Deformation probes for light nuclei in their collisions at relativistic energies
Hai-Cheng Wang Song-Jie Li Lu-Meng Liu Jun Xu Zhong-Zhou Ren




School of Physics Science and Engineering, Tongji University, Shanghai 200092, China

School of Physics Science and Engineering, Tongji University, Shanghai 200092, China

Physics Department and Center for Particle Physics and Field Theory, Fudan University, Shanghai 200438, China
[Correspond to ]junxu@tongji.edu.cn
School of Physics Science and Engineering, Tongji University, Shanghai 200092, China

[Correspond to ]zren@tongji.edu.cn
School of Physics Science and Engineering, Tongji University, Shanghai 200092, China


§ ABSTRACT 
We have investigated the performance of anisotropic flows ⟨ v_n^2 ⟩, transverse momentum fluctuations ⟨δ p_T^2 ⟩, and their correlations ⟨ v_n^2 δ p_T ⟩ in central collisions at relativistic energies as probes of deformation parameters β_n of colliding nuclei, if these nuclei are light nuclei with large β_n and different configurations of α clusters. The effects from higher-order β_n terms are illustrated by derived relations based on the overlap of two nuclei with uniform density distributions and by dynamic simulations of collisions of heavy nuclei whose density distributions are of a deformed Woods-Saxon (WS) form. While the linear relations between β^2_n, ⟨ v_n^2 ⟩, and ⟨δ p_T^2 ⟩ and that between β^3_n and ⟨ v_n^2 δ p_T ⟩ can be violated for extremely large β_n, they are mostly valid for realistic values of β_n, as long as the density distribution of colliding nuclei can be described by a deformed WS form. However, these linear relations are generally not valid with more realistic density distributions of light nuclei with α clusters, and the amount of deviation depends on the detailed α-cluster configurations. Care must be taken when one tries to extract the deformation of light nuclei, and specific probes for α-cluster structures in these nuclei are very much needed.

Deformation probes for light nuclei in their collisions at relativistic energies
    Zhong-Zhou Ren
    September 9, 2024
================================================================================





§ INTRODUCTION


Understanding the density distribution in finite nuclei, especially their shapes, is a fundamental goal of nuclear physics. Comparing with traditional low-energy experiments, it has been realized recently that relativistic heavy-ion collisions can serve as an alternative way of achieving this goal (see, e.g., Ref. <cit.>). Compared to collisions of spherical nuclei, collisions of deformed nuclei provide more possible geometries and configurations, which may lead to different areas and shapes of the overlap region, and thus enhance the anisotropic collective flow and affect the transverse momentum spectrum <cit.>. The situation becomes more clear in central collisions, where various probes have been proposed to extract the deformation of colliding nuclei, e.g., a linear correlation is observed between ⟨ϵ_n^2 ⟩ and β_n^2 <cit.>, where ϵ_n represents the nth-order anisotropy coefficient of the overlap region and β_n is the nth-order deformation parameter of the colliding nuclei. Since the initial anisotropy in coordinate space is eventually transformed into the final anisotropy in momentum space, the square of the final nth-order anisotropic flow ⟨ v_n^2 ⟩ is expected to be linearly correlated with β_n^2 as well. There are also other probes such as the transverse momentum fluctuation ⟨δ p_T^2 ⟩ originated from the fluctuation in the initial overlap area characterized by ⟨δ d_⊥^2 ⟩ <cit.>, as well as the correlation between the anisotropic flow and the transverse momentum fluctuation <cit.>. These probes have been used to successfully extract the deformation parameters of ^96Ru <cit.>, ^96Zr <cit.>, ^197Au <cit.>, and ^238U <cit.>. It was proposed that light nuclei such as ^20Ne could have a huge deformation which can be measured through their collisions <cit.>. The recent ^16O+^16O collisions under analysis by the STAR Collaboration also provide a good opportunity to investigate the structure of light nuclei.

So far the linear relations between β_n^2 and ⟨ϵ_n^2 ⟩ (⟨ v_n^2 ⟩) or ⟨δ d_⊥^2 ⟩ (⟨δ p_T^2 ⟩) <cit.> work well generally based on studies by assuming that the density distribution of colliding nuclei is of a deformed WS form with small β_n. On one hand, whether the linear relation is valid for large values of β_n^2 such as those for light nuclei needs further investigations. On the other hand, it has been shown that light nuclei are generally formed by different configurations of α clusters <cit.>, and their density distribution can no longer be described by a deformed Woods-Saxon form. The typical 3-α cluster and 4-α cluster structure in ^12C and ^16O, respectively, have been under hot investigation for a long time <cit.>, and α clusters may have linear-chain and triangle configurations in ^12C, and linear-chain, tetrahedron, square, and Y-shape configurations in ^16O <cit.>. ^20Ne is largely deformed as proposed in Ref. <cit.>, and it also has a special internal structure with five α clusters <cit.>. Another candidate of light nuclei could be ^10Be, which has a life time long enough to be used for heavy-ion experiments, and it could be composed of two α clusters and two valence neutrons <cit.>. These different configurations of light nuclei may affect anisotropic flows in their collisions <cit.>. It is of interest to extract the deformation or the α-cluster structure of these nuclei with proper probes in their collisions at relativistic energies.

The purpose of the present paper is to investigate whether the probes used to extract the deformation of colliding nuclei work for largely deformed nuclei and for light nuclei with different α-cluster configurations. If not, how large is the error in extracting β_n from these probes for nuclei whose density distribution can not be described by a deformed WS form. Section <ref> presents how the density distributions of ^12C, ^16O, ^20Ne, and ^10Be are obtained from the microscopic cluster model with Brink wave function <cit.>, and reviews briefly the structure of a multiphase transport (AMPT) model <cit.> used to simulate the collision dynamics of these nuclei at relativistic energies. Section <ref> first illustrates the performance of the probes for the deformation of colliding nuclei by using ^96Zr+^96Zr collisions, with the density distribution of ^96Zr parameterized by a deformed WS form, and then investigate the difference in the probes using density distributions of a deformed WS form and more realistic α-cluster configurations for light nuclei. We conclude and outlook in Sec. <ref>.




§ THEORETICAL FRAMEWORK



In this section, we will first present the framework of calculating the density distributions of ^12C, ^16O, ^20Ne, and ^10Be with α-cluster structures, and then briefly review the AMPT model used for simulating the collisions of these light nuclei at relativistic energies. The deformation probes for colliding nuclei in their central collisions at relativistic energies will be discussed, and numerical relations related to these probes with simple assumptions will be derived.



 §.§ A microscopic cluster model


In order to obtain the density distribution of light nuclei, we adopt the following Hamiltonian

    Ĥ   =   ∑^A_i=1 E_i^
      + ∑_i<j V^NN(r_ij) 
       +   ∑_i<j V^Cou(r_ij) + ∑_i<j V^ls(r_ij).

