Two or three dimensions

The band-structure code, called BAND, also uses STO basis sets with STO fit functions or numerical atomic orbitals. Periodicity can be included in one, two, or three dimensions. No geometry optimization is available for band-structure calculations. The wave function can be decomposed into Mulliken, DOS, PDOS, and COOP plots. Form factors and charge analysis may also be generated. 

Crystal (we tested Crystal 98 1.0) is a program for ah initio molecular and band-structure calculations. Band-structure calculations can be done for systems that are periodic in one, two, or three dimensions. A separate script, called LoptCG, is available to perform optimizations of geometry or basis sets. 

The macmolplt graphics package is designed for displaying the output of GAMESS calculations. It can display molecular structures, including an animation of reaction-path trajectories. It also may be used to visualize properties, such as the electron density, orbitals, and electrostatic potential in two or three dimensions. 

Parabolic Equations in Two or Three Dimensions Computations become much more lengthy when there are two or more spatial dimensions. For example, we may have the unsteady heat conduction equation... 

The first detailed book to describe the practice and theory of stereology was assembled by two Americans, DeHoff and Rhines (1968) both these men were famous practitioners in their day. There has been a steady stream of books since then a fine, concise and very clear overview is that by Exner (1996). In the last few years, a specialised form of microstructural analysis, entirely dependent on computerised image analysis, has emerged - fractal analysis, a form of measurement of roughness in two or three dimensions. Most of the voluminous literature of fractals, initiated by a mathematician, Benoit Mandelbrot at IBM, is irrelevant to materials science, but there is a sub-parepisteme of fractal analysis which relates the fractal dimension to fracture toughness one example of this has been analysed, together with an explanation of the meaning of fractal dimension , by Cahn (1989). 

Band gap engineetring confined hetetrostruciutres. When the thickness of a crystalline film is comparable with the de Broglie wavelength, the conduction and valence bands will break into subbands and as the thickness increases, the Fermi energy of the electrons oscillates. This leads to the so-called quantum size effects, which had been precociously predicted in Russia by Lifshitz and Kosevich (1953). A piece of semiconductor which is very small in one, two or three dimensions - a confined structure - is called a quantum well, quantum wire or quantum dot, respectively, and much fundamental physics research has been devoted to these in the last two decades. However, the world of MSE only became involved when several quantum wells were combined into what is now termed a heterostructure. 

Wool [32] has considered the fractal nature of polymer-metal and of polymer-polymer surfaces. He argues that diffusion processes often lead to fractal interfaces. Although the concentration profile varies smoothly with the dimension of depth, the interface, considered in two or three dimensions is extremely rough [72]. Theoretical predictions, supported by practical measurements, suggest that the two-dimensional profile through such a surface is a self-similar fractal, that is one which appears similar at all scales of magnification. Interfaces of this kind can occur in polymer-polymer and in polymer-metal systems. 

More than 90% of the rocks and minerals found in the earth s crust are silicates, which are essentially ionic Typically the anion has a network covalent structure in which Si044-tetrahedra are bonded to one another in one, two, or three dimensions. The structure shown at the left of Figure 9.15 (p. 243), where the anion is a one-dimensional infinite chain, is typical of fibrous minerals such as diopside, CaSi03 - MgSi03. Asbestos has a related structure in which two chains are linked together to form a double strand. 

In addition to the size of the molecules and their distribution, the shapes or structures of individual polymer molecules also play an important role in determining the properties and processability of plastics. There are those that are formed by aligning themselves into long chains of molecules and others with branches or lateral connections to form complex structures. All these forms exist in either two or three dimensions. 

Kinetic expressions for appropriate models of nucleation and diffusion-controlled growth processes can be developed by the methods described in Sect. 3.1, with the necessary modification that, here, interface advance obeys the parabolic law [i.e. is proportional to (Dt),/2]. (This contrasts with the linear rate of interface advance characteristic of decomposition reactions.) Such an analysis has been provided by Hulbert [77], who considers the possibilities that nucleation is (i) instantaneous (0 = 0), (ii) constant (0 = 1) and (iii) deceleratory (0 < 0 < 1), for nuclei which grow in one, two or three dimensions (X = 1, 2 or 3, respectively). All expressions found are of the general form... 

However, the application potential can be fnlly exploited only if suitable methods to control the structure and to prepare ordered particle arrays over macroscopic dimensions are available. Various methods have been tried to organize mesoscale particles in two or three dimensions. Among them are methods already being used for the organization of molecules at interfaces, while other methods were developed especially for the organization of particles ... 

