Periodical Structures

In order to apply the laser interference structuring method, the configuration of laser beams that produce desired interference pattern and hence energy distribution on the surface of the sample has to be calculated prior to the experiment. Calculating the configuration of electromagnetic waves that reproduce a desired interference pattern is an inverse problem and its solution is not known in general. 

Each laser beam with wave vector kj contributes a new degree of freedom to the interference pattern. With relatively few laser beams, the geometry and symmetry of the interference pattern remain simple. More complicated structures will in general require more generating laser beams for the construction of the interference pattern.  

Miitzel et al have presented a numerical method to solve the inverse problem in the general case that the laser beams are not in phase. For the laser configuration used in the present study, all laser beams are in phase and have the same angular frequency. Therefore, the mathematical procedure of solving the inverse problem can be simplified in the following way. 

Since the pattern is described in terms of individual beams with wave vectors kj, it is convenient to examine the pattern in k-spac by applying the discrete Fourier transformation (FT) to the desired interference pattern (see Figs. 3(a) and 3(b)). To reproduce the target pattern, the most significant points (vectors kj) in Fourier space must be used. These vectors contain all the information which is necessary to reproduce the pattern and are determined by the Ewald-sphere. The centre of the sphere is given by the coordinate k ky) which are the mean value of the kxi and kyi coordinates of all the -vectors respectively. There after, the angles of each partial beam parallel to the interference plane j) can be calculated as indicated in Fig. 3(c).  

In our experimental setup the laser beams are not confined to 2D-space, but strike the target with a certain angle aj (Fig. 2). The projection of the wave vector onto the diffraction plane must then be equal to the wave vector in 2D-space given in Fig. 3(c). This constraint defines the values of the angles of/ between the normal to the interference plane (Fig. 2(e)). In addition, the angles aj of the beams are related to each other by 

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Diffraction is not limited to periodic structures [1]. Non-periodic imperfections such as defects or vibrations, as well as sample-size or domain effects, are inevitable in practice but do not cause much difSculty or can be taken into account when studying the ordered part of a structure. Some other forms of disorder can also be handled quite well in their own right, such as lattice-gas disorder in which a given site in the unit cell is randomly occupied with less than 100% probability. At surfaces, lattice-gas disorder is very connnon when atoms or molecules are adsorbed on a substrate. The local adsorption structure in the given site can be studied in detail. 

Lattice models have been studied in mean field approximation, by transfer matrix methods and Monte Carlo simulations. Much interest has focused on the occurrence of a microemulsion. Its location in the phase diagram between the oil-rich and the water-rich phases, its structure and its wetting properties have been explored [76]. Lattice models reproduce the reduction of the surface tension upon adsorption of the amphiphiles and the progression of phase equilibria upon increasmg the amphiphile concentration. Spatially periodic (lamellar) phases are also describable by lattice models. Flowever, the structure of the lattice can interfere with the properties of the periodic structures. 

Two point defects may aggregate to give a defect pair (such as when the two vacanc that constitute a Schottky defect come from neighbouring sites). Ousters of defects ( also form. These defect clusters may ultimately give rise to a new periodic structure oi an extended defect such as a dislocation. Increasing disorder may alternatively give j to a random, amorphous solid. As the properties of a material may be dramatically alte by the presence of defects it is obviously of great interest to be able to imderstand th relationships and ultimately predict them. However, we will restrict our discussion small concentrations of defects. 

Parameters measured Surface topography (rms roughness, rms slope, and power spectrum of structure) scattered light line shape of periodic structure (width, side wall angle, height, and period)... 

A,/2 for topography characterization much smaller for periodic structure characterization (A, is the laser wavelength used to illuminate the sample)... 

Topography characterization of nominally smooth surftices process control when characterizing periodic structure can be applied in situ in some cases rapid amenable to automation... 

K. C. Hickman, S. M. Caspar, S. S. H. Naqvi, K. P. Bishop, J. R, McNeil, G. D. Tipton, B. R, Stallard, and B. L. Draper. Use of Diffraction From Latent Images to Improve Lithogrophy Control. Presented at the SPIE Technical Conference 1464 Symposium on I.C. Metrology, Inspection, and Process Control, San Jose, CA, 1991, Proc. SPIE. 1464, pp. 245-257, 1991. Another application is presented of scattering characterization and modeling from periodic structures for process control. 

Structurally, carbon nanotubes of small diameter are examples of a onedimensional periodic structure along the nanotube axis. In single wall carbon nanotubes, confinement of the stnreture in the radial direction is provided by the monolayer thickness of the nanotube in the radial direction. Circumferentially, the periodic boundary condition applies to the enlarged unit cell that is formed in real space. The application of this periodic boundary condition to the graphene electronic states leads to the prediction of a remarkable electronic structure for carbon nanotubes of small diameter. We first present... 