The summation in the above Hamiltonian is over the total nucleon number A. The first term represents the kinetic energy in the center-of-mass (c.m.) frame, the second term is Volkov No.2 force <cit.> representing the effective nucleon-nucleon interaction, the third term is the Coulomb interaction, and the fourth term is the spin-orbit interaction, with r⃗_ij being the relative coordinates between nucleon i and nucleon j. The form of the effective nucleon-nucleon interaction can be expressed as

    V^NN(r_ij)
      = ( V_1 e^-α_1 r^2_ij - V_2 e^-α_2 r^2_ij )
      ( W - M P̂_σP̂_τ + B P̂_σ - H P̂_τ ),

where P̂_σ and P̂_τ are the spin and isospin exchange operator, respectively, and V_1 = -60.650, V_2 = 61.140, α_1 = 0.980  ^-2, and α_2 = 0.309  ^-2 are used in all calculations. For the calculation of ^12C, ^16O, and ^10Be, we use W = 0.4, M = 0.6, and B = H = 0.125, determined from the phase shift data of α-nucleon and α-α scatterings as well as the binding energy of deuteron <cit.>, and they are also used in Ref. <cit.>. For the calculation of ^20Ne, we use W = 0.38, M = 0.62, and B = H = 0, as used in Ref. <cit.>. The G3RS force is
used for the spin-orbit interaction with the form

    V^ls(r_ij) = u_ls(e^-α_3 r^2_ij - e^-α_4 r^2_ij) L⃗·S⃗P̂_31,

with u_ls=1600 MeV, α_3=5 fm^-2, and α_4=2.778 fm^-2 as in Ref. <cit.>, and P̂_31 being the operator which projects the two-nucleon system to the (S=1, T=1) state.

For ^12C, ^16O, and ^20Ne, their ground states can be described with the Bloch-Brink wave function, namely, the creation operator C^†_α of α clusters acting on the vacuum state | vac⟩, i.e.,

    | Φ^Brink⟩ = (C^†_α)^n
          | vac⟩,

with n=A/4 being the number of α clusters, and

    C^†_α    = ∫ d^3 r_1 ⋯ d^3 r_4 
       ×ϕ(r⃗_1 - R⃗) a^†_σ_1, τ_1(r⃗_1)
          ⋯ϕ(r⃗_4 - R⃗) a^†_σ_4, τ_4(r⃗_4).

In the above, the spatial part of a single particle has a Gaussian form

    ϕ(r⃗ - R⃗) ∝exp[- (r⃗ - R⃗)^2/2 b^2],

where the Gaussian width b is fixed to be 1.46 fm for all nucleons <cit.>, and a^†_σ, τ is the creation operator for nucleon with spin σ and isospin τ.

For ^10Be, we follow the framework in Ref. <cit.>, where the ground state is described by the Tohsaki-Horiuchi-Schuck-Röpke (THSR) wave function, i.e.,

    | Φ^THSR_^10Be⟩
          = (C^†_α)^2 (c^†_n)^2
          | vac⟩
          ,

where C^†_α and c^†_n are the creation operators of α clusters and valence neutrons, respectively. The creation operator C^†_α of α clusters in the THSR framework is reformulated as

    C^†_α    = ∫ d R⃗ 𝒢_α(R⃗)
            ∫ d^3 r_1 ⋯ d^3 r_4 
       ×ϕ(r⃗_1 - R⃗) a^†_σ_1, τ_1(r⃗_1)
            ⋯ϕ(r⃗_4 - R⃗) a^†_σ_4, τ_4(r⃗_4),

according to the α-condensate picture <cit.>. Here a Gaussian container 𝒢_α(R⃗) is used to confine the motion of α clusters, i.e.,

    𝒢_α(R⃗)
                = exp(
                -R_x^2 + R_y^2/β_α,xy^2
            -R_z^2/β_α,z^2),

where β_α,xy and β_α,z are parameters to be optimized by the variational principle in the calculation.
The creation operator c^†_n of valence neutrons is formulated as

    c^†_n =
            ∫ d R⃗ 𝒢_n(R⃗)
            ∫ d^3 rϕ(r⃗ - R⃗) a^†_σ, τ(r⃗).

In the above, 𝒢_n(R⃗) is the Gaussian container for valence neutrons, which is modulated by a phase factor <cit.> to reproduce the negative parity of the π-orbit, i.e.,

    𝒢_n(R⃗)
                = exp(
            -R_x^2 + R_y^2/β_n,xy^2
            -R_z^2/β_n,z^2)         e^i m ϕ_R⃗.

Here m is the magnetic quantum number, which is set to be +1 and -1 for the two valence neutrons to achieve an overall zero angular momentum, and ϕ_R⃗ is the azimuthal angle of the neutron position R⃗ in the spherical coordinate system.

The above wave functions should then be further antisymmetrized, and they are calculated in the c.m. frame of the nuclei. To restore the rotational symmetry, we further project the wave functions through

    | Ψ^J_M⟩
      = 𝒫̂^J_M,K | Φ^⟩,

where the corresponding angular momentum projection operator is expressed as <cit.>

    𝒫̂^J_M,K =
      2J+1/16π^2 ∫^2π_0 dϕ∫^π_0 d θ sin(θ) 
    ×∫^4π_0 dγ  D^J*_M,K(ϕ, θ, γ)
      R̂(ϕ, θ, γ),

with D^J_M,K(ϕ, θ, γ) being the Wigner rotation matrix, R̂ being the rotation operator, and ϕ, θ, and γ being the three Euler angles. J and M are respectively the quantum number of the total angular momentum and that in the third direction after projection, and K is the one before projection. Since the ground states of ^12C, ^16O, ^20Ne, and ^10Be are all 0^+ states, both J and M are set to be 0. The wave function | Ψ^J_M⟩ after the angular momentum projection is used to calculate the energy of the ground state, and the variational principle is used in order to find the minimum energy with the optimized distance among nucleons for specific configurations as shown in Fig. <ref>. For ^12C, we consider typical triangle and linear-chain configurations, as shown in Fig. <ref> (a) and Fig. <ref> (b). For ^16O, we consider typical tetrahedron and linear-chain configurations, as shown in Fig. <ref> (c) and Fig. <ref> (d). The distance parameter d in the configurations of ^12C and ^16O is varied in order to get the minimum energy. For ^20Ne, we take the structure of a tetrahedron configuration for four α clusters and another one under the bottom of the tetrahedron <cit.>, as shown in Fig. <ref> (e), and the distance parameters d and d_5 are varied. For ^10Be, as shown in Fig. <ref> (f), values of parameters for the Gaussian container are all varied to achieve the ground-state energy.



With the optimized distance for a specific configuration obtained, we use the wave function before the angular momentum projection to calculate the density distribution

    ρ(a⃗)
      = ⟨Φ | ∑_i=1^A δ(r⃗_i - a⃗) | Φ⟩/⟨Φ | Φ⟩
      = ∑_i=1^A ρ_i(a⃗),

where the density for the ith nucleon at position a⃗ can be calculated from

    ρ_i(a⃗)
         =   ⟨Φ | δ(r⃗_i - a⃗) | Φ⟩/⟨Φ | Φ⟩
       =   1/(2 π)^3∫ d^3 k e^-i k⃗·a⃗⟨Φ | e^i k⃗·r⃗_i | Φ⟩/⟨Φ | Φ⟩.

