Branched flow is a universal phenomenon in which treebranch-like filaments form through traveling waves or particle flows in irregular mediums. Branched flow of high-current relativistic electron beams (REBs) in porous materials has been recently discovered [Jiang et al., Phys. Rev. Lett. 130, 185001 (2023)]. REB branching is accompanied by extreme beam focusing, up to a hundred times the initial value, at predictable caustic locations. The energy coupling efficiency between the beam and porous material surpasses that in homogeneous targets by two orders of magnitude. This paper examines REB branching, focusing on how beam parameters (e.g., Lorentz factor and density) and characteristics of the porous materials (e.g., pore size, skeleton thickness, and density) influence branching patterns. Analyses of the dynamics of individual beam electrons are also provided. The findings pave the way for further understanding REB branching and its potential applications in the future.

Branched flow refers to a phenomenon wherein treebranch-like filaments emerge as waves or particle flows propagate through an irregular medium.1 The medium's irregularity manifests through an effective potential V, which is randomly uneven, spatially smooth, weak ( V rms E k, where Ek represents the kinetic energy of the traveling flow), and long-range correlated (with a correlation length lc exceeding the flow wavelength).2–4 In such conditions, sudden and significant momentum changes in the flow are absent, but accumulated small changes profoundly influence the morphologies and dynamics of the flow. In particular, caustics and filaments form as the flow undergoes bending and bundling at favorable locations. For example, instead of smoothly spreading, tsunami waves refract and form branching strands due to fluctuations in the ocean height.5 Impurities within semiconductors can induce branched flows of current-carrying electrons.6 Cosmic rays form branches when traversing inhomogeneous interstellar dust clouds.7 Light undergoes branched flow in complex media characterized by refractive index fluctuations.8–10 

Recently, we reported that high-current relativistic electron beam (REB) propagation in porous materials (e.g., foam) can cascade into thin and dense branches at a length scale d0 given by
d 0 l c 1 / 3 n b 0 2 / 3 γ 2 / 3 ,
(1)
thereby bringing the branched flow phenomenon to high-energy-density physics.11 Here, lc is the average pore size of the foam and n b 0 and γ denote the initial density and Lorentz factor of the REB, respectively. Notably, REB branching differs from beam–plasma instabilities, as it arises from the microscopic skeleton-and-pore heterogeneity of the foam. The compensating return current, located within skin layers of the skeletons, induces magnetic fields in the unevenly distributed pores. As a consequence, the beam electrons undergo successive scattering by these fields, leading to branched flow. Furthermore, REB branching is accompanied by a significant increase in beam–target energy coupling efficiency, which can be two orders of magnitude larger than that in homogeneous plasmas with similar average density.

In this paper, we present a comprehensive investigation into the branched flow of REBs, focusing on two aspects: (1) the impact of beam and foam parameters on the branching patterns and (2) the dynamics of individual beam electrons. The paper is organized as follows: Sec. II introduces pore-resolved particle-in-cell (PIC) simulations, providing detailed results on the branched flow pattern. Since the branched flow pattern is determined by self-generated fields, Sec. III delves into the discussion on how REB and foam parameters affect these fields. Section IV presents the influence of interaction parameters on the branched flow pattern. In Sec. V, we present detailed analyses of the dynamics of individual beam electrons. The paper concludes with discussions in Sec. VI.

In early simulations regarding REB transport in foam, the foam is usually simplified as a homogeneous continuum with a volumetric average density.12–16 That is, the foam's possible fine structuring is neglected. A likely reason for the assumption is that the foam considered is rather dense, such that its internal structure is smaller than the spatial scale of interest. Another reason might be the computational limitations at that time, which may have hindered the resolution of such fine structures. While results from the continuum assumption agree reasonably well with experiments using dense foam, this assumption proves inadequate for low-density foams characterized by spongelike sub-micrometer-sized intertwining solid-density skeletons with micrometer-sized empty pores in between.17 These microstructures can significantly affect macroscopic results, necessitating their consideration in simulations.18–25 

