Three dimensions

One more application area is composite materials where one wants to investigate the 3D structure and/or reaction to external influences. Fig.3a shows a shadow image of a block of composite material. It consists of an epoxy matrix with glass fibers. The reconstructed cross-sections, shown in Fig.3b, clearly show the fiber displacement inside the matrix. The sample can be loaded in situ to investigate the reaction of matrix and fibers to external strain. Also absorption and transmission by liquids can be visualized directly in three-dimensions. This method has been applied to the study of oil absorption in plastic granules and water collection inside artificial plant grounds. 

The coin-tap test is a widely used teclinique on thin filament winded beams for detection of disbonded and delaminated areas. However, since the sensitivity of this teclinique depends not only on the operator but also on the thickness of the inspected component, the coin-tap testing technique is most sensitive to defects positioned near the surface of the laminate. Therefore, it was decided to constructed a new scaimer for automated ultrasonic inspection of filament winded beams. A complete test rig illustrated in figure 6 was constructed in order to reduce the scanning time. While the beam rotates the probe is moved from one end to the other of the beam. When the scarming is complete it is saved on diskette and can then be evaluated on a PC. The scanner is controlled by the P-scan system, which enables the results to be presented in three dimensions (Top, Side and End view). 

The shear viscosity is an important property of a Newtonian fluid, defined in terms of the force required to shear or produce relative motion between parallel planes [97]. An analogous two-dimensional surface shear viscosity ij is defined as follows. If two line elements in a surface (corresponding to two area elements in three dimensions) are to be moved relative to each other with a velocity gradient dvfdx, the required force is... 

The treatment of translational motion in three dimensions involves representation of particle motions in tenns... 

All the theory developed up to this point has been limited in the sense that translational motion (the continuum degree of freedom) has been restricted to one dimension. In this section we discuss the generalization of this to three dimensions for collision processes where space is isotropic (i.e., collisions in homogeneous phases, such as in a... 

In all of these stmctures the atomic positions are fixed by the space group syimnetry and it is only necessary to detennine which of a small set of choices of positions best fits the data. According to the theory of space groups, all stmctures composed of identical unit cells repeated hi three dimensions must confomi to one of 230 groups tliat are fomied by coinbinmg one of 14 distinct Bmvais lattices with other syimnetry operations. 

Haddon R C 1988 ii-eleotrons in three dimensions Accounts Chem. Res. 21 243-9... 

Terril R H ef a/1995 Monolayers in three dimensions NMR, SAXS, thermal and eleotron hopping studies of alkanethiol stabilized gold olusters J. Am. Chem. Soc. 117 12 537... 

Hosteler M J ef a/1996 Monolayers in three dimensions synthesis and eleotroohemistry of w-funotionalized alkanethiolate stabilized gold oluster oompounds J. Am. Chem. Soc. 118 4212... 

As with any vector, the above nonzero coupling vectors (w,.y (Rx), i j) can be decomposed, due to an extension beyond three dimensions [26] of the... 

James F. Leathrum and John A. Board. The parallel fast multipole algorithm in three dimensions. Technical report. Dept, of Electrical Engineering, Duke University, Durham, 1992. 

D. Beglov and B. Roux. Numerical solutions of the hypernetted chain equation for a solute of arbitrary geometry in three dimensions. J. Chem. Phys., 103 360-364, 1995. 

ISlS/Draw has no genuine molecular visualization tool. The rotate tool changes only the 2D rotate tool into a 3D rotate tool which rotates 2D structures in three dimensions. In order to visualize chemical structures in different styles and perspectives, it is necessary to paste the drawing, e.g., to the ACD/3D Viewer. 

Anuther concept that is extremely powerful when considering lattice structures is the fi i i/imca/ lattice. X-ray crystallographers use a reciprocal lattice defined by three vectors a, b and c in which a is perpendicular to b and c and is scaled so that the scalar juoduct of a and a equals 1. b and c are similarly defined. In three dimensions this leads to the following definitions ... 

Irude model thus predicts that the dispersion interaction varies as 1//. wo-dimensional Drude model can be extended to three dimensions, the result being ... 

For a fluid, with no underlying regular structure, the mecin squared displacement gradually increases with time (Figure 6.9). For a solid, however, the mean squared displacement typically oscillates about a mean value. Flowever, if there is diffusion within a solid then tliis can be detected from the mean squared displacement and may be restricted to fewer than three dimensions. For example. Figure 6.10 shows the mean squared displacement calculated for Li+ ions in Li3N at 400 K [Wolf et al. 1984]. This material contains layers of LiiN mobility of the Li" " ions is much greater within these planes than perpendicular to them. 

Tiic Langevin dipole method of Warshel and Levitt [Warshel and Levitt 1976] i.itermediate between a continuum and an explicit solvation model. A three-dimension... 

Gresho, P. M., Lee, R. L. and Sani, R. L., 1980. On the time-dependent solution of the incompressible Navier-Stokes equations in two and three dimensions. In Recent Advances in Numerical Methods in fluids, Ch. 2, Pineridge Press, Swansea, pp. 27-75. 

In the case of a polyatomic molecule, rotation can occur in three dimensions about the molecular center of mass. Any possible mode of rotation can be expressed as projections on the three mutually perpendicular axes, x, y, and z hence, three moments of inertia are necessar y to give the resistance to angular acceleration by any torque (twisting force) in a , y, and z space. In the MM3 output file, they are denoted IX, lY, and IZ and are given in the nonstandard units of grams square centimeters. 

The above partiele in a box model for motion in two dimensions ean obviously be extended to three dimensions or to one. 

For two and three dimensions, it provides a erude but useful pieture for eleetronie states on surfaees or in erystals, respeetively. Free motion within a spherieal volume gives rise to eigenfunetions that are used in nuelear physies to deseribe the motions of neutrons and protons in nuelei. In the so-ealled shell model of nuelei, the neutrons and protons fill separate s, p, d, ete orbitals with eaeh type of nueleon foreed to obey the Pauli prineiple. These orbitals are not the same in their radial shapes as the s, p, d, ete orbitals of atoms beeause, in atoms, there is an additional radial potential V(r) = -Ze /r present. However, their angular shapes are the same as in atomie strueture beeause, in both eases, the potential is independent of 0 and (j). This same spherieal box model has been used to deseribe the orbitals of valenee eleetrons in elusters of mono-valent metal atoms sueh as Csn, Cun, Nan and their positive and negative ions. Beeause of the metallie nature of these speeies, their valenee eleetrons are suffieiently deloealized to render this simple model rather effeetive (see T. P. Martin, T. Bergmann, H. Gohlieh, and T. Lange, J. Phys. Chem. 6421 (1991)). 

To describe the orientations of a diatomic or linear polyatomic molecule requires only two angles (usually termed 0 and ([)). For any non-linear molecule, three angles (usually a, P, and y) are needed. Hence the rotational Schrodinger equation for a nonlinear molecule is a differential equation in 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. 

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