in the exited strings and mini-jet partons which all will fragment into partons, and the partons undergo a cascade process till reaching a so-called parton freezeout status, then the freeze-out partons coalesce into hadrons via a simple coalesence model, and then the hadrons participate in the final rescattering process. AMPT was extensively used in the community and can successfully describe various physics results in relativistic heavy-ion collisions at the RHIC and LHC energies [54,55].
During the initialization in the original version of AMPT, the initial nucleon coordinate space in nuclei is described by the HIJING model [56] with the Woods-Saxon distribution [54]. However, the α-clustered nuclear structure in the initial condition embodies via reconfiguring in HIJING as following [53,57,58]. For three αs in equilateral triangle structure placing the α at each vertice with side length l 3α and for four αs in tetrahedral structure placing at each vertice with side length l 4α , for 12 C and 16  Ratios of four-particle to two-particle cumulants plotted as a function of the total number of wounded nucleons for the Bose-Einstein condensation (BEC) case and uniform case [44]. Ratios of four-particle to two-particle cumulants for 9 Be + Pb (left) and 16 O + Pb (right) collisions, plotted as a function of the total number of wounded nucleons [50].
inherited from the EQMD calculation [40] and nucleons in the α follow the Woods-Saxon distribution. And the α-clustered 12 C and 16 [58]. Once it is possible to provide the reasonable initial stable nuclei with different intrinsic geometry structure, they were introduced to the initial state in HIJING process of AMPT model to investigate the α-cluster influence in relativistic heavy-ion collisions. The initial geometry in 12 C + 197 Au collisions was characterised by the eccentricity coefficients ε n calculated using equation (3). Since the initial fluctuation should be also taken into account, the initial coordinates of participants were used to calculate ε n and to reconstruct the so-called participant plane [9,11], where, Ψ n {PP} is the nth-order participant plane angle, r and ϕ part are coordinate position and azimuthal angle of participants in the collision zone at initial state, and the average · · · denotes density weighting. Then the harmonic flow coefficients with respect to participant plane can be defined as, Figure 7 from Ref. [58] showed the calculated second and third order eccentricity coefficients, ε 2 {PP} and ε 3 {PP} in 12 C + 197 Au collisions at √ s NN = 200 GeV and 10 GeV for different configurations of 12 C, i.e. the α-clustered chain and triangle structure as well as nucleon distribution in the Woods-Saxon distribution. At the top RHIC energy ( √ s NN = 200 GeV), the chain (triangle) structure enhanced the seconded (third) order geometry coefficient and ε 2 {PP} (ε 3 {PP}) increased with the increasing of the number of particles, N track , in the collisions. In the Woods-Saxon case, the geometry coefficients (ε 2 {PP} and ε 3 {PP}) slightly decreased with the increasing of N track , which indicated the fluctuation contribution gradually weakened from peripheral collisions to central collisions. These initial geometry properties were consistent with those in references [43,44] and comparable with the presented results in figures 4. With the help of AMPT model, the evolution of collisions was simulated dynamically from the initial nucleon-nucleon collisions, partonic interaction to the hadronic rescattering and the initial geometry asymmetry is transferred to the final momentum space. Figure 8 presented the ratios of ε 3 {PP}/ε 2 {PP} and v 3 {PP}/v 2 {PP} which reflected the initial geometry properties: v 3 {PP}/v 2 {PP} from triangle structure configuration of 12 C displayed an increasing trend with the increasing of N track , that from the Woods-Saxon configuration kept a flat pattern in a low p T region and that from the chain structure configuration of 12 C also kept flat pattern and lower than that from the Woods-Saxon configuration. It indicated that the flow coefficient ratio of v 3 {PP}/v 2 {PP} was sensitive to the initial geometry properties. Via comparing the results from the Woods-Saxon case and the α-clustered triangle case, the ratio of v 3 {PP}/v 2 {PP} could be taken as a probe to distinguish the exotic nuclear structure in heavy-ion collisions, note that the chain structure of 12 was always considered in excited state.
A light nucleus with the intrinsic geometry structure colliding against a heavy nucleus can be a feasible experimental project to be performed at RHIC or LHC from the above discussion [43,44,50,58]. The system scan project is also important for investigating the effects from intrinsic geometry and fluctuations, which involves centrality dependence (such as the above introduction), asymmetrical collision systems (light nuclei + different-size nuclei), and symmetrical collision systems. That work [53] proposed three cases of collision systems: case I, 16 O nucleus (with or without α-cluster) + ordinary nuclei (always in the Woods-Saxon distribution) in the most central collisions; case II, 16 O nucleus (with or without α-cluster) + 197 Au nucleus collisions for centrality dependence, and case III, symmetric collision systems (namely, 10 B + 10 B, 12 C + 12 C, 16 O + 16 O (with or without α-cluster), 20 Ne + 20 Ne, and 40 Ca + 40 Ca) in the most central collisions. In that work the linear mode in the initial geometry eccentricity coefficients and flow coefficients were calculated [15,53]. And for higher-order initial geometry eccentricity coefficients and flow coefficients, the relationship for linear mode can be described by v L n ∝ ε L n (see Refs. [15] and references therein for details).  20 Ne, 40 Ca and 96 Zr), middle panels (Case II: (e), (f), (g), and (h)) are the results for the centrality dependence of the 16 O (with or without α-cluster) + 197 Au collisions, and lower panels (Case III: (i), (j), (k), and (l)) represent the symmetric collision systems from small to large ones in the most central collisions, i.e. 10 B + 10 B, 12 C + 12 C, 16 O + 16 O (with or without α-cluster), 20 Ne + 20 Ne, and 40 Ca + 40 Ca. [53] By using the AMPT model, the three-case collision systems were simulated and the initial geometry coefficients and the final flow coefficients are presented in figure 9 and 10, respectively [53]. From figure 9, ε 2 , ε 3 and ε L 4 kept the decreasing trend with the increasing of the N part (number of participants corresponding to centrality) which confirmed that the fluctuation contribution became less in central collisions or in larger collisions system. ε 3 {PP} in the collision systems with tetrahedron structure 16  from the system dependence from 10 B + 10 B, 12 C + 12 C, 20 Ne + 20 Ne to 40 Ca + 40 Ca collisions on panel (j) in figure 10. After undergoing the dynamical evolution in AMPT via partonic interaction and hadronic rescattering, the initial geometry properties were still survived and transferred to the momentum space by analyzing the flow coefficients as shown in figure 10.