The initial nucleons in relativistic heavy-ion collisions are sampled according to the above density distributions, and the later dynamics is modeled by the string-melting version of the AMPT model, to be discussed in the next subsection. It is noteworthy that we first sample the nucleons within each α cluster, and then use the corresponding distance parameters to construct the specific configuration, so in this way the α-cluster structure is preserved in the sampling.



 §.§ A multiphase transport model


In the AMPT model <cit.>, the initial particle production in relativistic heavy-ion collisions is modelled by a heavy ion jet interaction generator (HIJING) model <cit.>, where the Lund string fragmentation function

    f(z) ∝ z^-1 (1-z)^a exp(-b m_⊥^2/z)

is used to describe the momentum spectrum of the produced particles, with z being the light-cone momentum fraction of the produced hadron of transverse mass m_⊥ with respect to that of the fragmenting string, and a and b being two paramters. In the string-melting version, these particles are converted to their valence quarks and antiquarks at the same spatial coordinates. Partons do not undergo scatterings until they have propagated for a given formation time. The later dynamics of these partons is described by Zhang's parton cascade (ZPC) model <cit.>, where two-body elastic scatterings between partons are simulated using the following differential cross section

    dσ/dt≈9πα_s^2/2(t-μ^2)^2,

with t being the standard Mandelstam variable for four-momentum transfer, α_s being the strong coupling constant, and μ being the screening mass in the partonic matter. After the kinetic freeze-out of these partons, quarks and antiquarks are converted to hadrons via a spatial coalescence model. The later dynamics of the hadronic phase is described by a relativistic transport (ART) model <cit.> with various hadronic elastic and inelastic scattering and decay channels.

In the present study, we set the values of the parameters to be a=0.5 and b=0.9 GeV^-2 in the Lund string fragmentation function [Eq. (<ref>)], and α_s=0.33 and μ=3.2 fm^-1 in the parton scattering cross section [Eq. (<ref>)]. These parameters have been shown to reproduce the particle multiplicity and anisotropic flows in Au+Au collisions at √(s_NN)=200 GeV <cit.> and Pb+Pb collisions at √(s_NN)=2.76 TeV <cit.> reasonably well.



 §.§ Probes for deformation of colliding nuclei



In this subsection, we give a brief discussion of the probes for the deformation of colliding nuclei in relativistic heavy-ion collisions. Let's first assume that the nucleus density distribution can be approximately described by an axial symmetric deformed WS form, i.e.,

    ρ(r,θ) = ρ_0/1+exp[(r-R(θ))/a].

In the above, ρ_0 is the normalization constant, a is the diffuseness parameter, and

    R(θ)=R_0[1+∑_nβ_n Y_n,0(θ)]

is the deformed radius, with R_0 being the average radius, β_n being the deformation parameters, and Y_n,0 being the spherical harmonics.

For a given density distribution with axial symmetry, the deformation parameter β_n^⋆ can be calculated from

    β_n^⋆ = 4π Q_n/3 A R_rms^n,

where A is the total nucleon number, R_rms is the root-mean-square (rms) radius, and

    Q_n = ∫ρ(r) r^n Y_n,0(θ) d^3 r

is the intrinsic multipole moment. An axial symmetric density distribution with a β_n in Eq. (<ref>) generally leads to a different β_n^⋆ <cit.>. The previous studies, e.g., Refs. <cit.>, mostly extract β_n in Eq. (<ref>) rather than β_n^⋆ in Eq. (<ref>).

In the present study, we focus on the anisotropic flows ⟨ v_n^2 ⟩, transverse momentum fluctuations ⟨δ p_T^2 ⟩, and their correlations ⟨ v_n^2 δ p_T ⟩ as deformation probes in relativistic heavy-ion collisions, where ⟨...⟩ represents the event average. The nth-order anisotropic flow v_n originates from the nth-order anisotropy coefficient ϵ_n of the overlap region with respect to the event plane Φ_n. For tip-tip relativistic heavy-ion collisions at zero impact parameter, i.e., with symmetric axis head-on, ϵ_n can be formally expressed as

    ϵ_n e^inΦ_n=-∫ r^n sin^n(θ) e^inϕρ(r,θ)d^3r/∫ r^n sin^n(θ)  ρ(r,θ)d^3r.

For other collision configurations, the orientations of the colliding nuclei can be generated by incorporating the Wigner rotation matrix (see Appendix <ref>). ϵ_n for an arbitrary collision configuration can also be expressed as

    ϵ_n e^inΦ_n=-∫ r^n_⊥ e^inϕρ_⊥ (r_⊥,ϕ) r_⊥ dr_⊥ dϕ/∫ r^n_⊥ρ_⊥(r_⊥,ϕ)r_⊥ dr_⊥ dϕ,

where ρ_⊥(r_⊥, ϕ) = ∫ρ dz is the transverse nucleon density, with ρ being the nucleus density distribution at the given orientation. The transverse momentum fluctuation ⟨δ p_T^2 ⟩ originates from the fluctuation ⟨δ d_⊥^2⟩ of the overlap's inverse area d_⊥ = 1/√(x^2 y^2) <cit.>, where (...) represents the average value of a quantity in one event. For tip-tip relativistic heavy-ion collisions at zero impact parameter, x^2 and y^2 can be formally expressed as

    x^2   =   ∫ r^2 sin^2(θ) cos^2(ϕ) ρ(r,θ)d^3r/∫ρ(r,θ)d^3r,
    y^2   =   ∫ r^2 sin^2(θ) sin^2(ϕ) ρ(r,θ)d^3r/∫ρ(r,θ)d^3r,

respectively. For other collision configurations, again, the orientations of the colliding nuclei can be generated by incorporating the Wigner rotation matrix (see Appendix <ref>). The correlation between the anisotropic flow and the transverse momentum fluctuation ⟨ v_n^2 δ p_T ⟩ thus originates from that between the anisotropic coefficient and the overlap's inverse area ⟨ϵ_n^2 δ d_⊥⟩.

Following the procedure in Refs. <cit.>, we consider a uniform density distribution with a sharp edge, i.e., a=0 in Eq. (<ref>). Here we slightly go beyond the derivation in Refs. <cit.> by considering more higher-order β_n terms in the expansion of the numerator in Eqs. (<ref>), (<ref>), and (<ref>). With only nonzero β_2 and β_3 in Eq. (<ref>) as considered in the present study, we can get the following approximate relations

    ⟨ϵ_2^2 ⟩   =   0.477β_2^2+0.172β_2^3+0.161β_2β_3^2+O(β_n^4),
    ⟨ϵ_3^2 ⟩   =   0.539β_3^2+0.452β_2^2β_3+O(β_n^4),
    ⟨δ d_⊥^2⟩   =   0.25/R_0^4×(7.954β_2^2-4.301β_2^3+5.352β_2β_3^2)
          +O(β_n^4), 
    ⟨ϵ_2^2 δ d_⊥⟩   =   0.05/R_0^2×(-8.602β_2^3-3.875β_2^4-8.682β_2^2β_3^2)
          +O(β_n^5),
    ⟨ϵ_3^2 δ d_⊥⟩   =   0.05/R_0^2×(-7.849β_2β_3^2-0.440β_2^2β_3^2-3.813β_3^4)
          +O(β_n^5).


