High-symmetry Lattices Research Articles

Diffractive Nonlocal Metasurfaces (Laser Photonics Rev. 16(8)/2022)

Diffractive Nonlocal Metasurfaces In article number 2100633, Adam Overvig and Andrea Alù present a perspective on the emerging field of diffractive nonlocal metasurfaces, which are able to tailor both the spatial and spectral properties of light through deliberately engineered perturbations in high symmetry lattices. Highly customizable scattering rate, phase, and polarization properties, tailored over subwavelength scales, enable highly selective optical responses, offering new avenues for rationally designing ultrathin photonic devices.

A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level wavefunctions on the torus

A “modified” variant of the Weierstrass sigma, zeta, and elliptic functions is proposed whereby the zeta function is redefined by ζ(z) ↦ ζ̃(z) ≡ ζ(z)−γ2z, where γ2 is a lattice invariant related to the almost-holomorphic modular invariant of the quasi-modular-invariant weight-2 Eisenstein series. If ωi is a primitive half-period, ζ̃(ωi) = πωi*/A, where A is the area of the primitive cell of the lattice. The quasiperiodicity of the modified sigma function is much simpler than that of the original, and it becomes the building-block for the modular-invariant formulation of lowest-Landau-level wavefunctions on the torus. It is suggested that the “modified” sigma function is more natural than the original Weierstrass form, which was formulated before quasi-modular forms were understood. For the high-symmetry (square and hexagonal) lattices, the modified and original sigma functions coincide.

Evaluation and regularization of phase-modulated Eisenstein series and application to double Schlömilch-type sums

In the study of periodic media, conditionally convergent series are frequently encountered and their regularization is crucial for applications. We derive an identity that regularizes two-dimensional phase-modulated Eisenstein series for all Bravais lattices, yielding physically meaningful values. We also obtain explicit forms for the phase-modulated series in terms of holomorphic Eisenstein series, enabling their closed-form evaluation for important high symmetry lattices. Results are then used to obtain representations for the related double Schlömilch-type sums, which are also given for all Bravais lattices. Finally, we treat displaced lattices of high symmetry, expressing them in terms of origin-centered lattices via geometric multi-set identities. These identities apply to all classes of two-dimensional sums, allowing sums to be evaluated over each constituent of a unit cell that possesses multiple inclusions.

Minimum Vertex-type Sequence Indexing for Clusters on Square Lattice

An effective indexing scheme for clusters that enables fast structure comparison and congruence check is desperately desirable in the field of mathematics, artificial intelligence, materials science, etc. Here we introduce the concept of minimum vertex-type sequence for the indexing of clusters on square lattice, which contains a series of integers each labeling the vertex type of an atom. The minimum vertex-type sequence is orientation independent, and it builds a one-to-one correspondence with the cluster. By using minimum vertex-type sequence for structural comparison and congruence check, only one type of data is involved, and the largest amount of data to be compared is n pairs, n is the cluster size. In comparison with traditional coordinate-based methods and distance-matrix methods, the minimum vertex-type sequence indexing scheme has many other remarkable advantages. Furthermore, this indexing scheme can be easily generalized to clusters on other high-symmetry lattices. Our work can facilitate cluster indexing and searching in various situations, it may inspire the search of other practical indexing schemes for handling clusters of large sizes.

Analysis of magneto-optical properties for three-dimensional photonic crystals in high-symmetry arrangement doped by metamaterials and uniaxial materials