, one can find that the flow coefficients which characterised the momentum asymmetry reflected the initial geometry coefficients which denoted the initial state of the collision system. To further compare three cases, the ratio of anisotropic flow coefficients v 3 /v 2 was presented in figure 11 which all presented difference from the configuration of 16 O beam, namely α-clustered type gave the nontrivial system dependence in comparing with the Woods-Saxon type. It indicated that the case I and case III for the most central collisions could reduce the geometry distribution influence from the overlap region as much as possible, and then it suggested that v 3 /v 2 could be taken as a probe to identify the α-clustered structure of 16 O. Therefore the cases I and III were proposed as a potential scenario of a system scan experiment project at RHIC or LHC.
After the influence from the intrinsic geometry of the α-clustered light nuclei beam was introduced in relativistic heavy-ion collisions via analyzing the transformation from the initial eccentricity to the final collective flow in the above paragraph, the fluctuation method is discussed for determining how it is sensitive to the intrinsic geometry and the system size (larger fluctuation in small systems) or to both. The authors [43] presented the event-by-event statistical properties of the fireball via the observables of average ellipticity ε 2 , triangularity ε 3 , and their scaled standard deviations σ (ε 2 )/ε 2 , σ (ε 3 )/ε 3 as a function of the number of wounded 40 Fig. 11 Ratio of v 3 /v 2 as a function of N part : the case I, i.e. 16 O + ordinary nuclei at b = 0 fm (a), the case II, i.e. 16 O + Au at different centralities (b), the case III, i.e. symmetric collisions at b = 0 fm (c). The red or black lines (symbols) represent 16 O w/ or w/o α-cluster structure. [53] nucleons N w as shown in figure 12. In uniform (unclustered) case the scaled standard deviations σ (ε 2 )/ε 2 , σ (ε 3 )/ε 3 presented ignorable N w dependence with similar value of specific centrality as on right panel in figure 12. In BEC (clustered) case, however, σ (ε 2 )/ε 2 and σ (ε 3 )/ε 3 taken on the opposite N w dependence trend. The flow fluctuation quantified with scaled standard deviation should be approximately equal to the eccentricity fluctuation quantified in the same way, If the relationship was taken into account, it can be concluded that the eccentricity coefficients and the scaled standard deviations were sensitive to the initial geometry properties and could be measured with the standard flow analysis methods.
In reference [60], fluctuation and correlation of the azimuthal anisotropies in 12 C + 197 Au collisions at √ s NN = 200 GeV were explored by using AMPT model. Properties of the initial eccentricity and final harmonic flow fluctuation in the collisions with 12 C beam via the Woods-Saxon configuration and triangular α-clustering configuration were investigated via the scaled variance (scaled standard deviations), skewness, and kurtosis, The flow fluctuations were defined in the same way as equation (12). In that work, it was found that R ε 3 and R v 3 were sensitive to the triangle structure of 12 C which was consistent with the conclusion in reference [43] since there was different in the exact value from two different models. The skewness and kurtosis of the eccentricity fluctuation, namely S ε n and K ε n presented a significant difference in both magnitude and trend as a function of N track for third-order, however, this difference disappeared for flow fluctuation, namely S v n and K v n . It was argued that could be from the source evolution that in the final state of AMPT the v n (n = 2, 3) distributions were nearly Gaussian thus the skewness and kurtosis of elliptic and triangular flow fluctuations especially at large N track are almost consistent with zero within statistical uncertainties [60]. The correlations of the initial eccentricities with final flow harmonics were investigated via the Pearson coefficient to quantify the strength of the correlation defined as, where Ψ {PP} was the participant plane angle, Ψ was the phase of the flow coefficient v n (e.g., Ψ was the event-plane angle if v n was calculated based on the eventplane method). Figure 13 showed the calculated C v n ,ε n as a function of N track in 12 C + 197 Au collisions with different geometry configuration of 12 C [60]. The increasing trend of C v n ,ε n with the increasing of N track indicated stronger linear v n − ε n correlation at larger N track (central collisions). In comparison, the correlation between v 2 and ε 2 presented comparable results in the Woods-Saxon and triangle α-clustered configuration whereas triangle α-clustered configuration gave stronger v 3 − ε 3 correlation than the Woods-Saxon configuration especially at large N track . The photon flow observable from the α-clustered 12 C and 197 Au collisions could be another potential probe to investigate the initial geometry properties [61]. By using a hydrodynamic model with different structure configurations of 12 C beam, the transverse momentum spectra and anisotropic flow coefficients v n of thermal photons were calculated in 12 C + 197 Au collisions at √ s NN = 200 GeV. Comparing the results from unclustered with α-clustered 12 C, the work [61] found that the slope of thermal photon spectra was not sensitive to the orientations of collisions, however, the elliptic (v 2 ) and triangular flow (v 3 ) coefficients of direct photons for α-clustered 12 C structure were significantly larger and predominantly formed by the QGP radiation. Also a strong anti-correlation between initial spatial ellipticity and triangularity was observed.