To elucidate the impact of these microstructures on the propagation of REBs, we perform pore-resolved two-dimensional (2D) PIC simulations using the epoch code.26 As shown in Fig. 1, in the simulation, the porous material is SiO2 foam. The foam's internal structures are modeled by random Voronoi cells.27 For definitiveness, the skeleton thickness l d = 1 μ m and average pore size l c = 8  μm. The skeletons consist of solid-density Si and O atoms, with respective densities of n Si = 2.15 × 10 28 and n O = 4.3 × 10 28 m−3. For simplicity, the initial REB is mono-energetic and of uniform density n b 0 = 1.72 × 10 25 m−3, with a duration of 400 fs and Lorentz factor γ = 100. Detailed discussions regarding the impact of foam and REB parameters will follow. The simulation box spans from 0  < x < 100 μm and −15  < y < 15 μm. To resolve the skin depth of the solid skeletons, the spatial resolution Δ = 12.5 nm. There are 14 macroparticles per cell for both Si and O atoms and 100 for the beam electrons. In addition, over 300 macroelectrons per cell can be produced from the skeletons by field ionization. Periodic boundaries are used in the transverse direction. As discussed in Ref. 11 collisions are deemed negligible and are not included here.

FIG. 1.

Formation of branched flow patterns as a high-current relativistic electron beam propagates through a porous SiO2 foam along the x direction. The red to blue color bar is for the beam density (in units of n b 0, same below) at t = 333 fs. The initial density distribution of Si atoms in the foam (black) and self-generated azimuthal magnetic field Bz (red and blue, in Tesla) at t = 333 fs are shown beneath the beam density.

FIG. 1.

Formation of branched flow patterns as a high-current relativistic electron beam propagates through a porous SiO2 foam along the x direction. The red to blue color bar is for the beam density (in units of n b 0, same below) at t = 333 fs. The initial density distribution of Si atoms in the foam (black) and self-generated azimuthal magnetic field Bz (red and blue, in Tesla) at t = 333 fs are shown beneath the beam density.

Close modal

As shown in Fig. 1, REB propagation in the foam induces uneven fields in the pores. The latter, in turn, deflects the beam electrons, resulting in REB branching. Specifically, the REB breaks up into three narrow dense branches. These branches subsequently broaden, intersect, and evolve into a fluctuating pattern. Since the Lorentz force associated with the azimuthal magnetic field Bz is twice as large as that from electric fields (see discussion later and Ref. 11), Bz is more influential in shaping the dynamics of beam electrons. To visualize the electron motion, one may neglect the effects of electric fields and focus on the deflection caused by Bz for simplicity. The gyroradius of beam electrons, given by r g = γ m e c / e B z 41.6  μm, is significantly larger than the average pore size (or the characteristic length scale of Bz) of lc = 8  μm. Here, me and –e are the electron rest mass and charge, and c is the vacuum light speed. Therefore, the electrons can travel over several pores before significant deflection occurs.

Considering the REB dynamics from an energy perspective may provide additional insights. The effective potential strength of the uneven fields is expressed as V rms μ 0 e 2 c 2 l c 2 n b 0 / 8 3,11 and the initial kinetic energy of the beam electron reads E k , b 0 = γ m e c 2. Here, μ0 is the vacuum permeability. The ratio V rms / E k , b 0 0.03 1, indicating that the magnitude of the effective potential is sufficiently small to cause only forward scatterings of the REB. These weak scatterings over several peaks and valleys of the potential V lead to the formation of branched flows. This process is characterized by seemingly random, time-irreversible behavior, yet it exhibits deterministic and predictable tendencies. Since the distribution of pores, along with the associated effective potential, is isotropic, the transverse separations between two adjacent caustics are almost the same and match well with lc. Therefore, branched flow of REBs presents a new regime of beam–plasma interaction with typical features of complex systems, where patterns emerge from random yet correlated interactions.