If ⟨ v_n^2 ⟩, ⟨δ p_T^2 ⟩, and ⟨ v_n^2 δ p_T ⟩ are linearly correlated with ⟨ϵ_n^2 ⟩, ⟨δ d_⊥^2⟩, and ⟨ϵ_n^2 δ d_⊥⟩, respectively, similar relations between ⟨ v_n^2 ⟩, ⟨δ p_T^2 ⟩, and ⟨ v_n^2 δ p_T ⟩ and β_n are expected to be valid. For the detailed derivation to get Eqs. (<ref>)-(<ref>), we refer the reader to Appendix <ref>. In the lowest order, these relations are similar to those in Refs. <cit.>, while higher-order β_n terms as well as cross terms appear when we consider large deformation for light nuclei. These relations are useful for the illustration purpose, while one should keep in mind that there are a few approximations and limitations of these relations. First, we neglected event-by-event fluctuations and used uniform density distributions with sharp surfaces. Here the event-by-event fluctuations are caused by finite particle numbers and stochastic dynamics. Second, the derivation is only valid for not too large β_n but may fail for β_n ∼ 1.



§ RESULTS AND DISCUSSIONS



The purpose of the present study is to investigate whether the deformation probes work for light nuclei with both large β_n and special internal structures. In this section, we will first evaluate the performance of the deformation probes for nuclei with large β_2 or β_3, and then discuss how these deformation probes work for light nuclei with different α-cluster configurations.



 §.§ Validity of probes at large deformation









The performance of the deformation probes is evaluated with ^96Zr+^96Zr collisions at √(s_NN)=200 GeV, and the density distribution of ^96Zr is parameterized as Eqs. (<ref>) and (<ref>) with only nonzero β_2 and β_3. We fix R_0=5.02 fm and a=0.52 fm as in Ref. <cit.> while vary β_2 from 0 to 1.5 and β_3 from 0 to 0.9 to get different deformations of the colliding nuclei, and the dynamics of their collisions is described by the AMPT model. We select events at 0-5% centralities according to charged-particle multiplicities, and investigate the relations between ⟨ϵ_n^2 ⟩, ⟨ v_n^2 ⟩, ⟨δ d_⊥^2 ⟩, ⟨δ p_T^2 ⟩, and ⟨  v_n^2 δ p_T ⟩ and the deformation parameters β_n in the large ranges of β_2 and β_3. The nth-order anisotropic coefficient and the fluctuation of the overlap's inverse area are calculated from the coordinates of partons at t=0 in AMPT according to

    ϵ_n  =  √([∑_i r_⊥,i^n cos(nϕ_i)]^2+[∑_i r_⊥,i^n sin(nϕ_i)]^2)/∑_i r_⊥,i^n,





    δ d_⊥^2 = (d_⊥ - ⟨ d_⊥⟩)^2,

respectively, where r_⊥,i=√(x_i^2+y_i^2) and ϕ_i=arctan(y_i/x_i) are respectively the polar coordinate and polar angle of the ith particle in the transverse plane, and d_⊥ = 1/√(x^2 y^2) is the overlap's inverse area with (...) representing the average over all particles in one event. δ d_⊥ is linearly correlated with the deviation of the mean transverse momentum away from its event-averaged value δ p_T = p_T- ⟨p_T⟩ <cit.>. The anisotropic flows, the transverse-momentum fluctuation, and their correlation are calculated from the phase-space information of particles at the final stage in AMPT according to

    ⟨ v_n^2 ⟩   =   ⟨cos [n(φ_i-φ_j)] ⟩_i,j, 
    ⟨δ p_T^2 ⟩   =   ⟨ (p_T,i - ⟨p_T⟩) (p_T,j - ⟨p_T⟩) ⟩_i,j, 
    ⟨ v_n^2 δ p_T ⟩   =   ⟨cos[n(φ_i-φ_j)] (p_T,k - ⟨p_T⟩)⟩_i,j,k,

respectively. Here ⟨...⟩_i,j,... represents the average over all possible combinations of i,j,... for all events, and p_T,i=√(p_x,i^2+p_y,i^2) and φ_i=arctan(p_y,i/p_x,i) are, respectively, the momentum and its polar angle of the ith particle in the transverse plane. Particles at midpseudorapidities (|η|<2) and 0.2<p_T<3 GeV are selected for the calculation, with a pseudorapidity gap of |Δη|>0.5 used in calculating ⟨...⟩_i,j,... to remove the non-flow effect.

Figure <ref> displays the relations between β^2_n, ⟨ϵ_n^2 ⟩, and ⟨ v_n^2 ⟩ in central ^96Zr+^96Zr collisions from AMPT. As shown in the first column, a large β_n^2 leads to a large ⟨ϵ_n^2 ⟩ for both n=2 and 3, and the linear relation holds until β_2^2 ≳ 1 and β_3^2 ≳ 0.4. The slower increasing trend at large values of β_2^2 is inconsistent with the positive β_2^3 term in Eq. (<ref>), likely due to the fact that the derivation is not valid at β_2 ≳ 1. On the other hand, the non-zero value of ⟨ϵ_2(3)^2 ⟩ at β_2(3)^2=0 is due to the effect of finite particle numbers which leads to event-by-event fluctuations. The traditional linear relation between ⟨ϵ_n^2 ⟩ and ⟨ v_n^2 ⟩ is shown in the second column, which is seen to be slightly violated at ⟨ϵ_2^2 ⟩≳ 0.15 and ⟨ϵ_3^2 ⟩≳ 0.08. Consequently, the linear relation between β^2_n and ⟨ v_n^2 ⟩ is violated for β_2^2 ≳ 1 and β_3^2 ≳ 0.4. For a finite β_3(2), ⟨ϵ_2(3)^2 ⟩ is larger for a given β_2(3)^2, consistent with the positive β_2β_3^2 term in Eq. (<ref>) and the positive β_2^2β_3 term in Eq. (<ref>), while this generally does not lead to larger anisotropic flows ⟨ v_2(3)^2 ⟩. Considering the largest β_2 and β_3 for light nuclei obtained in the present study (see Table <ref>), the linear relation between β^2_n and ⟨ v_n^2 ⟩ is mostly valid, as long as the density distribution can be approximately described by a deformed WS form. We have also compared the relations between ⟨ϵ_n^2 ⟩, ⟨ v_n^2 ⟩, and β^⋆_n^2, with β_n^⋆ calculated from Eq. (<ref>). It is interesting to see that the linear relations of ⟨ϵ_2(3)^2 ⟩∼β^⋆_2(3)^2 and ⟨ v_2(3)^2 ⟩∼β^⋆_2(3)^2 are preserved at extremely large β^⋆_2(3), although they have smaller slopes.