In another study at low energy [62], the γ-photon energy spectrum is reconstructed, which is in good agreement with the experimental data of 86 Kr + 12 C at E/A = 44 MeV within the framework of the modified EQMD model. The directed and elliptic flows of free protons and direct photons were investigated by considering the α-clustering structure of 12 C. The difference in the collective flows between different configurations of 12 C is observed. As the previous work [61], this study also indicated that the collective flows of direct photons are sensitive to the initial configuration.
As a short summary for the above parts, the intrinsic geometry from α-cluster arrangement in light nuclei 7,9 Be, 12 C and 16 O aroused people's attention to investigate the effect from the initial geometry asymmetry and fluctuations and further to understand how the initial geometry properties transfer to the final momentum space. It was found the α-clustered geometry structure contributed to not only the collective flows, especially to triangular flow v 3 , but also to the flow fluctuations, which was sensitive to the initial geometry characteristic.
Additionally, the effect of α cluster in relativistic collisions via di-hadron azimuthal correlation was also present [63]. A collision system scan involving α-clustered 12 C and 16 O is studied by using a multiphase transport model in the most central collisions at √ s NN = 6.37 TeV. By comparing the correlation functions of different configurations and extracting root-mean-square width and kurtosis in the away-side region, the results show substantial distinction in correlation functions between Woods-Saxon distribution and α-clustered structures. In indicates di-hadron azimuthal correlation could be seen a potential probe to distinguish α-clustered nuclei.
Two-particle intensity interferometry was proposed and developed by Hanbury Brown and Twiss (HBT) in the 1950s [64] and throughout the last decades, the HBT method had been extensively applied in heavy-ion collisions to study the evolution of the fireball [65]. Via the HBT method, two-particle momentum correlation provided a unique way to obtain direct information about the space-time structure. By using AMPT model, the azimuthal angle dependences of the HBT radii relative to the second-and third-order participant plane from π − π correlations were calculated in 12 C + 197 Au collisions at √ s NN = 200 GeV. And the Woods-Saxon distribution, chain and triangle α-clustering configuration were taken into account and the comparison among the three cases was discussed in reference [57]. It was proposed the ratio of the third-to the second-order HBT radii R 2 o(s),3 /R 2 o(s),2 could be taken as as a probe for the three configurations. The definition of HBT correlation function and HBT radii could be found in Ref. [57] and references therein. The hadronic rescattering time t hs was discussed in the evolution of fireball, which was not mentioned above. Figure 14 presented the time evolution of the ratio of the third-to the second-order HBT radii R 2 s(o),3 /R 2 s(o),2 with two K T bins [57]. The ratio of R 2 s(o),3 /R 2 s(o),2 showed clearly momentum dependence and more stable with high K T which was considered due to high momentum pions likely from the center of the source at earlier stage. The key point was that the order of R 2 s(o),3 /R 2 s(o),2 for different configurations of 12 C did not change with the hadronic rescattering time t hs , i.e. the hadronic rescattering process did not destroy the geometry properties inherited from the initial state.
It was suggested that the strong electromagnetic fields were produced in relativistic heavy-ion collisions [67]. The strong electromagnetic field was considered in nocentral heavy-ion collisions and resulted in possible chiral magnetic effects [68,69]. If the intrinsic initial condition was taken into account, the fields would be affected by the different initial configurations of the nucleus [66,69]. The electromagnetic fields in the collision system involving 12 C + 197 Au, 24 Mg + 197 Au, 64 Cu + 197 Au, and 197 Au + 197 Au at √ s NN = 200 GeV [66] were calculated by the Linard-Wiechert potential defined in [66], and the α-clustered 12 C with chain and triangle structure was introduced in the calculations. Figure 15 showed the centrality dependence (denoted by impact parameter b) of the x-component of electric field ( E x /m 2 π ) (left panel), and y-component of magnetic field (-B y /m 2 π ) (right panel) at the center of the collision zone in the mentioned systems [66]. The inset displayed E x /m 2 π (left panel) and -B y /m 2 π (right panel) for different initial configurations of 12 C [66]. It indicated the chain structure of 12 C affected the fields significantly. And the triangle structure of 12 C reduced E x /m 2 π in peripheral collisions rapidly and the effect with two K T bins [57]. could be ignored for -B y /m 2 π by comparing with the results from the Woods-Saxon configuration. It suggested that the system scan project sheds light on hints of the exotic nuclear structure of 12 C [66]. Of course, detailed theoretical efforts should be paid attention to the electromagnetic effects as a probe to distinguish the initial intrinsic geometry properties.

Influence of neutron skin effects
The geometry structure from the neutron skin effect could not be detected directly from the centrality classification but may affect the multiplicity distribution as a function of impact parameter [29,70,71], dynamical process, such as electroweak processes [29,[72][73][74] and final collectivities [71,75].

