A-Type Lattices

One layer of a simple cubic lattice, (o) A layer of spheres in which a given sphere (shaded) has four nearest neighbors (lightly shaded spheres) and a coordination number of 4. The square unit cell (solid lines connecting the centers of four spheres) contains one-fourth of each of the four spheres and therefore a total of 1(= 4 X J) sphere. The given sphere is in four different unit cells (numbered), (b) A portion of the space lattice for the layer and a corresponding unit celt Three other unit cells are shown with dashed lines. 

Note that the d terms cancel and that the fraction of space occupied is 0.52 that is, about half of the space is occupied by the spheres. 

we can calculate the density of such a configuration. Remember that each sphere represents an atom, ion, or molecule for which we can calculate a mass. If we assume that the sphere is an atom and that its mass, as commonly calculated in general chemistry, is its atomic weight (AW) divided by Avogadro s number (in units of grams/mole divided by atoms/mole, which equals grams/atom), then the expression for the density in a simple cubic lattice is 

Name Coord. no. No. of spheres per unit cell Spheres touching along Fraction of space occupied by spheres Density expression 

The relationship between sphere diameter and cell edge in a body-centered cubic unit cell, (o) The 

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The transport process within the zeolite pore system involves the passage of sorbate molecules through the windows between adjacent cavities. For molecules with critical diameters similar to the free aperture of the window ( 4.2 A for type A zeolites), an activated diffusion process is to be expected, and a molecule at the center of the window may be identified as the transition state. For the A-type lattice the following expression for the limiting diffusivity may be derived from absolute rate theory (14)... 

A second A-type lattice is called body-centered cubic (bcc) and, as the name implies, differs from the simple cubic lattice in that a second sphere is placed in the center of the cubic cell. A unit cell is shown in Figure 7.8b. Whereas the eight spheres at the corners are only one-eighth in the unit cell, the center sphere is completely incorporated in the body of the cell and therefore has a coordination number of 8. 

This completes our detailed description of the four most important A-type lattices and their unit cells. An analysis of Table 7.2 shows that more than 80% of the elements crystallize in one of these four lattices. In addition, a number of molecular substances in which the individual molecules closely approximate spheres (for example, CH4, HCl, and H2) assume these structures. There are actually 14 possible A-type lattices. First formulated by M. A. Bravais in 1850, these are still... 

AB-type lattices are those in which the spheres representing the atoms, ions, or molecules are of two different sizes. The most common example of these lattices are ionic crystals in which the anion is larger than the cation. In this case, it is best to picture the anions forming an A-type lattice and the cations fitting into the holes in that lattice. To the extent that the crystal is purely ionic, the packing assumed by the anions will be in large measure determined by the relative sizes of the two species. That is, the holes in the anionic lattice must he of the proper size to accommodate the cations adequately. The first topic we need to cover, then, is the number and type of holes present in A-type lattices. 

Two views of an octahedral hole, (fl) A three-dimensional and (b) cross-sectional view of a cation occupying an octahedral hole in the A-type lattice of anions. The radius-ratio r /r characteristic of an octahedral hole is calculated to be 0.414. 

Cesium chloride has a radius ratio of 1.08 because, using Shannon-Prewitt radii, the cesium cation is larger than the chloride anion. In this case, we should actually calculate r lr (= 0.93) and assume that the cations form the A-type lattice and the chlorides fill the appropriate holes. Note that 0.93 falls in the cubic hole/C.N. = 8 range of Table 7.4. As shown in Figure 7.21e, the cesium cations form a simple cubic lattice, and the chloride anions occupy the cubic holes. Alternatively, the chloride anions can be pictured as forming the A-type lattice with the cesium cations in the cubic holes. Using the solid lines as the unit cell, note that the coordination number of both the cation and anion is 8. Note also that there is a total of one [8( )] chloride per unit cell and, of course, one cesium cation in the body consistent with a 1 1 stoichiometry. Table 7.9 shows that the greatest correlation (100%) between the known structure and calculated radius ratios occurs for the CsCl structure. 