In order to make practical use of the physical properties of nanoparticles, whether individual or collective, one has to find a way to address them. If we leave out the near field techniques, this in turn requires that the particles be monodisperse and organized in two or three dimensions. It is therefore necessary to imagine techniques allowing the self-organization and even, ideally, the crystallization of nanoparticles into super-lattices. 

In principle, the calculation of bonding in two or three dimensions follows the same scheme as outlined for the chain extended in one dimension. Instead of one lattice constant a, two or three lattice constants a, b and c have to be considered, and instead of one sequential number k, two or three numbers kx, ky and k- are needed. The triplet of numbers k = (kx, ky, kz) is called wave vector. This term expresses the relation with the momentum of the electron. The momentum has vectorial character, its direction coincides with the direction of k the magnitudes of both are related by the de Broglie relation [equation (10.5)]. In the directions a, b and c the components of k run from 0 to nja, njb and n/c, respectively. As the direction of motion and the momentum of an electron can be reversed, we also allow for negative values of kx, ky and kz, with values that run from 0 to —nja etc. However, for the calculation of the energy states the positive values are sufficient, since according to equation (10.4) the energy of a wave function is E(k) = E(—k). 

Flow in a porous medium in two or three dimensions is important in situations such as the production of crude oil from reservoir formations. Thus, it is of interest to consider this situation briefly and to point out some characteristics of the governing equations. 

If electron-pair, or covalent, bonding is periodic in two or three dimensions, crystals result. The most important case is the carbon-carbon bond. If it is extended periodically in two-dimensions the result is graphite in three-dimensions it is diamond. Other elements that form electron-pair bonds are Si, Ge, and a-Sn. Some binary compounds are A1P (isoelectronic with Si),... 

The active sites from which a reaction in a solid spreads are known as nuclei. It is known that nuclei may grow in one, two, or three dimensions, and each case leads to a different form of the rate law. If the nuclei form in random sites in the solid (or perhaps on the surface), the rate laws are known random nudeation rate laws that have the form... 

In general, these defect-free modulated structures can, to a first approximation, be divided into two parts. One part is a conventional structure that behaves like a normal crystal, but a second part exists that is modulated5 in one, two, or three dimensions. The fixed part of the structure might be, for example, the metal atoms, while the anions might be modulated in some fashion. The primary modulation might be in the position of the atoms, called a displacive modulation (Fig. 4.35a). Displacive modulations sometimes occur when a crystal structure is transforming from one... 

For well-posed quadratic objective functions the contours always form a convex region for more general nonlinear functions, they do not (see tlje next section for an example). It is helpful to construct contour plots to assist in analyzing the performance of multivariable optimization techniques when applied to problems of two or three dimensions. Most computer libraries have contour plotting routines to generate the desired figures. 

Equation (A.7) is referred to as the inner product, or dot product, of two vectors. If the two vectors are orthogonal, then xTy = 0. In two or three dimensions, this means that the vectors x and y are perpendicular to each other. 

In nanocrystalline materials, the electrons are confined in regions having one, two or three dimensions (Fig. 1) when the relative dimension is comparable with the De... 

The pore types FAU, MFI, MOR, and LTA discussed above contain basically all micropore types possible in zeolites. There are accessible and nonaccessible cages (FAU and LTA) and straight or meandering channels in one, two, or three dimensions, which may be either isolated or connected to each other (MFI and MOR). 

The growth of the nuclei then occurs in one, two or three dimensions creating rods (fibrils), discs or spheres (spherulites). The development of crystallinity (Vc) under isothermal conditions with time ( ) is generally analysed according to... 

Nano-structures comments on an example of extreme microstructure In a chapter entitled Materials in Extreme States , Cahn (2001) dedicated several comments to the extreme microstructures and summed up principles and technology of nano-structured materials. Historical remarks were cited starting from the early recognition that working at the nano-scale is truly different from traditional material science. The chemical behaviour and electronic structure change when dimensions are comparable to the length scale of electronic wave functions. Quantum effects do become important at this scale, as predicted by Lifshitz and Kosevich (1953). As for their nomenclature, notice that a piece of semiconductor which is very small in one, two- or three-dimensions, that is a confined structure, is called a quantum well, a quantum wire or a quantum dot, respectively. 

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