Lead-lead collisions
The right-hand panel of Fig. 2 presents the results in the case of symmetric Pb + Pb collisions. As earlier, sufficiently far away from the midrapidity, |y| 0, we can approximate where now x = ξ − Currently, the most accurate experimental measurements of W ± production in lead-lead collisions come from ATLAS Collaboration [45]. The data for W + /W − ratio are plotted in Fig. 3 and compared to the calculation described in this letter. The data, as well as the MCFM computation, include all W ± events within the kinematic region restricted by the lepton transverse momentum p T > 25 GeV, pseudorapidity interval |η ± | < 2.5, missing transverse momentum p missing T > 25 GeV, and the transverse mass of the ± ν system m T > 40 GeV. While our calculation is consistent with the data, χ 2 /N data ≈ 0.6 (N data is the number of data points), the measurements are clearly not accurate enough and have a too coarse centrality categorization to draw conclusions to any direction at this stage.

Summary and outlook
expected to induce in production of W ± bosons at t proton density is known to fall off more rapidly than trons towards the surface of neutron-rich nuclei like as we have demonstrated, correlates with the sign of t W ± boson. Thus, the W ± production could be use mark different experimental definitions of centrality We stress that in this paper we have considered the ce a purely geometric viewpoint neglecting e.g. possible the effect that could be caused by event-by-event Therefore, our results should be taken as first, roug of the expected systematics if a given experimenta selection method truly reflects the collision geometry. The effects caused by neutron skin are, of course, n W ± production but related phenomena are to be exp the case of high-p T photons and charged hadrons wh be measured at lower center-of-mass energies like th at the Relativistic Heavy-Ion Collider (RHIC). In the cas nucleus deeply-inelastic scattering, the neutron skin s differently the neutral-and charged-current reaction neutron-skin effect could serve as a handle to study ity also at planned deeply-inelastic scattering experim Electron-Ion Collider [58] or LHeC [59]. Fig. 16 The centrality dependence of W + /W − ratio as measured by the ATLAS Collaboration [76] in comparison with the results in reference [72].
The electroweak probe of the ratio of W ± was proposed in Ref. [72] to investigate the neutron skin effect in peripheral p + Pb collisions. In the Glauber modeling of the collisions, the geometry condition was considering as following for lead (Pb) nuclei: the density of nucleons in a spherical nucleus A was described in 2pF distributions by equation (6) and the density of protons and neutrons differs by fixing different parameters of a n , a p , d n and d p if the neutron skin was taken into account. Considering the cross section in hard process dσ hard nn = dσ hard pp = dσ hard pn and the production of inclusive charged leptons l ± from W ± → l ± ν decays (see [72] and references therein), the calculation reproduced the centrality dependence of W + /W − ratio as measured by the ATLAS Collaboration [76] as shown in figure 16 and suggested that the W ± production could be used to benchmark different experimental definitions of centrality at the LHC. Ref. [74] extended a Monte Carlo algorithm to investigate the neutron skin effect to the ratio of W + /W − in p + Pb collisions at LHC. Except the configuration of neutron skin in Pb nucleus was taken into account, the model also introduced the color fluctuation effects and the fluctuations of the number of collisions at a given impact parameter. It was found that the effect is a factor of 2 smaller than the original estimate [72] and suggested to extend the calculation of the W + /W − ratio in peripheral Pb-Pb collisions as in reference [29].
The neutron skin effect in direct-photon was investigated in reference [73]. The total number of participant nucleons could be affected by the neutron skin thickness, especially in peripheral collisions, which was due to the nucleon-nucleon collisions, namely pp, nn and pn collision. When the photon is emitted 'directly' in a hard scattering, the neutron skin would contribute to the inclusive prompt photon production in nucleus-nucleus collisions where the nucleus was assumed a neutron skin nucleus. Reference [73] calculated the 'central-to-peripheral ratio' (R cp ) of prompt photon production with and without accounting for the neutron skin effect and it was found the neutron skin caused a characteristic enhancement of the ratio, in particular at forward rapidity in Pb+Pb collisions at LHC. The neutron-skin effect in direct-photon was also investigated in reference [77] for the reactions induced by the neutron-rich projectile 50 Ca with 12 C and 40 Ca target in low-intermediate energy by the quantum molecular dynamics model. The results show that more direct hard photons are produced with the increasing of the neutron skin thickness. Meanwhile, it was found that central-to-peripheral ratio of yield R cp and the rapidity dependence of multiplicity and multiplicity ratio R cp are apparently sensitive to neutron skin thickness. In reference [29] with extending the study of reference [72], the results claimed that it would be demanding to unambiguously expose the neutron skin effect with direct photons [29]. In this calculation it considered the direct photons in either in the hard process or by the fragmentation of high-p T partons from the hard process and presented the uncertainties from the parton distribution function (PDF) and 2pF in details. Instead a probe for the neutron skin was proposed by a ratio between the cross sections for negatively and positively charged high-p T hadrons [29]. 148 Page 6 of 7 Eur. Phys. J. C (2017) Fig. 4 The expected statistical precision for 1 nb −1 (light vertical bars) and for 10 nb −1 (darker vertical bars) nucleon-nucleon luminosity in 70-80% (green) and 90-100% (blue) centrality classes. The shaded boxes show the 2pF uncertainty as in the lower panels of Fig. 3. The Kretzer FFs have been used observation is that with akk08 FFs the ratio in p+p actually turns negative at highp T , caused by the cross section for h − becoming negative. This clearly unphysical result implies that the considered kinematic region is out of the validity region of akk08. The results using dss and kretzer are not that different in p+p collisions but for n+n collisions almost a factor of two difference at the very highest values of p T is observed. These differences between the FF analyses generate some further theoretical uncertainty for the considered observable. Turning this around, a measurement of h − / h + in p+Pb or Pb+Pb collisions would clearly provide additional constraints for future FF analyses (modulo the possible finalstate effects in Pb+Pb).
