Three-dimensional periodic structures

A study of the external symmetry of crystals naturally leads to the idea that a single crystal is a three-dimensional periodic structure i.e., it is built of a basic structural unit that is repeated with regular periodicity in three-dimensional space. Such an infinite periodic structure can be conveniently and completely described in terms of a lattice (or space lattice), which consists of a set of points (mathematical points that are dimensionless) that have identical environments. 

Gates, B., Yin, Y, and Xia, Y, Eabrication and characterization of porous membranes with highly ordered three-dimensional periodic structures, Chem. Mater, 11, 2827, 1999. 

Xia Y, Lu Y, Kamata K, Gates B, Yin Y (2003) Macroporous materials containing three-dimensionally periodic structures. In Yang P (ed) Chemistry of nanostructured materials. 

Li, Q. Lewis, J. A. Nanoparticle inks for directed assembly of three-dimensional periodic structures. Adv. Mater., 2003, 15, 1639-1643. 

The concept of defects came about from crystallography. Defects are dismptions of ideal crystal lattice such as vacancies (point defects) or dislocations (linear defects). In numerous liquid crystalline phases, there is variety of defects and many of them are not observed in the solid crystals. A study of defects in liquid crystals is very important from both the academic and practical points of view [7,8]. Defects in liquid crystals are very useful for (i) identification of different phases by microscopic observation of the characteristic defects (ii) study of the elastic properties by observation of defect interactions (iii) understanding of the three-dimensional periodic structures (e.g., the blue phase in cholesterics) using a new concept of lattices of defects (iv) modelling of fundamental physical phenomena such as magnetic monopoles, interaction of quarks, etc. In the optical technology, defects usually play the detrimental role examples are defect walls in the twist nematic cells, shock instability in ferroelectric smectics, Grandjean disclinations in cholesteric cells used in dye microlasers, etc. However, more recently, defect structures find their applications in three-dimensional photonic crystals (e.g. blue phases), the bistable displays and smart memory cards. 

It has been demonstrated that an electrohydrod5mamic jet printing process is suitable for the fabrication of three-dimensional periodic structures, and thus has great potential in advanced biomedical applications, such as ceU alignment (62). 

Bragg Diffraction Scattering of X rays from a three-dimensional periodic structure such as a crystal, caused by waves reflecting from different crystal planes. 

The blue phases of types BPI and BPn are modeled as regular networks of disclination lines with periodicity of order p. Indeed, the three-dimensional periodic structure of these phases is revealed in their nonzero shear moduli, their ability to grow well-faceted monocrystals and Bragg reflection in the visible part of the spectrum (which is natural since p is of the order of a few tenths of a micron). The third identified phase, BPIH, that normally occurs between the isotropic melt and BPII, is less understood. It might be a melted array of disclinations. Note that although most blue phases have been observed in thermotropic systems, double-twist geometries are relatively frequently met in textures of biological polymers, like DNA. 

The situation is envisaged in which the total energy of a box of atoms can be calculated, and one wants to obtain the excess energy of an interface which has been constructed within the box. The simplest situation is if a static calculation has been made and the atomic positions are relaxed to the structure of minimum energy. However, free energy calculations are also feasible. Periodic boundary conditions parallel to the interface are employed, and perhaps also three dimensional periodicity, which implies that two boundaries per box are necessary. These technicalities as well as the method for calculating energies will not be discussed further here. 

These modifications constitute important sub-cases of the case of positional disorder for which only some characterizing points of the structure maintain long-range three-dimensional periodicity (indicated as case i in Sect. 2.1). 

Examples of Positional Disorder with Long-Range Three-Dimensional Periodicity Maintained Only for Some Characterizing Points of Structure... 

Modulated structures, corresponding to substances in which there is an average structure with a three-dimensional periodicity showing modulated perturbations. 

The advantage of being able to record diffraction intensities over a range of incident beam directions makes CBED readily accessible for comparison with simulations. Thus, CBED is a quantitative diffraction technique. In past 15 years, CBED has evolved from a tool primarily for crystal symmetry determination to the most accurate technique for strain and structure factor measurement [16]. For defects, large angle CBED technique can characterize individual dislocations, stacking faults and interfaces. For applications to defect structures and structure without three-dimensional periodicity, parallel-beam illumination with a very small beam convergence is required. 

The recommendations embodied in this document are concerned with the terminology relating to the structure of crystalline polymers and the process of macromolecular crystallization. The document is limited to systems exhibiting crystallinity in the classical sense of three-dimensionally periodic regularity. The recommendations deal primarily with crystal structures that are comprised of essentially rectilinear, parallel-packed polymer chains, and secondarily, with those composed of so-called globular macromolecules. Since the latter are biological in nature, they are not covered in detail here. In general, macromolecular systems with mesophases are also omitted, but crystalline polymers with conformational disorder are included. 

Photonic band gap crystals can be dehned as long-range ordered structures whose relative permittivity varies as a spatially periodic function. These three-dimensional periodic structnres have a feature size comparable to or shorter than the wavelength of visible light. 

The three-dimensional periodic electron-density distribution in a single crystal can be represented by a three-dimensional Fourier series with the so-called structure factors Fhkl as Fourier coefficients ... 

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