Layer Symmetry Research Articles

The Aspects of Dimensionality in the Analysis of Raman Spectra of 2D Materials

Two-dimensional crystals are a class of materials in which an individual layer is indefinitely extended in two dimensions. Each layer is the molecular unit in the crystalline lattice. The nature of interlayer/intermolecular interaction is poorly understood. However, it defines physical properties such as carrier mobility, thermal conductivity, photoluminescence, etc. The vibrational properties of 2D materials are described based on triperiodic space groups. However, the triperiodic space groups do not apply to the description of these structures. The 17 two-dimensional groups in triperiodic space groups do not allow the existence of a third dimension. Therefore, if structures are not planar, the approach is inappropriate. The way to describe such structures is using the diperiodic groups in three dimensions. The well-known 2D materials are graphite, graphene, chalcogenide of Ga, Mo, W, and other elements. The crystalline As2S3 belongs to a class of 2D materials. The unit cell contains two layers with ten atoms in each layer, and it is similar to a structure of metal chalcogenide (MoS(Se)2, WS(Se)2). A bond length of 2.24 A is within the layer, and 3.56 A is between the layers. This interlayer/intralayer bond length ratio (= 1.59) corresponds to a significant difference in bond strengths. Center-zone optical phonons in crystalline As2S3 have been investigated by Raman scattering in a wide temperature range for two polarizations in the layer plane (ac). The crystal's and individual layers' symmetry are different, defining different selection rules for optical phonons. According to the crystallographic data, crystal symmetry is monoclinic with 20 atoms in a unit cell and is P21/n (C2h 5). The axis b, perpendicular to the layers, is the only unique axis of the crystal. Therefore, all phonons in the ac plane should be indistinguishable. The layer symmetry is orthorhombic in the diperiodic space group in three dimensions, in which a, b, and c axes are axes of symmetry with a space group of Pnm21 (C2v 7). The intensity of Raman bands in the layer plane shows strong polarization dependence, indicating layer symmetry's dominance. Interlayer interaction is weak, and as a result, low-frequency Raman bands appear in the spectra. The position of these bands defines the strength of interlayer interaction. The data presented provides an understanding of the structural motives of 2D As2S3 and may predict the optical/electronic properties of similar 2D materials based on Raman spectra.

A stability evaluation method for deep-seated toppling in the upper Lancang river, Southwestern China

Deep-seated toppling in the upper reaches of the Lancang River, southwest China involves deformations exceeding 100 m in depth. The slope deformation is initiated by river downcutting and evolves distinctive characteristics with a depth of river incision. In this study, we propose a system for evaluating the stability of deep-seated toppled slopes in different evolutionary stages. This system contains identification criteria for each evolutionary stage and provides the corresponding stability evaluation methods. Based on the mechanical and kinematic analysis of slope blocks, the specific stage of slope movement can be identified in the field through outcrop mapping, in situ tests, surface displacement monitoring, and adit and borehole explorations. The stability evaluation methods are established based on the limiting equilibrium theory and the strain compatibility between the undisturbed zone and the toppled zone. Finally, several sample slopes in different evolution stages have been investigated to verify the applicability and accuracy of the proposed stability evaluation system. The results indicate that intense tectonic activity and rapid river incision lead to a maximum principal stress ratio exceeding 10 near the slope surface, thus triggering widespread toppling deformations along the river valley. When considering the losses of joint cohesion during the further rotation process, the safety factor of the slope drops by 7%–28%. The self-stabilization of toppling deformation can be recognized by the layer symmetry configuration after the free rotation of the deflected layers. Intensely toppled rock blocks mainly suffer sliding failures beyond the layer symmetry condition. The factor of safety of the K73 rockslide decreased from 1.17 to 0.87 by considering the development of the potential sliding surface and the toe-saturated zone.

Regulation of cluster synchronization in multilayer networks of delay coupled semiconductor lasers with the use of disjoint layer symmetry.

In real-world complex systems, heterogeneous components often interact in complex connection patterns and could be schematized by a formalism of multilayer network. In this work, the synchronization characteristics of multilayer network composed of semiconductor lasers (SLs) are investigated systematically. It is demonstrated that the interplay between different layers plays an important role on the synchronization patterns. We elucidate that the performance of cluster synchronization could be facilitated effectively with the introduction of disjoint layer symmetry into network topology. Intertwined stability of clusters from different layers could be decoupled into independent, and the parameter spaces for stable synchronization are extended significantly. The robustness of our proposed regulation scheme on operation parameters is numerically evaluated. Furthermore, the generality of presented theoretical results is validated in networks with more complex topology and multiple layers.