The h − / h + ratios in Pb+Pb collisions at √ s NN = 5.0 TeV for centrality classes 70-80 and 90-100% are shown in the upper right panel of Fig. 3 together with the MB result with dss and kretzer FFs. The nPDF effects, including the cen-To estimate the achievable experimental precision h − / h + ratios discussed above, we multiply the cro tions from Eq. (2) by the nominal Pb-Pb nucleon-n luminosity of L nn = 1 nb −1 (also with L nn = 10 nb −1 would correspond to the luminosity targeted after th Long Shutdown 2 [39]). 4 From the resulting number of N we compute the relative statistical uncertainty by 1 To have better statistics, we consider here the rapidity b |η| < 3 (excluding the mid-rapidity to have a larger from the neutron skin) and suitably wide p T interva results are shown in Fig. 4 where the total statistical tainty follows from combining the statistical uncert for h + and h − quadratically. As can be seen from Fig realistically measurable region is p T < 200 GeV/c 70-80% bin and p T < 100 GeV/c for the 90-100% b would expect that systematic uncertainties are small i ratios like they are e.g. in the case of W charge asymm

Summary and outlook
We have studied the impact of the NS effect to directand charged-hadron production in Pb+Pb collisions LHC. In the case of photon production the NS effect h 10% impact on R γ PbPb though the uncertainties in the and their spatial dependence are of the same order o larger than the expected effect. With R γ CP some of the uncertainties cancel out making the NS effect more parent. Also, going to larger rapidities decreases the uncertainties, but still the smallness of the NS effect a ambiguities due to the centrality dependence of the makes the direct-photon production a challenging obse to study the NS effect.
The bulk properties, such as charged particle multiplicity, collective motion and chiral phenomena, were the popular observables to investigate the properties of quark-gluon plasma created in ultra-relativistic heavy-ion collisions. It became a key point in recent years if the nuclear structure affects these observables, or in other word, whether a certain observable could be a probe to explore the exotic nuclear structure.
The ratio between the cross sections for negatively and positively charged highp T hadrons [29], h − /h + , was proposed to investigate the neutron skin effect in Pb+Pb collisions at LHC. With considering the uncertainties from the production mechanism and model parameters, the centrality dependence from the neutron skin effect was also clearly visible in the proposed observable [29] as shown in figure 17 where the statistical uncertainty was also estimated as did for experiments. Recently Ref. [70] developed an algorithm of the neutron skin thickness based on the symmetry energy to investigate the effect where two different nuclear energy density functionals, namely the standard Skyrme-Hartree-Fock (SHF) model [78] and the extended SHF (eSHF) model [79], were employed to describe the equation of state (EOS) and the properties of finite nuclei. Inputting the calculated neutron and proton density distributions into the transport models, AMPT-sm, AMPT-def, UrQMD [80], and Hijing, separately, as the initial state, they simulated the 96 [70]. To quantify the splitting of the N ch tails, the relative N ch difference between Ru + Ru and Zr + Zr collisions was defined as, R = 2 N ch RuRu − N ch ZrZr N ch RuRu + N ch ZrZr where the tracking efficiency could be cancelled out mostly [70]. Figure 18 showed a relatively weak model dependence of the R as a function of the neutron skin thickness of Zr ∆ r np . It suggested that measurements in the future experiment could constrain the neutron skin thickness, the symmetry energy and the parameters in EOS via comparing with the calculated results [70]. Recently the calculation [81] was developed to provide a method for direct measurement of the neutron skin thickness, ∆ r np , by using net-charge multiplicities in ultra-peripheral collisions of those isobars. The calculations in references [29,70,81] demonstrated that the neutron skin could affect the charged particle multiplicity distribution since the model, collision system and energy were different and all proposed to perform the experimental measurement to reveal the the neutron skin effect in the collisions.
Ref. [71] proposed to determine the neutron skin type by 96 44 Ru + 96 44 Ru and 96 40 Zr + 96 40 Zr collisions at √ s NN = 200 GeV. It was found that the Ru+Ru/Zr+Zr ratios of the N ch distributions and ε 2 in mid-central collisions were exquisitely sensitive to the neutron skin type (skin versus halo) which was clarified by the parameters in the Woods-Saxon distribution. Figure 19 clearly showed the separation tail of multiplicity ratio and the eccentricity ratio due to the neutron skin type (skin vs halo) and the eccentricity ratio pattern would be reflected at the final momentum asymmetry as observables in experiment.

Influence of the deformation in isobaric collisions
The deformation could result in the complexity of the initial geometry in ultrarelativistic heavy-ion collisions. This complexity results in the uncertainty to un-monotonically increases with Δr np . This is because, with increasing Δr np , the difference between Ru and Zr densities increases. This results in an increasing difference in N ch between Ru þ Ru and Zr þ Zr collisions. Figure 4 further shows that the value of R has a relatively weak model dependence. This includes even HIJING whose tail distribution is significantly narrower than the other models (cf. Fig. 2), which can already be discriminated by data measurements as aforementioned. In what follows, we further decipher what R entails by using nondynamical but intuitive geometrical models, namely, the GLAUBER and TRENTO models.
In a GLAUBER model [54,55], it is postulated that N ch ∝ ½ð1 − xÞN part =2 þ xN bin , where x is the so-called hard-component fraction. The N part and N bin differ between the two isobar collisions, in a manner sensitive to the nuclear density parameters. This yields a nonvanishing R dependence of Δr np , as shown by the dynamical model results. In addition, the different contribution weights from N part and N bin to N ch , characterized by the x parameter, also affect the R. The four models giving similar R as a function of Δr np may be indicative of their similar x values. This is not surprising because all these models can approximately describe the centrality dependence of N ch observed by the PHOBOS experiment, which suggested an x value of 0.1 [56]. In order to investigate the sensitivity of R to the x value, we show in Fig. 4 results from the GLAUBER model [57][58][59] with x ¼ 0.1 and a significantly larger value of x ¼ 0.2 (which can be taken as an extreme). Indeed, the x ¼ 0.1 results fall within the range of the four dynamical models. The x ¼ 0.2 results have a stronger sensitivity to Δr np , indicating that the N bin is more sensitiv nuclear density distribution.
