Non-periodic structures

Solvent flatness. On average, protein crystals contain about 50% solvent, which on an atomic scale usually adopts a random, non-periodic structure within the crystal and hence is featureless within the averaged unit cell. Therefore, if we know the location of the solvent regions within a macro-molecular crystal, we already know a considerable part of the electron density (i.e. the part that is flat and featureless), and flattening the electron density of the solvent region can improve the density of our macromolecule of interest. 

Therefore, in most cases the scanning force microscope gives a lattice image similar to diffraction techniques. Visualisation of non-periodic structures or lattice defects, which means the true atomic resolution is exceptional and practically not attainable for polymers [58,236,246]. The smallest defects observed by conventional SFM are linear dislocations whose lengths exceed the contact diameter [247-249]. To approach the true atomic resolution, the aperture must be decreased as far as possible by using sharper tips and operating at lowest measurable forces to minimise the contact area. For example, to achieve the contact... 

Clark ES, Weeks JJ, Eby RK (1980) Diffraction from non-periodic structures, the molecular conformation of polytetrafluoroethylne (phase II). In French AD, Gardner KH (eds) Fiber diffraction ACS symposium series 141. American Chemical Society, Washington, DC, p 183... 

We note that many non-periodic structures, including Fibonacci ones, have a specific feature, which consists of the following. The constituent layers (A and B) of different optical density (and hence different thickness) occur in different number inside the structure, contrary to periodic multilayers. When the number of optically more dense layers is greater than that of optically less dense ones, the overall thickness diminishes compared to periodic stack w ith the same number of layers. This allows to place more layers into the same thickness which results in an increase of dispersion in Fibonacci stmctures in comparison with periodic ones of the same length as will be shown below. 

A very important aspect of setting up a neural network potential is the choice of the input coordinates. Cartesian coordinates cannot be used as input for NNs at all, because they are not invariant with respect to rotation and translation of the system. Since the NN output depends on the absolute numbers fed into the NN in the input nodes, simply translating or rotating a molecule would change its energy. Instead, some form of internal coordinates like interatomic distances and bond angles or functions of these coordinates should be used. To define a non-periodic structure containing N atoms uniquely, 3N-6 coordinates are required. However, for NNs redundant information does not pose a problem, and sometimes the complete set of N(N— l)/2 interatomic distances is used." ... 

This method has been applied to silicon, a frequently studied model system for the construction of potentials, by training the NN to tight binding and DFT energies.The potential can be used for periodic and non-periodic structures. 

Periodic structures can also be imaged by phase contrast (defocus) techniques, but these require care as artefacts are easily produced, especially from non-periodic structures, and it has been suggested that some reports concerning polyurethanes have been misinterpreted (Roche and Thomas 1981). 

Abstract We summarize an ab-initio real-space approach to electronic structure calculations based on the finite-element method. This approach brings a new quality to solving Kohn Sham equations, calculating electronic states, total energy, Hellmann-Feynman forces and material properties particularly for non-crystalline, non-periodic structures. Precise, fully non-local, environment-reflecting real-space ab-initio pseudopotentials increase the efficiency by treating the core-electrons separately, without imposing any kind of frozen-core approximation. Contrary to the variety of well established k-space methods that are based on Bloch s theorem... 

We introduced a method to solve Kohn Sham equations and to calculate electronic states and other properties of non-crystalline, non-periodic structures with fully non-local real-space environment-reflecting ab-initio pseudopotentials using finite elements, together with some preliminary results of our program. We believe this... 

At first sight, the mechanical behaviour of metallic foams appears as the simplest among competing GDL solutions, as their structure offers homogenised properties. This is only partially achievable, however, due to their non-periodic structure and the effect of the number of struts [8]. Non-negligible anisotropy, non-uniform deformation patterns and considerable scatter in the measurements of identical samples are reported [8, 96]. Nevertheless, simple relations based on analytical considerations on the possible deformation modes of simplified unit... 

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. 

The SSW form an ideal expansion set as their shape is determined by the crystal structure. Hence only a few are required. This expansion can be formulated in both real and reciprocal space, which should make the method applicable to non periodic systems. When formulated in real space all the matrix multiplications and inversions become 0(N). This makes the method comparably fast for cells large than the localisation length of the SSW. In addition once the expansion is made, Poisson s equation can be solved exactly, and the integrals over the intersitital region can be calculated exactly. 

The LSDA approach requires simultaneous self-consistent solutions of the Schrbdinger and Poisson equations. This was accomplished using the Layer Korringa-Kohn-Rostoker technique which has many useful features for calculations of properties of layered systems. It is, for example, one of only a few electronic structure techniques that can treat non-periodic infinite systems. It also has the virtue that the computational time required for a calculation scales linearly with the number of different layers, not as the third power as most other techniques. 

Unlike solid state -stacks, however, double helical DNA is a molecular structure. Here CT processes are considered in terms of electron or hole transfer and transport, rather than in terms of material conductivity. Moreover, the 7r-stack of DNA is constructed of four distinct bases and is therefore heterogeneous and generally non-periodic. This establishes differences in redox energetics and electronic coupling along the w-stack. The intimate association of DNA with the water and counterions of its environment further defines its structure and contributes to inhomogeneity along the mole-... 

It may be mentioned that in 2D and 3D the possible rotations (the symmetry axes) that superimpose an infinitely periodic structure on itself are limited to angles 360°/n with n = 1, 2, 3, 4 or 6. Notice that for non-periodic, noncrystalline, quasi-crystalline structures, other symmetry axes are possible. See 3.11.3 and Fig. 3.45 on quasi-periodic crystals. 

Figure 3.49. Projection from a 2D periodic square lattice to obtain (on Xn) a 1D non-periodic crystal. The projection space Xn is inclined with an irrational slope (1 It) to the square lattice axes. The window W (the strip) parallel to Xn is chosen so that a structure with a physically reasonable density is obtained (Kelton 1995). The points contained in the window are projected.

Structural relations between quasicrystals and other intermetallic phases. As discussed in several sections of the review published by Kelton (1995) on quasicrystals and related structures, numerous studies and observations indicate structural similarities between non-periodic quasicrystal phases with crystalline phases and also, on the other hand, with amorphous, glassy and liquid phases. 

These problems have of course different weights for the different metals. The high reactivity of the elements on the left-side of the Periodic Table is well-known. On this subject, relevant examples based on rare earth metals and their alloys and compounds are given in a paper by Gschneidner (1993) Metals, alloys and compounds high purities do make a difference The influence of impurity atoms, especially the interstitial elements, on some of the properties of pure rare earth metals and the stabilization of non-equilibrium structures of the metals are there discussed. The effects of impurities on intermetallic and non-metallic R compounds are also considered, including the composition and structure of line compounds, the nominal vs. true composition of a sample and/or of an intermediate phase, the stabilization of non-existent binary phases which correspond to real new ternary phases, etc. A few examples taken from the above-mentioned paper and reported here are especially relevant. They may be useful to highlight typical problems met in preparative intermetallic chemistry. 

Here the principles of constructing a 3D structure model from several HREM images of projections of inorganic crystals will be presented. Some of the principles may also be applied to non-periodic objects. A complex quasicrystal approximant v-AlCrFe is used as an example (Zou et al., 2003). Procedures for ab initio structure determination by 3D reconstruction are described in detail. The software CRISP, ELD. Triple and 3D-Map are used for 3D reconstruction. The 3D reconstruction method was demonstrated on the silicate mineral (Wenk et al. 1992). It was also applied to solve the 3D structures of a series mesoporous materials (Keneda etal. 2002). 

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