Abstract In this paper, we use a modified plane wave expansion (PWE) method to investigate the properties of photonic band gaps (PBGs) for the extraordinary mode in the three-dimensional (3D) photonic crystals (PCs) which are composed of the anisotropic dielectric (the uniaxial materials) spheres immersed in the homogeneous metamaterials (epsilon-negative materials) background with high-symmetry (body-centered-cubic) lattices, as the magneto-optical Voigt effects are considered. The equations for calculating the PBGs in the first irreducible Brillouin zone are theoretically derived. It is numerically illustrated that the anisotropic PBGs and two flattened band regions can be achieved. The influences of the ordinary-refractive index, extraordinary-refractive index, filling factor of dielectric spheres, electronic plasma frequency and cyclotron frequency on the magneto-optical properties of such 3D PCs also are studied in detail, respectively, and some corresponding physical explanations are given. The numerical results demonstrate that the anisotropy can open partial band gaps in the proposed PCs, and the complete PBGs can be obtained compared with the conventional PCs only containing the isotropic material with similar structures. The bandwidths of PBGs can be tuned by introducing the epsilon-negative materials into such PCs containing the uniaxial materials. The anisotropic PBGs can be manipulated by the parameters as mentioned above. As the proposed PCs with high-symmetry lattices, the complete PBGs can be obtained by introducing the uniaxial materials.

Directional scaling symmetry of high-symmetry two-dimensional lattices.

Two-dimensional lattices provide the arena for many physics problems of essential importance, a scale symmetry, which rarely exists as noticed by Galileo, in such lattices can help reveal the underlying physics. Here we report the discovery and proof of directional scaling symmetry for high symmetry 2D lattices, i.e., the square lattice, the equilateral triangular lattice and thus the honeycomb lattice, with aid of the function y = arcsin(sin(2πxn)), where the parameter x is either the platinum number or the silver number , which are related to the 12-fold and 8-fold quasiperiodic structures, respectively. The directions and scale factors for the symmetric scaling transformation are determined. The revealed scale symmetry may have a bearing on various physical problems modeled on 2D lattices, and the function adopted here can be used to generate quasiperiodic lattices with enumeration of lattice points. Our result is expected to initiate the search of directional scaling symmetry in more complicated geometries.

The anisotropic photonic band gaps in three-dimensional photonic crystals with high-symmetry lattices composed of metamaterials and uniaxial materials

In this paper, the properties of anisotropic photonic band gaps (PBGs) for three-dimensional (3D) photonic crystals (PCs) composed of tellurium (Te) spheres (the uniaxial materials) in homogeneous single-negative metamaterials (epsilon-negative materials) background with high-symmetry (simple-cubic) lattices are theoretically investigated based on the plane wave expansion method. The equations for calculating the anisotropic PBGs in the first irreducible Brillouin zone are theoretically deduced. The influences of the ordinary-refractive index, extraordinary-refractive index, filling factor of dielectric, the electronic plasma frequency, the dielectric constant of epsilon-negative materials on the anisotropic PBGs are also studied in detail, respectively, and some corresponding physical explanations are also given. The numerical results show that the anisotropy can open partial band gaps in simple-cubic lattices and the complete PBGs also can be achieved compared to such 3D PCs doped by the conventional isotropic materials. It also is shown that the anisotropic PBGs can be manipulated by the parameters as mentioned above. Introducing the uniaxial materials into 3D dielectric-epsilon-negative materials PCs can enlarge the PBGs, and also provide a way to obtain the complete PBGs as such kind of 3D PCs with high-symmetry lattices.

A comparative study of band Faraday effects in 3D magnetized photonic crystals with different high-symmetry lattices with uniaxial materials