Symmetry groups of two-way twofold and three-way threefold fabrics.

This work discusses the symmetry groups of two classes of woven fabrics, two-way twofold fabrics and three-way threefold fabrics. A method to arrive at a design of a fabric is presented, employing methods in color symmetry theory. Geometric representations of all possible layer group or diperiodic symmetry structures of the fabrics are derived. There are 50 layer symmetry groups corresponding to two-way twofold fabrics and 27 layer symmetry groups corresponding to three-way threefold fabrics.

High frequency multi-field continualization scheme for layered magneto-electro-elastic materials

Magneto-electro-elastic (MEE) heterogeneous materials with periodic microstructure are investigated, with special focus on the particular case of a layered MEE material. By exploiting the transfer matrix method in combination with the Floquet–Bloch boundary conditions, the frequency dispersion spectrum can be derived by solving an eigenproblem, involving a 12 × 12 transfer matrix with palindromic characteristic polynomial. In particular, by assuming the cubic symmetry of layers, the problems involving longitudinal and transverse elastic waves are uncoupled. This circumstance has the advantage of simplifying the treatment and the size of the subproblems to be solved, corresponding to a 4 × 4 transfer matrix for the longitudinal case, and a 8 × 8 transfer matrix of the transverse problem, both characterized by palindromic characteristic polynomials. Afterward, a dynamic multi-field continualization approach is proposed to investigate wave propagation in MEE layered periodic material by matching the Z-transform of the vector collecting the nodal fields to the two-sided Laplace transform of the same vector at the macroscale. The continualization technique allows identifying multi-field integral-type non-local continua as well as multi-field generalized gradient-type (higher-order) non-local continua. Finally, to verify the accuracy of the proposed approach, the dispersion curves derived from the continualization technique are compared with the curves provided by the Floquet–Bloch theory.

Design of dual-phase lattice materials with balanced modulus, strength and energy absorption properties based on Sudoku arranged reinforcement phase distribution

In this paper, the effect of the distribution pattern of the reinforcement phase on the mechanical properties was studied for the simple-cubic matrix dual-phase lattices given the same second phase proportion. Two reinforcement phases, the body centered cubic and the face centered cubic cells, were investigated. Specifically, a novel distribution pattern of reinforcement phase inspired from Sudoku algorithm was proposed to design the dual-phase lattices with balanced elastic modulus, strength and energy absorption properties. Comparisons were made for mechanical performances of five types of distributions, i.e., the layer distribution, column distribution, Sudoku distribution, random distribution and symmetry distribution through extensive finite element simulations. The results indicate that the scattered distribution of the reinforcement phase is beneficial to the investigated mechanical properties. To guide the design of dual-phase lattices, a parameter called degree of concentration is introduced to describe the distribution of reinforcement phase. In general, the mechanical properties are inversely related to degree of concentration. The proposed Sudoku dual-phase lattice possesses zero degree of concentration along any of the three principle directions, thus exhibits advantageous isotropic mechanical properties. This paper provides a new avenue for the design of multi-phase lattice materials.

Ultra-low friction and stiffness dependence of interlayer friction in graphite flakes under various rotation angles

The 60-degree rotational symmetry of graphene layers leads to remarkable physical and mechanical phenomena. One corresponds to ultralow sliding friction and friction coefficient between incommensurate graphene layers. A pick-up method was proposed to pick graphite flake to microsphere probe and achieved graphite and graphite tribo-pair. Ultralow friction was attained for various rotation angles, and the friction coefficient was kept at the range of 10−3. Density functional theory (DFT) calculation was employed to reveal the angular dependence of friction, and the calculations are partially consistent with the experimental results. The difference of angle-dependence ultralow friction between experimental results and DFT calculations was observed, which was greatly influenced by the stiffness of the system. Friction hysteresis, abnormal shape of pull-off curves and long jumps in stick-slip were further demonstrated the stiffness of the system had changed during the friction measurements. The contact geometry in our experiment was a finite, symmetric, nanometer-sized (less than 1 nm), flexible graphite flake and a rigid graphite surface. The tribological system offers an experimental platform to study interlayer friction and contact geometry.