On the other hand, the two-component particl tion model has drawbacks. The centrality depen N ch at ffiffiffiffiffiffiffi ffi s NN p ¼ 200 GeV and 2.76 TeV have essen same shape [60], whereas the hard-scattering cros hence the x value, should vary with energy. The experiment found that a simple wounded quark m successfully describe the centrality dependence Au þ Au collisions over 7.7-200 GeV [61] and small systems at 200 GeV [62]. Recent a anisotropy data by STAR in ultracentral U þ U cannot be described by the GLAUBER approach [63 TRENTO model was proposed [64]. In the TRENT particle production is only related to the reduced t , a gam tuation parameter k ¼ 1.4, and a Gaussian nucleo 0.6 fm, which were found to well describe the m data in heavy ion collisions [64,65]. The R calc the TRENTO model is shown in Fig. 4. The Δr np ence is weaker because only N part contributes to Considering all model results, the overall sprea wider at larger Δr np . This is because the nuclea difference between the Ru and Zr nuclei incre increasing Δr np , so does the model dependence Experimentally, the N ch can be measured ex precisely. The relative hN ch i difference in cent sions is immune to many experimental unce Figure 4 thus strongly suggests that the iso may determine Δr np relatively accurately. This is by the rather weak dependence of the R observa details of QCD physics on particle production. T contrast to the hadronic observables in previ energy studies, where strong model dependence a more precise determination [17,22]. The curre imental range of the 208 Pb Δr NP is indicated by on the top of Fig. 4. The 208 Pb Δr np calculate eSHF (SHF) to accommodate those low-energy ments are written above the band. Our results indicate that with a given measurement of R, the in the derived Δr np of 96 Zr can be as good as covered by the four dynamical models (AMPT-sm HIJING, AMPT-def) and the two static models (TRE GLAUBER), as illustrated by the lower band hypothetically R ¼ 0.006). This would be an ap improvement over the current constraint from 2 results shall thus provide a significant input constrain the symmetry energy, bearing importa cations to nuclear matter and neutron star EOS We have assumed spherical nuclei in our cal The main idea of our work is still valid with nuclei. There are a number of promising ways to d the nuclear deformity from heavy ion collisions We postpone such a study to a future work. terized from the DFT-calculated ones, and compares them to those calculated directly from the DFT densities. It is found that the DFT densities and the WS parameterized ones give essentially the same Ru+Ru/Zr+Zr ratios. The two densities do yield slightly different results for each individual Ru+Ru or Zr+Zr system, up to a couple of percent (see the insets), but these differences cancel in the Ru+Ru/Zr+Zr ratios. This indicates that the properly parameterized WS densities are indeed sufficient to study the N ch and 2 . Note that the ratios in Fig. 2 are somewhat larger than the ones from the halo-type densities in Fig. 1. This is because the neutron skin thickness of Zr from the DFT calculation is r np =0.16 fm, larger than the 0.12 fm used in Fig. 1. It implies that those ratios in non-central collisions can further be used to quantitatively measure the neutron skin thickness. The halo-type neutron densities from DFT indicate that Zr has a larger diffuseness parameter a and rms radii than Ru. These yield distinctive shapes of Ru+Ru/Zr+Zr ratios of the N ch distributions and 2 .
For this study we have generated 50 million events for each collision species in each density case. Total over 2 billion events have been collected for each isobar system by the STAR experiment at RHIC [5], so the statistical uncertainties would be negligible. Since most of the systematic uncertainties cancel in those ratios between the two isobar systems, our results can be readily compared against experimental data to yield valuable information on the neutron skin type and thickness. If experimental data, on the other hand, do not favor our DFT and halo-type neutron density predictions, but rather the skin-type neutron density ones, then an overhaul of our current knowledge of nuclear structure would be called for.

Effect of nuclear deformation
So far, we have focused on the discussion of spherical nuclei. The elliptic flow in most central collisions is extra sensitive to In the previous section, we have demonstrated that DFT densities can be adequately approximated by halo-type neutron densities for comparative studies of N ch and 2 in isobar collisions. We assume this preserves for deformed Ru and Zr; this assumption is important as it allows us to bypass the challenging DFT calculations to still investigate efficiently the effects of nuclear deformity.