In this paper, the band Faraday effects in three-dimensional (3D) magnetized plasma photonic crystals (MPPCs) composed of anisotropic dielectric (the uniaxial material) spheres immersed in uniform magnetized plasma background with various lattices including the face-centered-cubic (fcc), body-centered-cubic (bcc), and simple-cubic (sc) lattices are theoretically investigated by a modified plane wave expansion (PWE) method, as the Faraday effects of magnetized plasma are considered (the incidence electromagnetic wave vector is parallel to the external magnetic field at any time). The equations for calculating the anisotropic PBGs in the first irreducible Brillouin zone are theoretically deduced. The anisotropic PBGs and a flatbands region can be achieved as the uniaxial material introduced into 3D MPPCs. The 3D MPPCs with fcc, bcc, sc lattices will not exhibit a complete PBG unless the uniaxial material is introduced. The influences of the ordinary refractive index, extraordinary refractive index, filling factor, plasma frequency, and plasma cyclotron frequency (the external magnetic field) on the properties of anisotropic PBGs for 3D MPPCs with fcc, bcc, sc lattices are investigated in detail, respectively, and some corresponding physical explanations are also given. The numerical results show that the anisotropy can open partial band gaps in 3D MPPCs with fcc, bcc, sc lattices, and the complete PBGs can be obtained compared to 3D MPPCs containing of the conventional isotropic dielectric. It also is shown that the anisotropic PBGs can be tuned by the ordinary refractive index, extraordinary refractive index, plasma cyclotron frequency, filling factor, and plasma frequency, respectively. The larger relative bandwidth of complete PBG can be obtained by introducing the uniaxial material as 3D MPPCs are with high-symmetry lattices. This also provides a way to design the tunable MPPCs devices.

Multialiquot cell approach for the SDPD of high-symmetry compounds

Recently mentioned [1] inconsistency of figures of merit [2,3] when indexing high symmetry lattices turned to a conclusion the smaller cells of lower symmetry can be applied as building units when solving a structure by direct space methods. The approach is expected to decrease time necessary for simulated annealing of a structure solution and may appear particularly useful for large organic structures. The poster illustrates the approach based of example compounds with a reatively small pyrochlore structure. The indexing program suggests several possible solutions and the correct solution of highest symmetry among them. The repetition of the same lattice described by different cells should be considered as an indication of the correct indexing solution. The extension to indexing algorithms, eliminating lower symmetry cells for the same lattice described by high-symmetry cell, and the corrected figures of merit taking into account the number of equal proposal cells might be drawn as follows: Msm = ∑ M(20)i Fsm = ∑ F(20)i

Zero temperature phase diagram of the square-shoulder system

Particles that interact via a square-shoulder potential, consisting of an impenetrable hard core with an adjacent, repulsive, steplike corona, are able to self-organize in a surprisingly rich variety of rather unconventional ordered, three-dimensional structures. Using optimization strategies that are based on ideas of genetic algorithms, we encounter, as we systematically increase the pressure, the following archetypes of aggregates: low-symmetry cluster and columnar phases, followed by lamellar particle arrangements, until at high pressure values compact, high-symmetry lattices emerge. These structures are characterized in the isobaric-isothermal ensemble as configurations of minimum Gibbs free energy. Based on simple considerations, i.e., basically minimizing the number of overlapping coronas while maximizing at the same time the density, the sequence of emerging structures can easily be understood. In contrast to a previous contribution [G. J. Pauschenwein and G. Kahl, Soft Matter 4, 1396 (2008)], we present here a systematic investigation of this phenomenon, considering a short, an intermediate, and a large shoulder width.

Dislocation effects in powder diffraction

Owing to the lack of proper models, mainly for the strain broadening component due to dislocations, full exploitation of whole powder pattern modelling for the microstructural analysis of materials has been limited so far to high-symmetry lattices. An extension of the dislocation-induced broadening model, valid for any lattice symmetry, is proposed here. Dislocation effects are modelled through the description of the dislocation contrast factor in terms of a crystallographic invariant whose terms can be either calculated when the material constants and slip systems are known (allowing a dislocation density to be refined) or refined on the experimental data for an effective estimation of dislocation effects.

