Geometric effect

One of the major difficulties in assessing the importance of geometric effects on a quantitative basis is the lack of precise knowledge of the appropriate sizes of the atoms in a metallic alloy and the observation that the ligands of an atom in a complex metallic structure generally surround it at several different distances. Certainly any model regarding metallic atoms as incompressible billiard balls is quite inappropriate  

The precisely known parameters respecting any crystal structure which has been aceurately determined are the arrangement of the 

The geometry of any particular structure adopted by a binary phase A B(, can be represented as a diagram composed of straight lines, one for every independent interatomic distance in the structure. The diagram is drawn with a reduced strain parameter (Da — dp)/D as ordinate and the diameter ratio DJD of the two component atoms as abscissa. Here Da and are the diameters of the 

When the point for a phase lies on a particular contact line on the diagram the atoms just come in contact according to the sum of their assumed radii. When it lies below the line contact has not been made, and when it lies above the line that particular contact is compressed according to the assumed sizes of the atoms (i.e., they are closer than the sum of their radii for the appropriate coordination). Thus, assuming the radius ratio for any pair of atoms, it is clear that the cell dimensions of a phase formed by them are free parameters that determine which contacts are not made, just made, or compressed in a particular structure. 

The Laves phase structures are built up of interpenetrating icosahedra and Friauf polyhedra (CN 12 and 16). The phases were always believed to form for geometric reasons since at a radius ratio 

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Measure Wall Thickness This window is used for the dialog to calibrate the algorithm aceording formula (3) and for point wise measurements after calibration. The row Ideal indicates the nominal wall thickness used, IQI indicates the wall thickness values used for calibration and the detected optical density. Local can be used for noise reduction and compensation of geometric effects. 

The geometrical problem. This involves evaluating the geometrical effect in item (2). It requires calculation of the volume of the overlapping regions as a function of d and the coil dimensions, say, r. The mathematics of this step are tedious and add little to the polymer aspects of the theory. 

Also used for drops. Geometric effect (d,mp/dtar.k) is usually unimportant. Ref. 118 gives a variety of references on correlations. ... 

From the depth profile alone it is not possible to distinguish whether the profile shape is a result of diffusion of elements or of geometric effects. This example demonstrates the capacity of 3D SIMS to improve the information content of depth profiles. 

Dose is related to the amount of radiation energy absorbed by people or equipment. If the radiation comes from a small volume compared with the exposure distance, it is idealized as a point source (Figure 8.3-4). Radiation source, S, emits particles at a constant rate equally in all directions (isotropic). The number of particles that impact the area is S t Tr where Tr is a geometric effect that corrects for the spreading of the radiation according to ratio of the area exposed to the area of a sphere at this distance i.e. the solid angle - subtended by the receptor (equation 8.3-4). 

This discussion of geometric effects ignored the attenuation of radiation by material through which the radiation must travel to reach the receptor. The number of particles, dN, penetrating material, equals the number of particles incident N times a small penetration distance, dx, divided by the mean free path length of the type of particle in the type of material (equation 8.3-8). Integrating gives the transmission coefficient for the radiation (equation 8.3-9). 

James M. Whitney, Geometrical Effects of Filament Twist on the Modulus and Strength of Graphite Fiber-Reinforced Composites, Textile Research Journal, September 1966, pp. 765-770. 

In addition to purely energetical heterogeneity one should also take into account some basic aspects of possible heterogeneities resulting from geometrical effects. The simplest and yet experimentally quite important geometric effects are due to the finite size of crystallites. Experimental measurements ave clearly demonstrated that the size of typical crystallites may be quite small (of the order of 50-100 A [116,132] and quite large (of the order of 10 A [61]. 

Filler orientation is not the only consequence of geometric effects in strained systems with an anisodiametric filler. The appearance of normal stresses has been named as another such phenomenon [169,171,185] which, like the abnormal viscosity behavior of suspensions, may not be due to the elasticity of system components. 

Despite the individual complexities of the systems surveyed here, several simple rules must have emerged in the reader s mind. Aside from the omnipresent site-blocking geometric effects, chemical or classical promotion can be understood, at least qualitatively, in terms of two simple and complementary mles, already outlined is section 2.5 ... 

Figure 2. Spectral photometric variation versus source position (1-3/im and 3-5/rm ranges). Upper surfaces are only affected by the geometrical effect of the slit geometry, and lower surfaces include the diffraction effects.

However, in subsequent studies [23-25,88-90] it was demonstrated that in reality the particle deposition is not a purely geometric effect, and the maximum surface coverage depends on several parameters, such as transport of particles to the surface, external forces, particle-surface and particle-particle interactions such as repulsive electrostatic forces [25], polydispersity of the particles [89], and ionic strength of the colloidal solution [23,88,90]. Using different kinds of particles and substrates, values of the maximum surface coverage varied by as much as a factor of 10 between the different studies. 

Geometrical effects, related to the number and geometrical arrangement of the surface metal atoms participating in the formation of the essential surface intermediates of the reaction in question. For these, number of atoms (ensemble size) appeared to be particularly crucial. 

Besides electronic effects, structure sensitivity phenomena can be understood on the basis of geometric effects. The shape of (metal) nanoparticles is determined by the minimization of the particles free surface energy. According to Wulffs law, this requirement is met if (on condition of thermodynamic equilibrium) for all surfaces that delimit the (crystalline) particle, the ratio between their corresponding energies cr, and their distance to the particle center hi is constant [153]. In (non-model) catalysts, the particles real structure however is furthermore determined by the interaction with the support [154] and by the formation of defects for which Figure 14 shows an example. 

Figure 4.6 Geometric effects at the electrochemical Ni(lll)/H20 interface for adsorbed H2O. Open symbols are for a five-layer slab and filled symbols for a three-layer slab hence, slab size is shown to not influence greatly the outcome of these calculations [Taylor et al., 2006a],...

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