Given that the self-generated pore fields Bz and Ey result in REB branching, this section considers the influence of interaction parameters on Bz and Ey. As discussed in Ref. 11, Bz arises from current neutralization. The return current, induced within skin layers of skeletons, is of huge density several times larger than the beam current, namely, j r | j b 0 | l c / 2 l s exp ( 1 ). Here, j b 0 is the beam-current density, and ls is the skin length of the skeletons. Since the solid-density skeletons effectively shield fields outside skin layers, the magnetic field induced by the current is distributed in the vacuum pores. The beam and return currents contribute approximately equally to the magnetic field, with the total magnetic strength given by Ampére's law as | B z | μ 0 e n b 0 l c c / 2 n b 0 l c. In addition, the REB, being a bunch of electrons, provides the space charge for the electrostatic fields Ey in the pore regions. From Poisson's equation, one obtains E y e n b 0 l c / 4 ε 0 n b 0 l c. As both Bz and Ey originate from the REB itself, the timescale for their onset is l c / c (several femtoseconds), much shorter than the expansion timescale of the skeletons. The magnetic field Bz plays a more crucial role in scattering beam electrons ( | F y | = e ( c | B z | | E y | ) e c | B z | / 2), as the electrostatic and magnetic forces associated with the beam tend to cancel each other, leaving the net magnetic force provided by the return current to scatter the beam electrons.

The above-mentioned analysis demonstrates a linear scaling between both Bz and Ey with n b 0 and lc. This correlation is substantiated through a series of simulations with varied values of n b 0 and lc. Figures 2 and 3 illustrate the observed increase in Bz and Ey with increasing n b 0 and lc. In addition, in all the cases, the distributions of pore fields are quite similar. This further validates the association of pore fields with the skeleton-and-pore microstructure, indicating that the free evolution of these fields is constrained by the latter. Figure 4 for the field strengths at the same or equivalent positions shows the linear scaling of Bz and Ey with n b 0 and lc, agreeing with our analysis.

FIG. 2.

Distributions of (a)–(d) azimuthal magnetic field Bz (in Tesla) and (e)–(h) transverse electric field Ey (in V/m) in a selected region. From left to right, the initial density of the electron beam n b 0 = 3.44 × 10 24 , 6.88 × 10 24 , 1.376 × 10 24, and 1.72 × 10 25 m−3. In all the cases, lc keeps at 8 μ m.

FIG. 2.

Distributions of (a)–(d) azimuthal magnetic field Bz (in Tesla) and (e)–(h) transverse electric field Ey (in V/m) in a selected region. From left to right, the initial density of the electron beam n b 0 = 3.44 × 10 24 , 6.88 × 10 24 , 1.376 × 10 24, and 1.72 × 10 25 m−3. In all the cases, lc keeps at 8 μ m.

Close modal
FIG. 3.

Distributions of (a)–(c) azimuthal magnetic field Bz (in Tesla) and (d)–(f) transverse electric field Ey (in V/m). From left to right, the average pore size of the foam lc = 1.33, 2, 4, and 8 μ m.

FIG. 3.

Distributions of (a)–(c) azimuthal magnetic field Bz (in Tesla) and (d)–(f) transverse electric field Ey (in V/m). From left to right, the average pore size of the foam lc = 1.33, 2, 4, and 8 μ m.

Close modal
FIG. 4.

(a) Strengths of the azimuthal magnetic field Bz (red circles) and transverse electric field Ey (blue squares) at x = 20 and y = 3 μ m obtained from simulations for different initial beam density n b 0 with l c = 8 μ m. (b) Same as (a), but at x = 20 l c / 8 and y = 3 l c / 8 obtained from simulations with different lc. The solid lines are linear interpolations based on the simulation results.

FIG. 4.

(a) Strengths of the azimuthal magnetic field Bz (red circles) and transverse electric field Ey (blue squares) at x = 20 and y = 3 μ m obtained from simulations for different initial beam density n b 0 with l c = 8 μ m. (b) Same as (a), but at x = 20 l c / 8 and y = 3 l c / 8 obtained from simulations with different lc. The solid lines are linear interpolations based on the simulation results.

Close modal

Since the pore fields and the associated effective potential strength V rms depend on n b 0 and lc, the resulting branched flow patterns are affected by these interaction parameters. In addition, the Lorentz factor γ of the REB determines the dynamics of beam electrons under the pores fields, it is, thus, also crucial for the branched flow patterns. One can characterize the branched flow patterns by considering the distance d0 from the foam's front surface to the first caustics; thus, d0, in a sense, represents the transition from order to randomness. Figure 5 shows the branched flow patterns and associated variation strength of the beam density (namely, scintillation index) Σ = ( n b 2 / n b 2 ) 1 for different interaction parameters. It shows that d0 decreases with increasing n b 0 and lc but increases with higher γ, in good agreement with Eq. (1).

FIG. 5.