With a given β 2 , we determine R and a of the proton and neutron densities, respectively, by matching the r and r 2 (Eq. (5)) of the corresponding spherical DFT densities. We set β 2 = 0.16 for this initial study; the corresponding WS parameters are listed in Table 2. The finite β 2 reduces the a parameter but have negligible effect on R. Fig. 3 shows the Ru+Ru/Zr+Zr ratios of the N ch distributions and 2 with various combinations of Ru and Zr deformities. The N ch distribution ratio changes insignificantly due to the finite β 2 ; the change comes primarily from that in a, e.g. a finite β 2 for Zr reduces the a value yielding a larger N ch in Zr+Zr collisions compared to the spherical case and a smaller ratio of the N ch distributions. The 2 ratios, with the same β 2 value for both Ru and Zr, are almost identical for the spherical (β 2 = 0) and deformed (β 2 = 0.16) cases, except perhaps in very central collisions. If the two nuclei differ in β 2 , then the 2 ratio is significantly impacted in central (and semi-central) collisions, as expected [33]. However, the effect of deformation on the 2 ratio in peripheral and semiperipheral collisions is insignificant. This feature indicates that the shape of the 2 ratio in peripheral and semi-peripheral collisions can still be used to determine the neutron skin type and thickness as we demonstrated in the previous section with spherical nuclei. As aforementioned, 40 times more statistics have been collected by STAR than used in this simulation [5]. Even though experimental triggers suffer inefficiencies in peripheral collisions, it is significant only in very peripheral collisions (N ch 30), and even there the efficiency is larger than 20%. In other words, despite the large bias to the v ratio in (semi-)central collisions, our proposal to discrim- Fig. 19 The Ru+Ru/Zr+Zr ratios of (a) the N ch distributions and (b) the ε 2 in relativistic isobar collisions with various combinations of 96 Ru and 96 Zr nuclear deformities. The lower panels show the double ratios of the deformed/spherical (and spherical/deformed) to the spherical/spherical results [71].
derstand the transformation efficiency of asymmetry from the initial geometry to the final momentum space and the extraction of viscosity over entropy ratio (η/s) from the QGP created in ultra-relativistic heavy-ion collisions. In the passed two decades, amount of works raised in theory [16] and experiments [18] contributed to exploring the properties of collectivity in the fireball. Recently the RHIC-STAR Collaboration [82] reported the azimuthal anisotropy measurements of strange and multistrange hadrons in U+U collisions at √ s NN = 193 GeV and the deformation effect was studied in references [83], as well as the ATLAS [84] and the ALICE [85], respectively, reported measurements of the azimuthal anisotropy of hadrons produced in Xe+Xe collisions at √ s NN = 5.44 TeV and investigated in references [83]. Corre-lations between transverse momentum and collective flow coefficients or among the different order of the flow coefficients were investigated thoroughly and suggested as the tools to reveal the characteristic of the initial state of the collisions [86,87]. All of these efforts in experiments and theory contributed strong constraints on both the initial state geometry and medium response in ultra-relativistic heavy-ion collisions. Next a brief review presents for the recently popular studies in isobaric collisions with considering the deformation in 96 Zr) conducted in 2018 at the Relativistic Heavy-Ion Collider (RHIC) was motivated by the search for the chiral magnetic effect (CME) in quantum chromodynamics [88]. It was a basic problem how the initial state affects the CME signal and background from collective flow in the isobaric collision systems with similar size but different initial charge from protons. The spatial distribution of 96 44 Ru and 96 40 Zr in the rest frame could be written in the Woods-Saxon form as in equation (7).
Ref. [32] calculated the magnetic field and the initial geometry eccentricity coefficient in 96  GeV by using HIJING model in which the nuclear deformation was considered and formulated in equation (7). The results showed more than 10% difference in the CME signal and less than 2% difference in the elliptic-flow-driven backgrounds for the centrality range of 20-60%, which indicated that the isobaric collisions could provide an ideal tool to disentangle the CME signal from the background effects.
However, Ref. [89] gave a conservative prediction of the CME signal to background by employing the density functional theory (DFT) to calculate the proton and neutron distributions as the initial spatial state. The eccentricities and magnetic fields were calculated by using the Monte Carlo-Glauber model and the DFT densities were implemented in AMPT model to simulate the elliptic flow. It was found that the results respective to the participant and reaction planes resulted in comparable differences, in the two isobaric collision systems, for eccentricities or magnetic fields and the differences are insignificant with respect to the participant plane, which possibly weakened the power of isobaric collisions for the CME search [89]. Actually, very recently STAR Collaboration reported the first detailed results for searching for CME by Ru+Ru to Zr+Zr isobaric collisions and found no visible CME signal [90]. Of course, this is not the end of CME story. In contrary, the data needs to more high statistics and the background needs to be more clearly understood and separated. It's a key problem to understand the background in CME which is considered mainly from the collective flow. The quadrupole and octupole deformations introduced simultaneously for the isobar systems can reproduce the centrality dependence of the v 2 ratio qualitatively and v 3 ratio quantitatively [91]. It provides a new method to constraint or measure parameters of nuclear structure in relativistic heavy-ion collisions systems [91].
Ref. [92] developed a hydrodynamic framework which was initialized by the impact parameter-dependent Glasma model and employed UrQMD [80] transport model as an afterburner to simulate the ultra-relativistic heavy-ion collisions, as well as isobaric collision system. shows the probability distributions in b from FT) for an interaction to happen (N part 2). ity is linear in b up to 10 fm or so because at wo colliding nuclei are guaranteed to interact. e probability drops because not every encounter ast one nucleon-nucleon (NN) interaction. The at slightly smaller b in Ru+Ru than Zr+Zr ause Ru is slightly smaller than Zr from the DFT

A. Multiplicity distributions
ows the midrapidity charged hadron multiplicity tions from the AMPT-SM(DFT) simulations. ity counts all charged pions, charged kaons, proprotons within the pseudorapidity range of |η| < erse momentum range of p T > 0.2 GeV/c. The are almost identical between Ru+Ru and Zr+Zr e average multiplicities N ch for MB Ru+Ru  and Zr+Zr collisions are 65.97 and 65.15, respectively. The slightly larger N ch and larger N ch tail in Ru+Ru than Zr+Zr collisions is because of the smaller effective radius of Ru than Zr from the DFT mass densities [8]. With smaller radius, the Ru+Ru collision zone is smaller to start with, and the energy density should be higher and the number of binary collisions larger. Figure 4 shows the ratio of the N ch distribution in Ru+Ru to that in Zr+Zr collisions. The DFT density result is shown in red where the ratio curves up at large N ch because of the larger N ch tail in Ru+Ru (see Fig. 3). Also shown for comparison is the ratio from the WS-R Q densities, which is consistent with the calculation in Ref. [5]. The trend of the ratio in the WS-R Q case is the opposite to the DFT case, because the Ru charge radius is larger than the Zr's in the WS-R Q . These opposite behaviors are a decisive discriminator for the relative mass radii between the Ru and Zr nuclei. Also shown in Fig. 4 is the ratio from simulations using the WS-R DFT , which give a similar trend as AMPT-SM(DFT) that directly used DFT densities. This confirms the conclusion that the tail behavior in the ratio is mainly due to the ordering of the nuclear mass radii.