In AB -type structures, there are two types of atoms, ions, or molecules. The larger spheres are usually visualized to form an A-type lattice, and the smaller ones occupy some fraction of the holes (cubic, octahedral, or tetrahedral) in that lattice. Which holes are occupied is predicted by the radius ratio of the two spheres. For ionic crystals, the radius ratio is usually calculated as the radius of the cation over that of the anion. Ionic radii are derived from high-resolution X-ray studies. 

A part of the NaCl structure is reproduced below. Which A-type lattice do the Cl anions assume Describe this lattice in terms of a layering scheme ABCD and so on. What type of holes are occupied by the Na cations Label all unoccupied tetrahedral and octahedral holes in the figure. 

A-type lattice and the Ni atoms occupy the octahedral holes. How many NiAs formula units are there per unit cell in this structure ... 

Table 7.1 164 A-type lattices (coordination numbers, no. of spheres per unit cell, fraction of space occupied, and density expression)...

As in crystals, defects in liquid crystals can be classified as point, line or wall defects. Dislocations are a feature of liquid crystal phases where tliere is translational order, since tliese are line defects in tliis lattice order. Unlike crystals, tliere is a type of line defect unique to liquid crystals tenned disclination [39]. A disclination is a discontinuity of orientation of tire director field. 

Iron(II) bromide [7789-46-0] FeBr2, can be prepared by reaction of iron and bromine ia a flow system at 200°C and purified by sublimation ia oitrogea or uader vacuum. Other preparative routes iaclude the reactioa of Fe202 with HBr ia a flow system at 200—350°C, reactioa of iroa with HBr ia methanol, and dehydration of hydrated forms. FeBr2 crystallizes ia a layered lattice of the Cdfy type and has a magnetic moment of... 

A similar effect occurs in highly chiral nematic Hquid crystals. In a narrow temperature range (seldom wider than 1°C) between the chiral nematic phase and the isotropic Hquid phase, up to three phases are stable in which a cubic lattice of defects (where the director is not defined) exist in a compHcated, orientationaHy ordered twisted stmcture (11). Again, the introduction of these defects allows the bulk of the Hquid crystal to adopt a chiral stmcture which is energetically more favorable than both the chiral nematic and isotropic phases. The distance between defects is hundreds of nanometers, so these phases reflect light just as crystals reflect x-rays. They are called the blue phases because the first phases of this type observed reflected light in the blue part of the spectmm. The arrangement of defects possesses body-centered cubic symmetry for one blue phase, simple cubic symmetry for another blue phase, and seems to be amorphous for a third blue phase. 

Boron creates an electron deficiency in the siHcon lattice resulting in a -type semiconductor forp—n junctions. Boron compounds are more commonly used as the dopant, however (see Boron hydrides). 

The parallelization of crystallites, occurring as a result of fiber drawing, which consists in assuming by crystallite axes-positions more or less mutually parallel, leads to the development of texture within the fiber. In the case of PET fibers, this is a specific texture, different from that of other kinds of chemical fibers. It is called axial-tilted texture. The occurrence of such a texture is proved by the displacement of x-ray reflexes of paratropic lattice planes in relation to the equator of the texture dif-fractogram and by the deviation from the rectilinear arrangement of oblique diffraction planes. With the preservation of the principle of rotational symmetry, the inclination of all the crystallites axes in relation to the fiber axis is a characteristic of such a type of texture. The angle formed by the axes of particular crystallites (the translation direction of space lattice [c]) and the... 

Some evolution types observed in our simulations are shown in Figs. 2-7. The simulations were performed for the same 2D alloy model as that used in Refs. , on a square lattice of 128x128 sites with periodic boundary conditions. The as-quenched distribution Ci(0) was characterized by its mean value c and small random fluctuations Sci = 0.01. The intersite atomic jumps were supposed to occur only between nearest neighbors and we used the reduced time variable t = <7,m-... 

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