Coincident site lattice defined

We assume in the following discussion that the solid surface under consideration is of the same chemical identity as the bulk, that is, free of any oxide film or passivation layer. Crystallization proceeds at the interfaces between a growing crystal and the surrounding phase(s), which may be solid, liquid, or vapor. Even what we normally refer to as a crystal surface is really an interface between the crystal and its surroundings (e.g., vapor, vacuum, solution). An ideal surface is one that is the perfect termination of the bulk crystal. Ideal crystal surfaces are, of course, highly ordered since the surface and bulk atoms are in coincident positions. In a similar fashion, a coincidence site lattice (CSL), defined as the number of coincident lattice sites, is used to describe the goodness of fit for the crystal-crystal interface between grains in a polycrystal. We ll return to that topic later in this section. 

Table 13 shows the features of compound tessellations (3, 6 [ 3, 6 ] to r = lOOA [Eqn. (12) assuming a = 5.3 A], each of which describes a coincidence-site lattice (CSL) (Ranganathan 1961) the multiplicity n of its mesh is termed coincidence index or Z factor and corresponds to the order of the subgroup of translation defining the two-dimensional CSL with respect to the hp lattice. As shown in Table 13, the minimal value of the E factor for the hp lattice is 7 (see also Pleasants et al. 1996). 

There are two well-known models of GBs that were developed primarily from studies of metals by considering the relative misorientation of the adjoining grains. These are the coincidence-site lattice (CSL) theory and the dis-placement-shift-complete lattice (DSCL). We first define two special quantities E and E Imagine two infinite arrays of lattice points (one array for each crystal) they both run throughout space and have a common origin. For certain orientations, a fraction of the points in each lattice will be common to both lattices. 

We have studied above a model for the surface reaction A + 5B2 -> 0 on a disordered surface. For the case when the density of active sites S is smaller than the kinetically defined percolation threshold So, a system has no reactive state, the production rate is zero and all sites are covered by A or B particles. This is quite understandable because the active sites form finite clusters which can be completely covered by one-kind species. Due to the natural boundaries of the clusters of active sites and the irreversible character of the studied system (no desorption) the system cannot escape from this case. If one allows desorption of the A particles a reactive state arises, it exists also for the case S > Sq. Here an infinite cluster of active sites exists from which a reactive state of the system can be obtained. If S approaches So from above we observe a smooth change of the values of the phase-transition points which approach each other. At S = So the phase transition points coincide (y 1 = t/2) and no reactive state occurs. This condition defines kinetically the percolation threshold for the present reaction (which is found to be 0.63). The difference with the percolation threshold of Sc = 0.59275 is attributed to the reduced adsorption probability of the B2 particles on percolation clusters compared to the square lattice arising from the two site requirement for adsorption, to balance this effect more compact clusters are needed which means So exceeds Sc. The correlation functions reveal the strong correlations in the reactive state as well as segregation effects. 

In the context of motion and flux, it is clear that the flux should be defined with respect to a reference frame (Chakraborty 1995). In crystalline silicates, because diffusion of oxygen and silicon can be much slower than that of other cations (with possible exception of Al), this can be achieved quite easily by using the fixed silicate lattice as a reference frame in which ions jump from site to site. This is the so-called lattice fixed frame, which commonly coincides with the laboratory frame. Note that the motion of a dilute isotope of oxygen (e.g. O) can still be treated in this frame. Pick s first law can readily be modified to take into account the variability of reference frames. 

Coincidence-IA (POL) p,q,r, and s are all rational numbers. A superceU, which in the case of Fig. 7(b) is a 2 x 2 array of primitive cells, defines the phase-coherent registry with the substrate. By convention, the supercell is defined by comers that coincide with substrate lattice points. If these sites are considered energetically preferred, this condition implies that the other overlayer lattice points on or within the perimeter of the supercell are less favorable. Consequently, if only the overlayer-substrate interface is considered, coincidence is less preferred than commensurism. Two alternative primitive unit cells are depicted here, constructed from different primitive lattice vectors. Though the matrix elements differ, the determinants and, therefore, the areas are identical. Note that the description of the unit cell with coinciding with [0,1] illustrates the reciprocal space criterion = ma [m = 1). 

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