Branched flow patterns (in unites of n b 0, same below) for (a1)–(a3) different initial beam densities at n b 0 = 3.44 × 10 24 , 6.88 × 10 24, and 1.38 × 10 25 m−3 (with γ = 100 and l c = 8 μ m); (b1)–(b3) different beam Lorentz factors γ = 50, 200, and 250 (with n b 0 = 1.72 × 10 25 m−3 and l c = 8 μ m); and (c1)–(c3) different average pore sizes lc = 4, 6, and 10  μ m (with γ = 100 and n b 0 = 1.72 × 10 25 m−3). Scintillation indices Σ = ( n b 2 / n b 2 ) 1 corresponding to the beam electron densities are presented by red curves. The SiO2 foams in (a) and (b) are of the same initial structure.

FIG. 5.

Branched flow patterns (in unites of n b 0, same below) for (a1)–(a3) different initial beam densities at n b 0 = 3.44 × 10 24 , 6.88 × 10 24, and 1.38 × 10 25 m−3 (with γ = 100 and l c = 8 μ m); (b1)–(b3) different beam Lorentz factors γ = 50, 200, and 250 (with n b 0 = 1.72 × 10 25 m−3 and l c = 8 μ m); and (c1)–(c3) different average pore sizes lc = 4, 6, and 10  μ m (with γ = 100 and n b 0 = 1.72 × 10 25 m−3). Scintillation indices Σ = ( n b 2 / n b 2 ) 1 corresponding to the beam electron densities are presented by red curves. The SiO2 foams in (a) and (b) are of the same initial structure.

Close modal

As d0 increases, we find that the branches are less likely to overlap with each other. Instead, they seem to be confined in the pores and propagate rather stably as individual strands, leading to a mild spatial evolution of Σ. This behavior can be interpreted in terms of the focusing angle θ of the REB, defined as tan θ = l c / 2 d 0 n b 0 2 / 3 l c 4 / 3 γ 2 / 3. We see the branches become collimated with decreasing n b 0 and lc and increasing γ, which agrees with Fig. 5. Collimated branches may lead to low-divergence x- and γ-rays.

It is worth mentioning that lc not only affects d0 and θ but also determines the number of the first caustics in the transverse direction. It is consistent with the universal feature of branched flows, where the average transverse separation between two adjacent caustics equals the correlation length of the effective potential1–4—corresponding, in our case, to lc. As lc approaches zero, the transverse separation between adjacent caustics tends to zero, and d0 approaches infinity. That is, REB branching is absent in continuum media.

Equation (1) reveals another unique feature of REB branching: the branched flow pattern is independent of the skeleton electron density np. This distinguishes REB branching from beam–plasma instabilities, as the latter can be sensitive to np.28–35 This feature of REB branching is observed under the condition that the skeleton is sufficiently dense to be considered rigid during REB propagation, i.e., n p n b 0. In this case, the pore fields and associated V rms exclusively depend on n b 0 and lc, as discussed in Sec. III. Therefore, the dynamics of beam electrons are unaffected by np. Figure 6 for simulation results with different skeleton atom densities confirms that the branching patterns remain almost identical despite substantial differences in the skeleton densities.

FIG. 6.

(a) and (c) Densities (in unites of n b 0, same below) of the beam electrons and (b) and (d) densities of the foam electrons at t = 333 fs. For (a) and (b), the initial foam's Si atom density n Si = 4.3 × 10 27 m−3. For (c) and (d), n Si = 2.15 × 10 27 m−3.

FIG. 6.

(a) and (c) Densities (in unites of n b 0, same below) of the beam electrons and (b) and (d) densities of the foam electrons at t = 333 fs. For (a) and (b), the initial foam's Si atom density n Si = 4.3 × 10 27 m−3. For (c) and (d), n Si = 2.15 × 10 27 m−3.

Close modal

Figure 7 illustrates the influence of skeleton thickness ld on the branching pattern. This effect stems from our model, where V rms is obtained under the assumption that ld approaches zero. Given that both Bz and Ey are effectively zero in the skeletons, an increase in ld results in decreasing V rms. As a result, d0 occurs at a longer distance in dense foam with thicker skeletons. In addition, finite ld gives rise to the formation of so-called superwires,36 which can propagate as stable thin filaments over extended distances, as indicated by the red arrows in Figs. 7(b1) and 7(c1). Superwires emerge when a certain beam electron and its neighboring ones in phase space coincidentally follow similar trajectories, maintaining stable manifolds over a finite time. As x x V and y y V are effectively zero in the skeletons, the stability matrix M of beam electrons with similar conditions is more likely to remain stable for thick skeletons. See Ref. 37 for the definition of M and detailed discussion. That is, superwires can form and propagate extensively in foams with thick skeletons.