There is observable difference in the intermediate N ch region between simulations directly using DFT densities and using WS densities with DFT effective radii. This may be used to tell whether the density distribution is DFT like or simply WS like with the correct ordering of the mass radii. In fact, the intermediate N ch regions of the ratios from the two WS densities appear unfeatured; however, that from the DFT density seems nontrivially structured. If experimentally confirmed, then it would constitute strong evidence for the validity of the DFT densities for the isobaric nuclei.
A very successful isobar run has just concluded in 2018. STAR has taken approximately 2 billion minimum bias events each for Ru+Ru and Zr+Zr collisions. This is a factor of 25 higher statistics compared to the AMPT statistics we have run for AMPT-SM(DFT). The data statistical error bars would be approximately a factor of 5 smaller than those 054907-3 to our AMPT simulation, and with largely canceled systematic uncertainties in the v 2 ratio, the two scenarios of DFT and WS nuclear densities can be decisively determined. Note that the difference in the DFT v 2 ratios between Fig. 9 and Fig. 10 is because of the smearing in b when the centrality is defined according to multiplicity.

V. SUMMARY
In this paper, we make predictions of isobaric 96 44 Ru + 96 44 Ru and 96 40 Zr + 96 40 Zr collisions using the string-melting version of the AMPT model with the nuclear density distributions calculated by the DFT. We present the charged hadron multiplicity (N ch ) distributions and the elliptic azimuthal anisotropies v 2 as a function of centrality. Emphases are put on th differences between the two isobaric systems.
We show that, while the charge radius of 96 44 Ru is l that of 96 40 Zr, the mass radius of 96 44 Ru from DFT i smaller than that of 96 40 Zr. Because of this, the ra N ch distributions in Ru+Ru to Zr+Zr collisions cu large N ch , opposite to the trend obtained using the WS densities with charge radii. This feature can be against isobaric data to decisively determine the rela radii of the isobaric nuclei.
With the same effective mass radii, while the m distribution ratio may lose the discriminating powe DFT and WS densities, the centrality dependence ratio in Ru+Ru to Zr+Zr collisions can decisively whether DFT density is more realistic than WS.
For the CME search, it is important to first dete initial conditions of the isobaric collisions. The impo our work lies in the testable predictions from seve nuclear density distributions by a commonly used model. With the large data statistics (billions of ev cumulated in the isobar run at RHIC and cance systematic uncertainties, our predictions should b decisively determine the initial conditions of the collisions and pave the way for further studies, pa in terms of the CME search. to 10% on the elliptic flow of both collision systems (Ru+Ru and Zr+Zr) related to deformation.
Ref. [75] introduced isospin-dependent nucleon-nucleon spatial correlations in the geometric description of both nuclei, deformation for 96 44 Ru and the so-called neutron skin effect for the neutron-rich isobar, i.e., 96 40 Zr in the SMASH model. The results suggested a significantly smaller CME signal to background ratio for the experimental charge separation measurement in peripheral collisions with the isobar systems than previously expected [75].
The charged hadron multiplicity distributions and collective flow were studied in the deformed isobaric collisions [33] by using AMPT model with the nuclear structure calculated by the density functional theory (DFT) [89]. Reference [33] presented the predictions for the charged hadron multiplicity distributions in 96 44 Ru+ 96 44 Ru and 96 40 Zr+ 96 40 Zr collisions at RHIC. The charged hadron ratio showed in figure 20 and it clearly presented different trend of the centrality dependence of the ratio with nuclei radii from the DFT calculation and the mass radii [32]. The deformation effect to collective flow was presented in figure 21 which clearly show the elliptic flow difference for the centrality range of 20-60%.

Summary
A brief introduction to ultra-relativistic heavy-ion collisions and the effects from different initial state. structures were presented. The efforts in theoretical and experimental works were focused on the initial geometry asymmetry effect involving the overlap zone of the collided nuclei in collisions, the fluctuations and the intrinsic nuclear structure. In particular, impacts from α-clustering structure, neutron skin and nuclear deformation were discussed by hydrodynamics, transport model and some hybrid models as input in the initialization. The probes to exploring the intrinsic nuclear structure, or say how the intrinsic nuclear structure affects the final observables, were proposed by amount of theoretical works, such as collective flow, HBT correlations and charged hadron or photon production, or chiral magnetic effect in the collisions at RHIC and LHC. Although there exists model dependence or different choice of parameters, some final observables were sensitive to the initial sate. These efforts in various theoretical survey as well as experimental observation contributed to understand the basic problem of nuclear structure in nuclear community, and the ultra-relativistic heavy-ion collisions could be taken as a potential platform to study the nuclear structure except the original aim of quark-gluon plasma.