Considering that some nuclei (e.g., ^20Ne) have both large β_2 and β_3, we discuss more explicitly the cross relation between β^2_3(2) and ⟨ϵ_2(3)^2 ⟩ as well as ⟨ v_2(3)^2 ⟩ in Fig. <ref>. It is seen that ⟨ϵ_2(3)^2 ⟩ increases also approximately linearly with increasing β_3(2), and for a finite β_2(3) the whole curves move to the upper side. This is again consistent with the positive β_2β_3^2 term in Eq. (<ref>) and the positive β_2^2β_3 term in Eq. (<ref>). For the final anisotropic flows, the relation between ⟨ v_2^2 ⟩ and β_3^2 is similar to that between ⟨ϵ_2^2 ⟩ and β_3^2, while the relation between ⟨ v_3^2 ⟩ and β_2^2 shows a different behavior. Note that we use the same scale for the y axis in Fig. <ref> as in Fig. <ref>, so it is seen that the correlation strength shown in Fig. <ref> is much weaker than that shown in Fig. <ref>.

Figure <ref> illustrates the relation between β^2_n, ⟨δ d_⊥^2 ⟩, and ⟨δ p_T^2 ⟩ as well as between β^3_n and ⟨ v_n^2 δ p_T ⟩ in central ^96Zr+^96Zr collisions from AMPT calculations. A large β^2_2 leads to large fluctuations of the initial overlap area characterized by ⟨δ d_⊥^2 ⟩, and the linear relation approximately holds until β_2^2 ≳ 1 and becomes saturated, as shown in Fig. <ref> (a). The saturation behavior is consistent with the negative coefficient of the β_2^3 term in Eq. (<ref>). The linear relation between ⟨δ d_⊥^2 ⟩ and ⟨δ p_T^2 ⟩ is also approximately valid from the AMPT dynamics, as shown in Fig. <ref> (b). Consequently, the linear relation between β^2_2 and ⟨δ p_T^2 ⟩ is approximately valid until β_2^2 ≳ 1 and then the increasing trend becomes slower, as shown in Fig. <ref> (c). We also illustrate the relation between β^3_2 and ⟨ v_2^2 δ p_T ⟩ in Fig. <ref> (d). The negative value of ⟨ v_2^2 δ p_T ⟩ is due to the fact that a larger (smaller) overlap area generally leads to a smaller (larger) p_T but a larger (smaller) v_2^2, and this is especially so for a larger β_2 (see typical cases for central tip-tip and body-body collisions). The linear relation between β^3_2 and ⟨ v_2^2 δ p_T ⟩ is valid until β_2^3 ≳ 0.5 and then the slope becomes much smaller for larger β_2. This behavior is inconsistent with the negative β_2^4 term in Eq. (<ref>), probably due to the event-by-event fluctuation or that the derivation is not applicable at too large β_2. While there are some linear relations between β^2_3 and ⟨δ d_⊥^2 ⟩ as well as ⟨δ p_T^2 ⟩ and between β^3_3 and ⟨ v_3^2 δ p_T ⟩, the correlation strength is rather weak compared to those for β_2, as can also be expected from Eq. (<ref>) where there is no β_3^3 term. These relations with a fixed finite β_3 or β_2 are also compared, and the qualitative behaviors are consistent with the above discussions. Again, the linear relations between β^2_n, ⟨δ d_⊥^2 ⟩, and ⟨δ p_T^2 ⟩ as well as those between β^3_n and ⟨ v_n^2 δ p_T ⟩ are mostly valid for reasonable values of β_2 and β_3 in realistic nuclei whose density distributions can be approximately described by a deformed WS form. The linear relations of ⟨δ d_⊥^2 ⟩∼β^⋆_2(3)^2, ⟨δ p_T^2 ⟩∼β^⋆_2(3)^2, and ⟨ v_2(3)^2 δ p_T⟩∼β^⋆_2(3)^3 are better preserved, and they have mostly smaller slopes.



 §.§ Validity of probes with α clusters










Now we investigate whether the deformation probes shown in the previous subsection work well for collisions of light nuclei with α clusters, and we perform AMPT simulations for central collisions of ^12C+^12C, ^16O+^16O, ^20Ne+^20Ne, and ^10Be+^10Be also at √(s_NN)=200 GeV. In order to carry out a fair comparison, we construct an axial symmetric deformed WS density distribution as Eq. (<ref>) for each configuration of light nuclei, with β_2 and β_3 in the deformed WS distribution adjusted to reproduce β_2^⋆ and β_3^⋆ calculated using Eq. (<ref>) from the realistic density distribution with α clusters. The values of the radius parameter R_0 and the diffuseness parameter a in the deformed WS distribution are determined in such a way that the values of ⟨ r^2 ⟩ and ⟨ r^4 ⟩ should be the same as those from the realistic density distribution with α clusters, where the lth-order moment of r is defined as ⟨ r^l ⟩ = ∫ρ(r⃗) r^l d^3 r/∫ρ(r⃗) d^3 r.

Figure <ref> compares the realistic density distributions for ^12C, ^16O, ^20Ne, and ^10Be with different α-cluster configurations as shown in Fig. <ref> and the deformed WS distributions with parameters determined by the realistic density distributions as described above. The values of β_2^⋆ and β_3^⋆ from realistic density distributions as well as those of β_2 and β_3 used in the parameterized deformed WS distribution for different cases are given in Table <ref>. Here the density distributions of the deformed WS form are axial symmetric with respective to the z axis. For the density distributions with realistic α-cluster structures, they are also axial symmetric with respective to the z axis, except the triangle-shaped ^12C and tetrahedron-shaped ^16O, for which the density distributions are plotted in the 3-α plane for a better vision. Obviously, the deformed WS distributions are quite different from the realistic ones with α clusters in most cases, and one may expect that they may lead to different values of the probes even if they have the same β_2^⋆ and β_3^⋆.

Figure <ref> displays how the deformation probes using deformed WS density distributions deviate from those using realistic density distributions with different α-cluster configurations for ^12C, ^16O, and ^10Be. The left three columns show relation between probes and β_2(3) in the deformed WS distribution, and the right three columns show relations between probes and β^⋆_2(3) calculated from Eq. (<ref>). For both parameterized WS distributions or realistic density distributions as shown in Fig. <ref>, these nuclei have either finite β_2 (β^⋆_2) or finite β_3 (β^⋆_3). For the WS density distributions, we calculate results from spherical and deformed density distributions with the same R_0 and a, as shown by black squares in Fig. <ref>. Basically, ⟨ v_2(3)^2 ⟩, ⟨δ p_T^2 ⟩, and ⟨ v_2(3)^2δ p_T ⟩ follow qualitatively linear relations with β_2(3)^2 or β_2(3)^3, consistent with results from ^96Zr as shown in Sec. <ref>, indicated by solid lines in Fig. <ref>. However, results from most realistic density distributions with α clusters deviate from these linear relations. The chain structure of ^12C and ^16O as well as the realistic density distribution for ^10Be lead to larger ⟨ v_2^2 ⟩, larger ⟨δ p_T^2 ⟩, and smaller ⟨ v_2^2δ p_T ⟩, and this can be understood from the larger asymmetries in z and r_xy directions for the realistic density distributions compared to those for a deformed WS form as shown in Fig. <ref>. The triangle structure of ^12C leads to a smaller ⟨ v_3^2 ⟩, a larger ⟨δ p_T^2 ⟩, and a smaller ⟨ v_3^2δ p_T ⟩, compared to a deformed WS distribution with the same finite β_3. The smaller ⟨ϵ_3^2 ⟩ from the triangle structure of ^12C can be intuitively understood, since the three α clusters actually form a plane, which generally does not lead to a large ϵ_3 of the overlap region for arbitrary collision configurations, while there are more collision configurations for a deformed WS distribution with a finite β_3 to have a large ϵ_3. The much larger ⟨δ p_T^2 ⟩ originating from the larger fluctuation of the inverse overlap's area ⟨δ d_⊥^2 ⟩ for the triangle structure of ^12C compared to the corresponding deformed WS distribution can also be understood in the similar way. The difference is smaller for the tetrahedron configuration of ^16O, since the density distributions from a deformed WS form and a realistic calculation are not quite different, as can be seen from Fig. <ref>. The relations between deformation probes and β_2(3)^⋆ show qualitatively similar behaviors except with smaller slopes. Here we don't see a robust deformation probe among those investigated in the present study that is only sensitive to the deformation parameters of colliding nuclei but insensitive to the existence of α clusters.