FIG. 7.

(a1)–(c1) Distributions of beam electron density (in unites of n b 0, same below). (a2)–(c2) Distributions of initial foam's Si atom density. (a3)–(c3) Distributions of azimuthal magnetic field Bz (in Tesla). From left to right, the skeleton thickness ld = 0.5, 2.3, and 4.8  μ m.

FIG. 7.

(a1)–(c1) Distributions of beam electron density (in unites of n b 0, same below). (a2)–(c2) Distributions of initial foam's Si atom density. (a3)–(c3) Distributions of azimuthal magnetic field Bz (in Tesla). From left to right, the skeleton thickness ld = 0.5, 2.3, and 4.8  μ m.

Close modal

In this section, we employ particle tracing methods to gain insight into REB branching. As shown in Fig. 8(a), the trajectories of individual electrons exhibit a quasi-ballistic nature. That is, the branched flow pattern develops linearly, allowing us to treat the beam electrons as single particles due to the absence of collective motion. Here, the term “linearly” refers to the fact that the uneven background fields remain quasistatic during pattern formation. In this sense, one may artificially divide the REB branching into two stages. The first is a nonlinear stage of the background fields induction by the return current, as a consequence of the collective response of foam electrons to the high-current REB. The second stage appears to be linear, during which the beam electrons undergo scattering by the uneven fields, behaving as individual particles.

FIG. 8.

(a) Trajectories of randomly selected 300 beam electrons. (b)–(f) Phase spaces ( p y , y) of randomly selected 4000 beam electrons at x = 0.8, 15.2, 25.2, 39.2, and 99.2  μ m, respectively. The corresponding histograms are shown on the right side. The color bar is for the initial y coordinates of the selected beam electrons.

FIG. 8.

(a) Trajectories of randomly selected 300 beam electrons. (b)–(f) Phase spaces ( p y , y) of randomly selected 4000 beam electrons at x = 0.8, 15.2, 25.2, 39.2, and 99.2  μ m, respectively. The corresponding histograms are shown on the right side. The color bar is for the initial y coordinates of the selected beam electrons.

Close modal

The formation of branching patterns from random scatterings is elucidating when observed in the phase space ( p y , y ) of the traced beam electrons. As shown in Fig. 8(b), the initial ( p y , y ) is nearly a straight line since the beam is initially mono-energetic. The corresponding beam density is also uniform, as indicated by the histogram on the right side. As the beam propagates, successive random kicks from the uneven fields cause the phase space ( p y , y ) to stretch and curve, resulting in a focused beam [see Fig. 8(c)]. As a result, three singularities, where y p y = , occur around x 25 μ m, as shown in Fig. 8(d). These singularities in the phase space correspond to caustics, where the beam converges to an extremely high density, as shown in the histogram and Fig. 1. The phase space undergoes folding after the caustics, giving rise to individual branches in the coordinate space. For each branch, the peripheral density is higher than that at the center [see Fig. 8(e)]. Eventually, the phase space evolves into random distributions through further random scatterings as well as overlapping of different branches [see Fig. 8(f)]. Figure 8(f) also reveals an intriguing aspect of REB branching wherein groups (or manifolds) of beam electrons at neighboring initial locations can extend over significant zones over time. This emphasizes the sensitivity of a single electron's motion to its initial condition. However, the resultant branched flow pattern, formed by the ensemble of beam electrons, adheres to well-defined statistical laws.2–4 