Layer symmetry and interlayer engineering of birnessites towards high-performance rechargeable aqueous Zn-MnO2 batteries

Rechargeable aqueous Zn-MnO2 batteries are promising candidates for large-scale energy storage systems, yet still plagued by the phase transition and structural collapse issues of MnO2 cathodes during cycling. Interlayer intercalation for the layered MnO2 turns out to be a viable alternative and become the mainstream structure design strategy. However, the characteristics of Mn octahedral layers are generally neglected. Herein, for the first time we elucidate apart from interlayer ions, how layer symmetry of birnessites exerts on the electrochemical performance. The Mn(II) ions stabilized hexagonal birnessite exhibits elevated charge storage performance than its monoclinic precursor, attributing to the layer cation vacancies generated after symmetry transformation, interlayer Mn(II) ions and nanosized morphology. A high specific capacity of 279 mAh g−1 at 1 C is achieved, as well as an outstanding long-term cycling stability with 97% retention over 8000 cycles. The reaction mechanism is comprehensively illustrated. This work previews a new gateway for the design of high-performance layered cathode materials by synergistic manipulation of the crystal structure of the layers and the interlayer environment.

Two-dimensional single crystal monoclinic gallium telluride on silicon substrate via transformation of epitaxial hexagonal phase

Van der Waals (vdW) epitaxial growth of large-area and stable two-dimensional (2D) materials of high structural quality on crystalline substrates is crucial for the development of novel device technologies. 2D gallium monochalcogenides with low in-plane symmetry stand out among the layered semiconductor materials family for next-generation optoelectronic and energy conversion applications. Here, we demonstrate the formation of large-area, single crystal and optically active 2D monoclinic gallium telluride (m-GaTe) on silicon substrate via rapid thermal annealing induced phase transformation of vdW epitaxial metastable hexagonal gallium telluride (h-GaTe). Stabilization of multilayer h-GaTe on Si occurs due to the role of the first layer symmetry together with efficient GaTe surface passivation. Moreover, we show that the phase transformation of h-GaTe to m-GaTe is accompanied by the strain relaxation between Si substrate and GaTe. This work opens the way to the fabrication of single-crystal 2D anisotropic semiconductors on standard crystalline wafers that are difficult to be obtained by epitaxial methods.

Effective Constitutive Parameters Extraction of Bianisotropic Metamaterials Using Bloch Modes

In this article, a method is proposed to extract the effective constitutive parameters of bianisotropic metamaterials using Bloch modes. We consider 3-D structures that are realized by stacking identical unit layers, where each unit layer is a 2-D periodic array of metallic or dielectric elements of arbitrary shape embedded in a background dielectric medium. Bianisotropic metamaterials do not satisfy the condition of reflection symmetry of unit layers considered in previous studies. The eigenvectors and eigenvalues of the generalized transfer matrix of a unit layer are related to the Bloch modes of the structure. When only two dominant Bloch modes exist, closed-form expressions are found for effective permittivity, permeability, and magnetoelectric tensors of the medium. The effects of nondominant Bloch modes are taken into account as interface surface impedance matrices. The calculated scattering parameters of a slab of bianisotropic metamaterial are shown to be in good agreement with the results obtained from full-wave electromagnetic simulations using commercial solvers.

Real-Space Observation of Ripple-Induced Symmetry Crossover in Ultrathin MnPS3.

Stacking order is expected to have a significant impact on the properties of van der Waals layered magnets, as it determines the crystallographic and magnetic symmetry. Recent synchrotron-based optical studies on antiferromagnetic MnPS3 have revealed a thickness-dependent symmetry crossover, suggesting possible different stackings in few-layer crystals from the bulk, which, however, has not been explicitly identified. Here, by using a combination of atomic-scale electron microscopy and theoretical calculations, we show that despite the bulk monoclinic stacking persists macroscopically down to bilayer, additional local rippling effect lifts the monoclinic symmetry of the few layers while preserving the trigonal symmetry of individual monolayers, leading to possible monolayer-like behavior in ultrathin MnPS3 samples. This finding reveals the profound impact of rippling on the microscopic symmetry of two-dimensional materials with weak interlayer interactions and raises the possibility of approaching the paradigmatic two-dimensional Néel antiferromagnetic honeycomb lattice in MnPS3 without reaching monolayer thickness.