Figure <ref> shows similar results as Fig. <ref> but for ^20Ne which has both finite β_2 and β_3. The left two columns show relation between probes and β_2(3) in the deformed WS distribution, and the right two columns show relations between probes and β^⋆_2(3) calculated from Eq. (<ref>). Using the density distribution of the WS form, qualitatively linear relations between ⟨ v_2(3)^2 ⟩, ⟨δ p_T^2 ⟩, and β_2(3)^2 for a fixed β_3(2) are observed, as shown by black squares and blue triangles as well as solid lines, and the behaviors are qualitatively consistent with those from ^96Zr+^96Zr collisions as shown in Sec. <ref>. Again, the relations between deformation probes and β_2(3)^⋆ show qualitatively similar behaviors, except with smaller slopes. Overall, for the 5-α cluster structure of ^20Ne, we found that the values of the resulting deformation probes are not too different from those obtained from the parameterized WS distribution within statistical error.



§ CONCLUSIONS



We have investigated how the probes of deformation parameters β_n of colliding nuclei in their collisions at relativistic energies, such as anisotropic flows ⟨ v_n^2 ⟩, transverse momentum fluctuations ⟨δ p_T^2 ⟩, and their correlations ⟨ v_n^2 δ p_T ⟩, work for light nuclei with large β_n and different α-cluster configurations. By assuming a uniform density distribution with a sharp surface, we have derived the relations between the above probes and β_n, where higher-order relations and cross relations are observed. The performance of these probes is also investigated with AMPT simulations of collisions of heavy nuclei by assuming they have large β_n. While the linear relations between β^2_n, ⟨ v_n^2 ⟩, and ⟨δ p_T^2 ⟩ and that between β^3_n and ⟨ v_n^2 δ p_T ⟩ can be violated for extremely large β_n, they are mostly valid for realistic values of β_n, as long as the density distribution of colliding nuclei can be described by a deformed WS form. However, using more realistic density distributions with α clusters for light nuclei, these probes can deviate from those using a deformed WS form with the same deformation parameters, and the amount of deviation can be different for different α-cluster configurations. For the tetrahadron structure of ^16O and the 5-α cluster structure of ^20Ne, it is difficult to distinguish the difference in the deformation probes from realistic density distributions and WS density distributions. Therefore, specific probes for α-cluster structures in these nuclei are very much demanded in future analysis. For other cases, no robust deformation probe among those investigated in the present study, which is only sensitive to the β_n of colliding nuclei but insensitive to the existence of α clusters, is observed, so care must be taken when one tries to extract the deformation of light nuclei.


This work is supported by the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDB34030000, the National Natural Science Foundation of China under Grant Nos. 12375125, 12035011, and 11975167, and the Fundamental Research Funds for the Central Universities.






§ RELATIONS BETWEEN INITIAL GEOMETRY AND DEFORMATION



Here we try to derive the relations between the initial geometry in relativistic heavy-ion collisions at zero impact parameter and the deformation parameters β_n of colliding nuclei. The initial geometry is characterized by the nth-order anisotropy coefficient ϵ_n, the fluctuation of the overlap's inverse area δ d_⊥^2, and their correlation ϵ_n^2 δ d_⊥. We start from a density distribution with a general deformed WS form

    ρ(r,θ, ϕ) = ρ_0/1+exp[(r-R(θ,ϕ))/a]

with R(θ,ϕ)=R_0[1+∑_l,mβ_l α_l,m Y_l,m(θ,ϕ)], but we take the limit of a → 0 so that the density distribution is uniform with a sharp surface. We basically follow the procedure in Refs. <cit.> but keep more higher-order β_l terms.

ϵ_n with respect to the event plane Φ_n in central tip-tip relativistic heavy-ion collisions can be formally expressed as

    ϵ_n e^inΦ_n=-∫ r^n sin^n(θ) e^inϕρ(r,θ,ϕ)d^3r/∫ r^n sin^n(θ)  ρ(r,θ,ϕ)d^3r.

By using Y_n^n = √((2n+1)!!/4π(2n)!!)sin^n(θ) e^inϕ and uniform density distribution within r ∈ [0, R_0(1+∑_l,mβ_l α_l,m Y_l,m)], the above equation can be further written as


    ϵ_n e^inΦ_n   =    -√(4π (2n)!!/(2n+1)!!)∫ (1+∑_l,mβ_lα_l,mY_l^m)^n+3Y_n^n sin(θ) dθ dϕ/∫ (1+∑_l,mβ_lα_l,mY_l^m)^n+3sin^n+1(θ) dθ dϕ
       ≈    -√(4π (2n)!!/(2n+1)!!)∫ [1+(n+3) ∑_l,mβ_lα_l,mY_l^m+(n+3)(n+2)/2(∑_l,mβ_lα_l,mY_l^m)^2]Y_n^n sin(θ) dθ dϕ/∫ [1+(n+3)∑_l,mβ_lα_l,mY_l^m] sin^n+1(θ) dθ dϕ.

It is seen that we consider additional β_l^2 terms in the expansion of the numerator, compared to Ref. <cit.>. By defining

    A_n=(n+3)Γ(1+1/2+n/2)/πΓ(1+n/2)√((2n)!!/(2n+1)!!)

and

    B_n=(n+3)(n+2)Γ(1+1/2+n/2)/2πΓ(1+n/2)√((2n)!!/(2n+1)!!)

and by rotating the two nuclei with the same Euler angles Ω=(α,β,γ), the above equation can be further written as

    ϵ_n e^inΦ_n =       -A_nβ _n∑_m α_n,mD_n,m^n-B_n ∫∑_l_1,m_1β_l_1α_l_1,m_1Y_l_1^m_1∑_l_2,m_2β_l_2α_l_2,m_2Y_l_2^m_2∑_m'D_n,m'^n Y_n^m'sin(θ) dθ dϕ
    
    =       -A_nβ _n∑_m α_n,mD_n,m^n 
          - B_n ∑_l_1,m_1,l_2,m_2,m'(β_l_1β_l_2α_l_1,m_1α_l_2,m_2 D_n,m'^n√((2l_1+1)(2l_2+1)(2n+1)/4π)([ l_1 l_2   n;   0   0   0 ]) ([ l_1 l_2   n; m_1 m_2  m' ])),

with D_n,m^n(α,β,γ) being the Wigner rotation matrix, and ([ j_1 j_2   j; m_1 m_2   m ]) representing the Wigner 3j-Symbol. Taking the square of the above expression leads to

    ϵ_n^2   =   A_n^2β _n^2(∑_m α_n,mD_n,m^n)(∑_m'α_n,m'D_n,m'^n)
       +   2A_nB_nβ _n ∑_l_1,m_1,l_2,m_2,m',mβ_l_1β_l_2α_n,mD_n,m^nα_l_1,m_1α_l_2,m_2 D_n,m'^n√((2l_1+1)(2l_2+1)(2n+1)/4π)([ l_1 l_2   n;   0   0   0 ]) ([ l_1 l_2   n; m_1 m_2  m' ]) 
       +   O(β_n^4).