Figures 9(a)–9(c) show the fields experienced by beam electrons. Indeed, in the foam target, these fields evolve randomly and are several orders of magnitude stronger than that of homogeneous targets at the same average density. Such strong and dynamic fields may lead to efficient radiation in foam targets. By analyzing the power spectral density, defined as PSD = F ( W ) 2 / C, one sees in Figs. 9(d) and 9(e) that the frequency densities of Bz and Ey are inversely proportional to f2. It further confirms the time-correlated characteristic, akin to Brownian noise,38 of the transverse fields. Here, F denotes to Fourier transform, W is a generalized signal corresponding to the fields experienced by beam electrons in this context, C serves as a normalizing constant, and f is the frequency of W. On the other hand, the frequency density of Ex appears relatively constant, namely, inversely proportional to f  0, as shown in Fig. 9(f). This suggests a negligible correlation in the longitudinal direction, resembling white noise characteristics.38 Therefore, even though the beam electrons experience random interactions in both longitudinal and transverse directions, branching exclusively manifests in the latter.

FIG. 9.

Evolution of (a) azimuthal magnetic field Bz, (b) transverse, and (c) longitudinal electric field Ey and Ex experienced by randomly selected 100 beam electrons as they propagate in the foam (red) and a homogeneous target (black) at the same average density of the foam for reference. The dark red curves correspond to that experienced by a typical beam electron. (d)–(f) show the corresponding power spectral densities (PSD) of Bz, Ey, and Ex in the case of foam target, respectively.

FIG. 9.

Evolution of (a) azimuthal magnetic field Bz, (b) transverse, and (c) longitudinal electric field Ey and Ex experienced by randomly selected 100 beam electrons as they propagate in the foam (red) and a homogeneous target (black) at the same average density of the foam for reference. The dark red curves correspond to that experienced by a typical beam electron. (d)–(f) show the corresponding power spectral densities (PSD) of Bz, Ey, and Ex in the case of foam target, respectively.

Close modal

REB branching in porous materials is a quite complicated process. This paper only addresses REBs with simplified “plane wave” profile, focusing on the impact of beam and foam parameters on the branched flow pattern. There are other important interaction parameters. For example, REB's temporal profile is directly related to the time-dependent n b 0 ( t ). Therefore, the location of the first caustics can vary over time as d 0 l c 1 / 3 n b 0 ( t ) 2 / 3 γ 2 / 3. In the case of a temporally Gaussian REB with sufficient duration, d0 first decreases with time and then increases with it. The initial emittance and energy spread of REBs can also affect the resulting pattern.11 Moreover, foams are not necessarily isotropic, i.e., lc can vary in different locations, modifying the correlation of induced pore fields and resulting REB dynamics.

For experimental detection of REB branching, one may consider irradiating porous foam targets with REBs, generated either through laser acceleration or conventional accelerators, and measure the rear surface transition radiation. The radiation profile should resemble that of the branched beam electrons. Since REB branching is more pronounced for large lc and diminishes when lc approaches zero, foams with large pores are more suitable for REB branching. Triacetate cellulose,39 polystyrene,40 and additive manufactured foams41 with micrometer-sized pores can be used to trigger REB branching, while carbon nanofoams42 with pore size around 10 nm can serve as reference cases where the dynamics of REBs may be quite similar to that in homogeneous targets.

Branched flow of REBs with unique features may open up new avenues in beam–plasma physics. Due to the pore-and-skeleton structures, the foam facilitates effective and volumetric heating,11 thereby attaining high-energy-density states. The heating mechanism for ions may depend on the skeleton thickness, probably transitioning from skeleton-normal sheath acceleration for thick skeletons to Coloumb explosion for thin skeletons. Moreover, there is a prospect of fabricating the foam target using nuclides of interest and harnessing the dense return current within the skeletons for nuclear excitation. In addition, REB propagation in foam experiences rapidly changing strong fields, offering promise for generating tunable bright radiations by adjusting beam and foam parameters. We believe that the journey to understanding this intriguing phenomenon has just begun.

This work was supported by the National Key R&D Program of China (Grant No. 2022YFA1603300), the National Natural Science Foundation of China (Grant Nos. 12205201, 12175154, 12005149, and 11975214), and the Shenzhen Science and Technology Program (Grant Nos. RCYX20221008092851073 and 20231127181320001). The epoch code is used under UK EPSRC Contract (Nos. EP/G055165/1 and EP/G056803/1).

The authors have no conflicts to disclose.

K. Jiang: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). T. W. Huang: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). R. Li: Investigation (equal); Validation (equal). C. T. Zhou: Funding acquisition (supporting); Project administration (lead); Resources (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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