Anomalous Second Harmonic Generation from Atomically Thin MnBi2Te4

MnBi2Te4 is a van der Waals topological insulator with intrinsic intralayer ferromagnetic exchange and A-type antiferromagnetic interlayer coupling. Theoretically, it belongs to a class of structurally centrosymmetric crystals whose layered antiferromagnetic order breaks inversion symmetry for even layer numbers, making optical second harmonic generation (SHG) an ideal probe of the coupling between the crystal and magnetic structures. Here, we perform magnetic field and temperature-dependent SHG measurements on MnBi2Te4 flakes ranging from bulk to monolayer thickness. We find that the dominant SHG signal from MnBi2Te4 is unexpectedly unrelated to both magnetic state and layer number. We suggest that surface SHG is the likely source of the observed strong SHG, whose symmetry matches that of the MnBi2Te4-vacuum interface. Our results highlight the importance of considering the surface contribution to inversion symmetry-breaking in van der Waals centrosymmetric magnets.

Correlated disorder and crystal structure of β-VOSO4

Pauflerite β-VOSO4 has recently been identified as a one-dimensional S = ½ Heisenberg system, of interest both from a fundamental point of view and a potential material for future spintronics applications. The observation of diffuse scattering in a synthetic β-VOSO4 provides a microscopic interpretation of the underlying correlated disorder, which is linked to the inversion of the short–long V—O distance pairs along VO6 chains, forming a local defect state. Direct Monte Carlo modeling indicates that such defects form thin layers with a positive inter-layer correlation, forming small domains with inverted vanadyl bonding patterns. Two-dimensional defects in anisotropic magnetic systems may perturb, or even destroy, long-range magnetic ordering leading to unusual interactions. In particular, the lack of inversion symmetry in the defect layers opens up the possibility for the Dzyaloshinskii–Moriya interaction (DMI) and, consequently, chiral magnetism localized in the defect planes. The defect β-VOSO4 structure, therefore, opens up new possibilities for the study of low-dimensional magnetic systems.

High manganese redox variability and manganate predominance in temperate soil profiles as determined by X-ray absorption spectroscopy

Manganese speciation is a key to understanding the fate of contaminants, nutrients, and organic matter in soils. To date, quantification of Mn species in bulk soils has been performed mainly by sequential extraction methods and rarely supported by spectroscopic analysis. In order to obtain quantitative information on the Mn species inventory of soils, we investigated 46 soil horizons (<2-mm fraction, 45.1–2,280 mg/kg Mn) of nine typical Central European soils (Cambisols, Chernozems, Luvisols, Podzol, Stagnosol) by chemical Mn analyses and Mn K-edge X-ray absorption spectroscopy, and related speciation results to major soil properties. Amounts of Mn2+, Mn3+, and Mn4+ and the average oxidation state of Mn were evaluated by linear combination fitting (LCF) of X-ray absorption near edge structure (XANES) spectra. Additionally, we used extended X-ray absorption fine structure (EXAFS) spectroscopy to identify and quantify major Mn species. For this, EXAFS spectra of 20 organic and mineral soil samples from five soils (Cambisols, Chernozem, Luvisol) were analyzed by LCF and shell fitting. XANES analyses revealed a high Mn redox variability in organic surface layers, with Mn2+ being most abundant (≤100%, x¯ = 54%), followed by Mn3+ (≤80%, x¯ = 32%) and Mn4+ (≤55%, x¯ = 14%). Mineral soil horizons contained significantly less Mn2+ (≤56%, x¯ = 23%), about equal quantities of Mn3+ (≤68%, x¯ = 31%), and were enriched in Mn4+ (≤89%, x¯ = 46%). EXAFS analyses implied the presence of six major Mn species groups: manganates, organically complexed Mn, Mn(III) oxyhydroxides, silicate-bound Mn, Mn oxides without tunnel- or layer structure, and physisorbed Mn. In litter horizons, Mn was mainly present in organic complexes (58–91%, x¯ = 78%) and as physisorbed Mn (≤15%), but individual horizons also comprised manganates, Mn(III) oxyhydroxides, and silicate-bound Mn. Manganates, likely mixtures of phyllomanganates with hexagonal layer symmetry and tectomanganates, dominated in all mineral soil horizons (37–94%, x¯ = 67%). Correlation analysis showed that manganates dissolve completely during dithionite-citrate and acid ammonium oxalate extractions, and suggested that Mn4+-rich manganates preferentially form under less acidic soil conditions, partly by oxidation of organically complexed Mn(II), and that they are enriched in the soil clay fraction. Mineral soil horizons also contained minor quantities of organically complexed Mn (≤39%, x¯ = 11%), silicate-bound Mn (≤30%, x¯ = 8%), Mn(III) oxyhydroxides (≤37%, x¯ = 7%), Mn oxides without tunnel- or layer structure (≤18%, x¯ = 5%), and physisorbed Mn (≤14%, x¯ <1%). The detection of Mn(III) oxyhydroxides such as feitknechtite (β-MnOOH) or groutite (α-MnOOH) as well as the spinel hausmannite (Mn3O4) in acidic soils is remarkable, since their formation is normally linked to neutral or alkaline pH conditions. Minor contributions of silicate-bound Mn indicate the release of Mn from primary minerals already at early stages of soil formation, and low concentrations of physisorbed Mn suggest that exchangeable Mn is rapidly converted into manganates in oxic soils. The predominance of manganates in mineral soils has far-reaching implications for the functioning of soils and biogeochemical element cycles, as these minerals play an important role in metal binding, plant nutrition, and redox-related processes.