In the present study, we only use finite β_2 and β_3 but β_n=0 for n>3. The event average here is equal to the rotational average ∫ (...) dΩ /(8π^2)=∫ (...) dαsin(β) dβ dγ/(8π^2), and then we can get numerically the relations as Eqs. (<ref>) and (<ref>) in the axial symmetric case. For independent rotation of the two nuclei, as shown in Ref. <cit.>, qualitatively similar relations apply except that the coefficients are a factor of 2 smaller.

Next, we calculate the fluctuation δ d_⊥^2 of the overlap's inverse area d_⊥ = 1/√(x^2 y^2). At zero impact parameter, x^2 in tip-tip collisions can be formally expressed as

    x^2   =   ∫ r^2 sin^2(θ) cos^2(ϕ) ρ(r,θ, ϕ)d^3r/∫ρ(r,θ, ϕ)d^3r
       =   3R_0^2/20π∫ (1+∑_l,mβ_lα_l,mY_l^m)^5 [1/3 -2/3√(π/5)Y_2^0+√(2π/15)(Y_2^2+Y_2^-2)] sin(θ) dθ dϕ
       ≈   3R_0^2/20π∫(1+5∑_l,mβ_lα_l,mY_l^m+10∑_l_1,m_1,l_2,m_2β_l_1β_l_2α_l_1,m_1α_l_2,m_2Y_l_1^m_1 Y_l_2^m_2) [1/3 -2/3√(π/5)Y_2^0+√(2π/15)(Y_2^2+Y_2^-2)] sin(θ) dθ dϕ
       =   R_0^2/5+3R_0^2/20π∫[ -10/3√(π/5)∑_l,mβ_lα_l,mY_l^mY_2^0+5√(2π/15)∑_l,mβ_lα_l,mY_l^m(Y_2^2+Y_2^-2)+10/3∑_l_1,m_1,l_2,m_2β_l_1β_l_2α_l_1,m_1α_l_2,m_2Y_l_1^m_1 Y_l_2^m_2. 
          . -20/3√(π/5)∑_l_1,m_1,l_2,m_2β_l_1β_l_2α_l_1,m_1α_l_2,m_2Y_l_1^m_1 Y_l_2^m_2Y_2^0  +10√(2π/15)∑_l_1,m_1,l_2,m_2β_l_1β_l_2α_l_1,m_1α_l_2,m_2Y_l_1^m_1 Y_l_2^m_2(Y_2^2+Y_2^-2) ]sin(θ) dθ dϕ.

In the above, the relation cos^2(ϕ) sin^2(θ)=1/3 -2/3√(π/5)Y_2^0+√(2π/15)(Y_2^2+Y_2^-2) is used, and we keep higher-order β_l terms compared to Ref. <cit.>. By rotating the two nuclei with the same Euler angles Ω and carrying out the integral of the spherical harmonics, the above equation can be further written as

    x^2   =   R_0^2/5+3R_0^2/20π[ -10/3√(π/5)∑_mβ_2α_2,mD_0,m^2+5√(2π/15)∑_mβ_2α_2,m(D_2,m^2+D_-2,m^2)+10/3∑_l,m,β_l^2α_l,m^2 .
          . -20/3√(π/5)∑_l_1,m_1,l_2,m_2,mβ_l_1β_l_2α_l_1,m_1α_l_2,m_2D_0,m^2√(5(2l_1+1)(2l_2+1)/4π)([ l_1 l_2   2;   0   0   0 ])([ l_1 l_2   2; m_1 m_2   m ]) .
          .+10√(2π/15)∑_l_1,m_1,l_2,m_2,mβ_l_1β_l_2α_l_1,m_1α_l_2,m_2(D_2,m^2+D_-2,m^2)√(5(2l_1+1)(2l_2+1)/4π)([ l_1 l_2   2;   0   0   0 ])([ l_1 l_2   2; m_1 m_2   m ]) ]
       =   R_0^2/5+R_0^2/√(20π)∑_mβ_2α_2,m[-D_0,m^2+√(3/2)(D_2,m^2+D_-2,m^2)]+R_0^2/2π∑_l,mβ_l^2α_l,m^2 
          +R_0^2/√(5π)∑_l_1,m_1,l_2,m_2,mβ_l_1β_l_2α_l_1,m_1α_l_2,m_2√(5(2l_1+1)(2l_2+1)/4π)([ l_1 l_2   2;   0   0   0 ])([ l_1 l_2   2; m_1 m_2   m ])[-D_0,m^2+√(3/2)(D_2,m^2+D_-2,m^2)].

Similarly, we can formally express y^2 in central tip-tip collisions as

    y^2   =   ∫ r^2 sin^2(θ) sin^2(ϕ) ρ(r,θ,ϕ)d^3r/∫ρ(r,θ,ϕ)d^3r
       ≈   R_0^2/5+R_0^2/√(20π)∑_mβ_2α_2,m[-D_0,m^2-√(3/2)(D_2,m^2+D_-2,m^2)]+R_0^2/2π∑_l,mβ_l^2α_l,m^2 
          +R_0^2/√(5π)∑_l_1,m_1,l_2,m_2,mβ_l_1β_l_2α_l_1,m_1α_l_2,m_2√(5(2l_1+1)(2l_2+1)/4π)([ l_1 l_2   2;   0   0   0 ])([ l_1 l_2   2; m_1 m_2   m ])[-D_0,m^2-√(3/2)(D_2,m^2+D_-2,m^2)],

for which we have used the relation sin^2(ϕ) sin^2(θ)=1/3 -2/3√(π/5)Y_2^0-√(2π/15)(Y_2^2+Y_2^-2) and keep higher-order β_l terms compared to Ref. <cit.>. Thus, up to the β_l^3 term, x^2 y^2 can be written as

    x^2 y^2   =   R_0^4/25-R_0^4/5√(5π)∑_mβ_2α_2,mD_0,m^2+R_0^4/20π∑_m_1,m_2β_2^2α_2,m_1α_2,m_2[D_0,m_1^2D_0,m_2^2-3/2 (D_2,m_1^2+D_-2,m_1^2)(D_2,m_2^2+D_-2,m_2^2)] 
          +R_0^4/5π∑_l,mβ_l^2α_l,m^2-R_0^4/√(20π^3)∑_l,m_1,m_2β_2β_l^2α_2,m_1α_l,m_2^2D_0,m_1^2  
          -2R_0^4/5√(5π)∑_l_1,m_1,l_2,m_2,mβ_l_1β_l_2α_l_1,m_1α_l_2,m_2√(5(2l_1+1)(2l_2+1)/4π)([ l_1 l_2   2;   0   0   0 ])([ l_1 l_2   2; m_1 m_2   m ])D_0,m^2
          +R_0^4/5π∑_l_1,m_1,l_2,m_2,m,m'β_2β_l_1β_l_2α_l_1,m_1α_l_2,m_2α_2,m'√(5(2l_1+1)(2l_2+1)/4π)([ l_1 l_2   2;   0   0   0 ])([ l_1 l_2   2; m_1 m_2   m ])
          ×[D_0,m^2D_0,m'^2-3/2 (D_2,m^2+D_-2,m^2)(D_2,m'^2+D_-2,m'^2)]+O(β_l^4).