In this paper, we review progress in the experimental detection and theoretical modeling of the normal modes of vibration of carbon nanotubes. Insofar as the theoretical calculations are concerned, a carbon nanotube is assumed to be an infinitely long cylinder with a mono-layer of hexagonally ordered carbon atoms in the tube wall. A carbon nanotube is, therefore, a one-dimensional system in which the cyclic boundary condition around the tube wall, as well as the periodic structure along the tube axis, determine the degeneracies and symmetry classes of the one-dimensional vibrational branches [1-3] and the electronic energy bands[4-12]. 

V. Triply Periodic Structures Generated from the Basic Model 702... 

In the Fourier space, the order parameter 0(r) for periodic structures is approximated by the Fourier series... 

V. TRIPLY PERIODIC STRUCTURES GENERATED FROM THE BASIC MODEL... 

In this section we characterize the minima of the functional (1) which are triply periodic structures. The essential features of these minima are described by the surface (r) = 0 and its properties. In 1976 Scriven [37] hypothesized that triply periodic minimal surfaces (Table 1) could be used for the description of physical interfaces appearing in ternary mixtures of water, oil, and surfactants. Twenty years later it has been discovered, on the basis of the simple model of microemulsion, that the interface formed by surfactants in the symmetric system (oil-water symmetry) is preferably the minimal surface [14,38,39]. 

For ordered periodic structures of a period A a dimensionless ratio between the two lengths, Ryy = Kyy /, provides an additional characteristic of the structure. On the basis of the results of Ref. 39, we can estimate this ratio for the periodic minimal surfaces. For simple minimal surfaces, P, D, or G [39], we find respectively = 0.306, Ryyi = 0.195, and Ryyi = 0.248. For more complicated periodic structures [39] its value can be even smaller than 0.1 for example, for the surface labeled GX5, Ryy = 0.073. 

The diffusion field just ahead of the solid front can be thought of as containing two ingredients a diffusion layer of thickness associated with global solute rejection, and modulations due to the periodic structure of the solid of extent A (A averaging approximation by Jackson and Hunt [137] seems justified. 

Transient length increases to about 10 steps periodic structures (with short periodicity) emerge. 

X (j)) 0.35 Transient length grows significantly a new kind of periodic structure with longer period appears, thereby increasing the spectrum of possible dynamical behaviors. 

Transient length increases to about 60 steps periodic structures of period equal to about 40 steps appear. 

H. Kogclnik, C. V. Shank. Stimulated emission in a periodic structure. Appl. Phys. Lett. 1971, IS,... 

Unlike the heterostructures whose periodic structure must be accurately controlled, the formation of nanocomposite structure is self-organized based upon thermodynamically driven spinodal phase segregation [118-121]. For the CVD... 

Structurally, plastomers straddle the property range between elastomers and plastics. Plastomers inherently contain some level of crystallinity due to the predominant monomer in a crystalline sequence within the polymer chains. The most common type of this residual crystallinity is ethylene (for ethylene-predominant plastomers or E-plastomers) or isotactic propylene in meso (or m) sequences (for propylene-predominant plastomers or P-plastomers). Uninterrupted sequences of these monomers crystallize into periodic strucmres, which form crystalline lamellae. Plastomers contain in addition at least one monomer, which interrupts this sequencing of crystalline mers. This may be a monomer too large to fit into the crystal lattice. An example is the incorporation of 1-octene into a polyethylene chain. The residual hexyl side chain provides a site for the dislocation of the periodic structure required for crystals to be formed. Another example would be the incorporation of a stereo error in the insertion of propylene. Thus, a propylene insertion with an r dyad leads similarly to a dislocation in the periodic structure required for the formation of an iPP crystal. In uniformly back-mixed polymerization processes, with a single discrete polymerization catalyst, the incorporation of these intermptions is statistical and controlled by the kinetics of the polymerization process. These statistics are known as reactivity ratios. 

The common periodic structures displayed by surfaces are described by a two-dimensional lattice. Any point in this lattice is reached by a suitable combination of two basis vectors. Two unit vectors describe the smallest cell in which an identical arrangement of the atoms is found. The lattice is then constructed by moving this unit cell over any linear combination of the unit vectors. These vectors form the Bravais lattices, which is the set of vectors by which all points in the lattice can be reached. 

The Si04 tetrahedra can be arranged into several silicate units, e.g. squares, six-or eight-membered rings, called secondary building blocks. Zeolite structures are then built up by joining a selection of building blocks into periodic structures. 

The same periodic structures can also be formed from alternating AIO4 and PO4 tetrahedra the resulting aluminophosphates are not called zeolites but AlPOs. Zeolites are made by hydrothermal synthesis under pressure in autoclaves, in the presence of template molecules such as tetramethylammonium, which act as structure directing agents. 

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