Stacking-order-dependent interlayer coupling in Janus WSSe/WS2 heterostructures

In transition metal dichalcogenide (TMDC) heterostructures, interlayer coupling plays a crucial role in the design and emergence of diverse properties. Here, we show the control of the interlayer coupling in Janus WSSe/WS2 heterostructures by changing the stacking order. Low-frequency Raman measurements revealed that heterostructures with the S/S interface show strong interlayer coupling compared to those with the S/Se interface. In the case of the S/S interface, photoluminescence (PL) from WS2 was quenched due to charge transfer in the type-II band alignment, while, interestingly, in the case of the S/Se interface, it was enhanced. The stacking-order dependence of the interlayer coupling and photoluminescence are attributed to the direction of interfacial electric field caused by broken out-of-plane mirror symmetry in a WSSe layer. In the case of the S/S interface, the interfacial electric field promotes the charge transfer as evidenced by strong PL quenching. On the contrary, in heterostructures with the S/Se interface, the charge transfer is blocked as the electric field prevents carrier transfer and modulates the band offset. Our results clearly show that Janus TMDCs with intrinsic fields can be effectively used to manipulate the interlayer interactions and optical properties in heterostructures.

Revisiting Covered Calls and Protective Puts: A Tale of Two Strategies

<h3>Background</h3> To examine the normative profile and determinants of retinal nerve fibre layer (RNFL) symmetry in a non-glaucoma, multiethnic Asian population. <h3>Methods</h3> Chinese, Malay and Indian adults aged ≥40 years were recruited from the Singapore Epidemiology of Eye Diseases study. Participants underwent standardised systemic and ocular examinations. RNFL thickness was obtained using a spectral-domain optical coherence tomography (Cirrus HD-OCT). RNFL symmetry (in %) was calculated based on Pearson correlation coefficient between the RNFL thickness profiles of the right and left eyes. Multivariable linear regression analysis was used to investigate the associations between ocular and systemic factors with RNFL symmetry. <h3>Results</h3> 4211 participants (1227 Chinese, 1245 Malays, 1739 Indians) were included. The mean RNFL symmetry was 86.7%±8.5% in Chinese, 84.7%±10.2% in Malays and 84.0%±10.7% in Indians. The fifth percentile limit of RNFL symmetry was 71.2% in Chinese, 65.0% in Malays and 62.0% in Indians. In multivariable analysis adjusting for age, gender, ethnicity, hypertension, diabetes, hyperlipidaemia, intereye absolute differences in intraocular pressure (IOP), axial length and disc area, Malays (β=−0.9; p=0.03) and Indians (β=−1.76; p&lt;0.001) were found to have lower RNFL symmetry compared with Chinese. Older age, greater intereye differences in IOP, axial length and disc area were significantly associated with lower RNFL symmetry (all p≤0.003). <h3>Conclusions</h3> In non-glaucoma individuals, intereye RNFL profile is less symmetric in Malays and Indians than that in Chinese. This also suggests that current commercial optical coherence tomography’s deployment of a single, universal RNFL symmetry cut-off for glaucoma detection is flawed, and ethnic-specific cut-off is warranted.