By definition d_⊥ = 1/√(x^2 y^2) can then be expressed as

    d_⊥   ≈   5/R_0^2{1+√(5/4π)∑_mβ_2α_2,mD_0,m^2-5/8π∑_m_1,m_2β_2^2α_2,m_1α_2,m_2[D_0,m_1^2D_0,m_2^2-3/2 (D_2,m_1^2+D_-2,m_1^2)(D_2,m_2^2+D_-2,m_2^2)]. 
          .-5/2π∑_l,mβ_l^2α_l,m^2+√(5/π)∑_l_1,m_1,l_2,m_2,mβ_l_1β_l_2α_l_1,m_1α_l_2,m_2√(5(2l_1+1)(2l_2+1)/4π)([ l_1 l_2   2;   0   0   0 ])([ l_1 l_2   2; m_1 m_2   m ])D_0,m^2 .
          .+√(125/16π^3)∑_l,m_1,m_2β_2β_l^2α_2,m_1α_l,m_2^2D_0,m_1^2-5/2π∑_l_1,m_1,l_2,m_2,m,m'β_2β_l_1β_l_2α_l_1,m_1α_l_2,m_2α_2,m'. 
          .×√(5(2l_1+1)(2l_2+1)/4π)([ l_1 l_2   2;   0   0   0 ])([ l_1 l_2   2; m_1 m_2   m ])[D_0,m^2D_0,m'^2 -3/2 (D_2,m^2+D_-2,m^2)(D_2,m'^2+D_-2,m'^2)] }+O(β_l^4).

Let's define the deviation of a quantity A away from its rotational average value as δ A = A - ⟨ A ⟩, where ⟨ A ⟩=∫ A dΩ/(8π)^2. In this way, δ d_⊥, (δ d_⊥)^2, and ϵ_n^2 δ d_⊥ can then be formally expressed as

    δ d_⊥   =   5/R_0^2{√(5/4π)∑_mβ_2α_2,mδ D_0,m^2-5/8π∑_m_1,m_2β_2^2α_2,m_1α_2,m_2δ[D_0,m_1^2D_0,m_2^2-3/2 (D_2,m_1^2+D_-2,m_1^2)(D_2,m_2^2+D_-2,m_2^2)]. 
          .+√(5/π)∑_l_1,m_1,l_2,m_2,mβ_l_1β_l_2α_l_1,m_1α_l_2,m_2√(5(2l_1+1)(2l_2+1)/4π)([ l_1 l_2   2;   0   0   0 ])([ l_1 l_2   2; m_1 m_2   m ])δ D_0,m^2.
          .+√(125/16π^3)∑_l,m_1,m_2β_2β_l^2α_2,m_1α_l,m_2^2δ D_0,m_1^2-5/2π∑_l_1,m_1,l_2,m_2,m,m'β_2β_l_1β_l_2α_l_1,m_1α_l_2,m_2α_2,m'. 
          .×√(5(2l_1+1)(2l_2+1)/4π)([ l_1 l_2   2;   0   0   0 ])([ l_1 l_2   2; m_1 m_2   m ])δ[D_0,m^2D_0,m'^2-3/2 (D_2,m^2+D_-2,m^2)(D_2,m'^2+D_-2,m'^2)]}+O(β_l^4),


    (δ d_⊥)^2   =   25/R_0^4{5/4π∑_m_1,m_2β_2^2α_2,m_1α_2,m_2δ D_0,m_1^2δ D_0,m_2^2-√(125/64π^3)∑_m,m_1,m_2β_2^3α_2,mα_2,m_1α_2,m_2.
          .×δ D_0,m^2δ[D_0,m_1^2D_0,m_2^2-3/2 (D_2,m_1^2+D_-2,m_1^2)(D_2,m_2^2+D_-2,m_2^2)].
          .+5/πβ_2∑_l_1,m_1,l_2,m_2,m,m'β_l_1β_l_2α_l_1,m_1α_l_2,m_2α_2,m'√(5(2l_1+1)(2l_2+1)/4π)([ l_1 l_2   2;   0   0   0 ])([ l_1 l_2   2; m_1 m_2   m ])δ D_0,m^2δ D_0,m'^2}+O(β_l^4),

and

    ϵ_n^2 δ d_⊥   =   5/R_0^2A_n^2{√(5/4π)β_2β_n^2∑_m_1,m_2,m_3α_2,m_1α_n,m_2α_n,m_3D_n,m_2^n D_n,m_3^nδ D_0,m_1^2.
          .-5/8πβ_2^2β_n^2∑_m_1,m_2,m_3,m_4α_2,m_1α_2,m_2α_n,m_3α_n,m_4δ[D_0,m_1^2D_0,m_2^2-3/2 (D_2,m_1^2+D_-2,m_1^2)(D_2,m_2^2+D_-2,m_2^2)]D_n,m_3^n D_n,m_4^n.
          .+√(5/π)β_n^2∑_l_1,m_1,l_2,m_2,m_3,m_4,mβ_l_1β_l_2α_l_1,m_1α_l_2,m_2α_n,m_3α_n,m_4. 
          .×√(5(2l_1+1)(2l_2+1)/4π)([ l_1 l_2   2;   0   0   0 ])([ l_1 l_2   2; m_1 m_2   m ])D_n,m_3^n D_n,m_4^nδ D_0,m^2.
          .+√(5/4π)(n+2) β_2β_n∑_l_1,m_1,l_2,m_2,m_3,m',mβ_l_1β_l_2α_n,mα_l_1,m_1α_l_2,m_2α_2,m_3. 
          .×√((2l_1+1)(2l_2+1)(2n+1)/4π)([ l_1 l_2   n;   0   0   0 ]) ([ l_1 l_2   n; m_1 m_2  m' ])D_n,m^nD_n,m'^nδ D_0,m_3^2}+O(β_l^5),

respectively.
Again, using finite β_2 and β_3 but β_n=0 for n>3 and taking the rotational average by integrating over the Euler angles, we can get numerically the relations as Eqs. (<ref>), (<ref>), and (<ref>) in the axial symmetric case. For independent rotation of the two nuclei, as shown in Ref. <cit.>, qualitatively similar relations apply except that the coefficients are a factor of 2 smaller.