Structure and order in soft matter: symmetry transcending length scale

Molecular crystals, surfactant assemblies, and block copolymers are properly classified as “soft matter,” but these classes of materials are usually regarded as distinct owing to their different properties and applications. Furthermore, the length scales of translation order in typical molecular crystals are typically at least one order of magnitude smaller than in surfactant assemblies and block copolymers, which often organize as high symmetry hexagonal and cubic microstructures. Whereas the structures of molecular crystals are generally viewed to be governed by intermolecular forces at short length scales that impose local order, surfactant assemblies and block copolymers are locally disordered even though they form ordered microstructures at longer length scales. Though historically the crystal structures of most molecular crystals have been characterized for the purpose of understanding molecular structure, the recent emergence of crystal engineering as a solid-state discipline has shifted the focus toward the elucidation of the factors responsible for crystal packing and strategies for crystal design. This coincides with a growing interest in surfactants and block copolymers having materials properties that can be tuned precisely at the molecular level. As such, it seems opportune to begin examining the structural relationships that connect these classes of materials. The intent of this highlight is to provoke interest in such comparisons through examples of molecular crystals that exhibit features which mimic microstructures observed in surfactant assemblies and block copolymers. These compounds appear to be characterized by molecular components with some degree of amphiphilic character and supramolecular structural elements that (i) enforce topologies that are predestined to form certain high symmetry structures, (ii) introduce conformational softness that permits curvature at small length scales, (iii) form aggregates (i.e., through hydrogen bonding) which effectively increase the length scale so that curved surfaces required for high symmetry lattices can be formed with minimal energetic penalties. These examples suggest that the relationship between length scale, energy, curvature, and crystal symmetry bear further examination.

High symmetry crystal supramolecularity: –XPh3 molecules in cubic latticesElectronic supplementary information (ESI) available: additional figures (Fig. S1–3) described in the text. See http://www.rsc.org/suppdata/nj/b1/b104324m/

The majority of molecular crystals in the Cambridge Structural Database involve relatively low symmetry molecules, engaged by intermolecular interactions with restricted symmetry, resulting in the preponderant occurrence of a small number of relatively low symmetry crystallographic space groups as expounded by Brock and Dunitz. We are examining some contrasting crystal structure types that contain higher symmetry molecules involved in high symmetry intermolecular motifs and generating high symmetry space groups. This paper considers molecules containing XPh3 groups, engaged in multiple phenyl embraces with local S6 () symmetry, generating crystal lattices with cubic symmetry. The conceptual framework for these analyses also involves the presence, symmetry, and occupancy of cavities in high symmetry lattices, with extensions to crystal trapping of unusual species such as [Cu2Br5]2− and [Cu8Br15]6−. The compounds included in the analyses of crystal packing and symmetry in this paper are (MePh3P+)2[Cu2Br5]2−, Ph3POGaCl3 , Ph3AsI2 , N(Ph-2-OMe)3 , Ph3SnCl, Ph3PAuCCAuPPh3·(C6H6)2 , Ph3PAu(B6H6)AuPPh3 , (Ph3PH+)2[SnCl6]2−, and (Ph3MeP+)6 [Cu8Br15]6−. Concerts of edge-to-face (EF) and offset-face-to-face (OFF) local interactions between phenyl groups generate bimolecular (EF)6 = 6PE with symmetry S6 , hexamolecular (EF)6 with symmetry S6 , trimolecular (EF)3 , hexamolecular (OFF)6 , and two-dimensional {(OFF)6}∞ . Crystal supramolecular analysis includes description of the asymmetric segments of intermolecular motifs, as well as the conventional asymmetric sector of the molecule.

Calculation of Ion Channeling Directions in Low Symmetry Optical Materials

Rutherford backscattering spectroscopy along ion channeling directions is useful for numerous applications in the characterization of all types of materials, due to its sensitivity to interstitial location, lattice quality, and composition. The determination of channeling directions and sample orientation in materials with high symmetry lattices is usually straightforward. However, many optical materials have relatively low crystal symmetry, which complicates the determination of likely directions for ion channeling. This is especially true of electrooptic materials, such as LiNbO3, which lack inversion symmetry. Therefore, an algorithm for the calculation of likely channeling directions in low symmetry lattices has been developed.