Research on nanosecond laser processing of basic unit for ridge surface: forming condition of microgroove and micro-ridge structure on TC4 surface

Frictional resistance significantly reduces the energy efficiency of aircraft. The ridge surface has a drag reduction effect, and its processing has become one of the important ways for improving the energy efficiency of aircraft. The micro-ridge and microgroove are the basic units of the ridge surface. A nanosecond laser can prepare them. However, numerous parameters and complex laser etching mechanisms make it difficult to determine their molding conditions, the forming quality of which is difficult to guarantee. This problem restricts the high-precision preparation of the ridge surface, which inevitably affects its drag reduction performance. Therefore, it is crucial to determine the high-quality molding conditions of the microgroove and micro-ridge to ensure the drag reduction effect of the ridge surface. Here, we focus on aviation titanium alloy TC4. Through an etching experiment of the straight section, the influence of parameters on the feature size of its cross-section profile is systematically studied. The recast layer scale, symmetry, and smoothness of the cross-section profile are used to evaluate the molding quality of microgrooves. The symmetry of the cross-section profile assesses the molding quality of micro-ridges. The forming parameter range and high-quality forming condition of microgrooves and the high-quality forming parameter range of micro-ridges are determined. Finally, the morphology evolution rule and mechanism of the nanosecond laser etching TC4 is revealed based on this research. The direct molding rules and parameter range of micro-ridges on the TC4 surface are reported and show a broadening of the nanosecond laser micromachining range. Moreover, this study will provide a process window for optimizing nanosecond laser processing for microgrooves and micro-ridges on the TC4 surface.

Association Analysis between Mechanical Symmetry and Requirements

This paper focuses on obtaining the association rules between the symmetry of different mechanical structure layers and requirements through data mining, then suggests the selection of symmetry schemes under different requirement conditions, based on the mined association rules and their strength. Firstly, a thousand symmetry structure cases, namely text data, are transformed into binary data. Then, the data analysis software RapidMiner is used to build an association rule mining model to obtain the association rules from symmetry to symmetry, from symmetry to requirements, from requirements to requirements and from requirements to symmetry. Among them, this paper focuses on the association rules with requirements as the premise and symmetry as the conclusion. Finally, according to the support and confidence of the association rules, the selection of symmetry under a single requirement, multiple requirements and decomposable requirements is discussed and summarized.

Polymorphism, polytypism and modular aspect of compounds with the general formula A2M3(TO4)4 (A = Na, Rb, Cs, Ca; M = Mg, Mn, Fe3+, Cu2+; T = S6+, P5+): order-disorder, topological description and DFT calculations.

The crystal structure of Na2Mn3(SO4)4 [unit-cell parameters a = 14.8307 (18), b= 9.9107 (18), c = 8.6845 (12) Å, space group Cmc21] displays order-disorder (OD) character and can be described using the OD groupoid family, more precisely a family of OD structures built up by two types of non-polar layers, with layer symmetry P(m)c21 (L2n+1 type) and P(b)cm (L2n type) (category IV). A new hypothetical MDO2 polytype has been proposed and the geometry optimization demonstrates its reasonability as another possible stable polytype. Compounds Na2Mn3-xMgx(SO4)4 with the unit-cell parameters a ∼ 29.2-29.7 Å, b ∼ 9.5-9.9 Å, c ∼ 8.7 Å and space group Pbca can be described in terms of modularity as a sequence of A, S1 and S2 modules:…|AS1AS2AS1AS2|… or (AS1AS2), together with MDO1 (AS1AS1) and MDO2 (AS2AS2). The crystal structures of itelmenite, NaCaFe3+3(PO4)4, and Ca2MgFe3+2(PO4)4 are crystal-chemical isotypic to Na2Mn3-xMgx(SO4)4 and should be considered as (A*S1A*S2) derivatives of the (AS1AS2)-type structure.

Crystal structure of a system with three-body interactions in strong confinement

Crystal structure of the Stillinger-Weber system in a narrow slit pore is studied. It is discovered that depending on the density the system forms several different crystal phases. Importantly, the system splits into several layers, and the symmetry of different layers can be different, which is not observed in systems with pair interaction only.