Crystal lattice A three-dimensional

Crystal Lattice A three-dimensional array of points related by translational symmetry. The translation can occur in three independent directions giving three independent base vectors. We can fully describe such a lattice by three vectors, a, b, c, and three angles, a, (3, y. The special property of a crystal lattice is that the lattice points are identical if we have an atom at or near one point, there must be an identical atom at the same position relative to every other lattice point. 

Diffraction methods depend on interference effects, and therefore obtaining structural information from the observed patterns also involves Fourier transformations. For crystal lattices a three-dimensional FT is used to convert between the recorded diffraction scattering pattern (in reciprocal space) and the crystallographic lattice (in real space) (Sections 3.4 and 10.2). Similarly, in gas-phase electron diffraction a one-dimensional FT converts between the (reciprocal space) diffraction data and the (real space) radial distribution curves, which are one-dimensional plots of increasing distances separating pairs of atoms in the stmcture. 

For a two-dimensional array of equally spaced holes the difftaction pattern is a two-dimensional array of spots. The intensity between the spots is very small. The crystal is a three-dimensional lattice of unit cells. The third dimension of the x-ray diffraction pattern is obtained by rotating the crystal about some direction different from the incident beam. For each small angle of rotation, a two-dimensional difftaction pattern is obtained. 

With x-rays, however, one can have his cake and eat it too That the two conditions given above can both be met for a curved crystal was - appreciated first by Du Mond and Kirkpatrick18 and put into practice first by Johansson.21 This situation exists because-the crystal is a three-dimensional lattice of exceedingly small spacing. It is therefore possible to bend the crystal until the Bragg planes have the radius R,... 

The reciprocal lattice of a mosaic crystal is a three-dimensional periodic system of points, each of which characterized by a vector Hhu = ha -l- kb -l-Ic, where a, b, c are axial vectors and h,k,l, are point indices. 

According to Eq. (1.16), the elastic coherent X-ray scattering amplitude is the Fourier transform of the electron density in the crystal. The crystal is a three-dimensional periodic function described by the convolution of the unit cell density and the periodic translation lattice. For an infinitely extended lattice,... 

Unlike the two-dimensional arrays in these examples, a crystal is a three-dimensional array of objects. If we rotate the crystal in the X-ray beam, a different cross section of objects will lie perpendicular to the beam, and we will see a different diffraction pattern. In fact, just as the two-dimensional arrays of objects we have discussed are cross sections of objects in the three-dimensional crystal, each two-dimensional array of reflections (each diffraction pattern recorded on film) is a cross section of a three-dimensional lattice of reflections. Figure 2.11 shows a hypothetical three-dimensional diffraction pattern, with the reflections that would be produced by all possible orientations of a crystal in the X-ray beam. 

A study of the external symmetry of crystals naturally leads to the idea that a single crystal is a three-dimensional periodic structure i.e., it is built of a basic structural unit that is repeated with regular periodicity in three-dimensional space. Such an infinite periodic structure can be conveniently and completely described in terms of a lattice (or space lattice), which consists of a set of points (mathematical points that are dimensionless) that have identical environments. 

A crystallographic plane (hkl) is represented as a light spot of constructive interference when the Bragg conditions (Equation 2.3) are satisfied. Such diffraction spots of various crystallographic planes in a crystal form a three-dimensional array that is the reciprocal lattice of crystal. The reciprocal lattice is particularly useful for understanding a diffraction pattern of crystalline solids. Figure 2.7 shows a plane of a reciprocal lattice in which an individual spot (a lattice point) represents crystallographic planes with Miller indices (hkl). 

After an aqueous dispersion of monodispersed spherical colloids was injected into the cell, a positive pressure was applied through the glass tube to force the solvent (water) to flow through the channels. The beads were accumulated at the bottom of the cell, and crystallized into a three-dimensional opaline lattice under continuous sonication. So far, we have successfully applied this approach to assemble monodispersed colloids (both polystyrene beads and silica spheres) into ccp lattices over areas of several square centimeters. This method is relatively fast opaline lattices of a few square centimeters in area could be routinely obtained within several days. This method is also remarkable for its flexibility it could be directly employed to crystallize spherical colloids of various materials with diameters between 200 nm and 10 pm into three-dimensional opaline lattices. In addition, this procedure could be easily modified to crystalhze spherical colloids with diameters as small as 50 nm. ... 

The regular arrangement of the components of a crystalline solid at the microscopic level produces the beautiful, characteristic shapes of crystals, such as those shown in Fig. 10.8. The positions of the components in a crystalline solid are usually represented by a lattice, a three-dimensional system of points designating the positions of the components (atoms, ions, or molecules) that make up the substance. The smallest repeating unit of the lattice is called the unit cell. Thus a particular lattice can be generated by repeating the unit cell in all three dimensions to form the extended structure. Three common unit cells and their lattices are shown in Fig. 10.9. Note from Fig. 10.9 that the extended structure in each case can be viewed as a series of repeating unit cells that share common faces in the interior of the solid. 

Modifications of phosphorous pentoxide. Phosphorus pentoxide forms three solid modifications, of which the metastable M form is the ordinary commercial P30g. This modification crystallizes as rhombohedra with a molecular lattice (P40 (,) and sublimes readily at 250°C and 10 mm. (Glixelll and Boratynskl). Above 260°C and even more quickly above 500°C, form M changes Into form R. The latter crystallizes in a three-dimensional atomic lattice of PO tetrahedra and is less volatile. A form S, which... 

Pure diamond is a crystalline form of only carbon atoms. The crystal lattice is three-dimensional, and all carbon atoms are attached to four other carbon atoms in a perfect tetrahedral geometry. Because the lattice repeats in all three dimensions, there is no easy way to distort the structure, making it a very hard material. The structure of graphite, for comparison, is many stacked-up layers of carbon atoms. These two-dimensional sheets can slide relative to one another, making graphite relatively soft in comparison to diamond. 

The concept of a crystal as a three-dimensional lattice consisting of periodically repeating units provides no information, per se, on the structure or size of these units or of the periodicity of their repetition. With substances of low molar mass, these units are often identical to the molecules themselves, and the intermolecular distance determines the periodicity of these molecular crystals. 

Diffraction by a crystal can be described as a superposition of the spherical wave contributions from all atoms. As already discussed, the description can be greatly simplified if we compare the crystal to a three-dimensional diffraction grating. The process can then be viewed as reflection at the lattice planes, followed by interference. This is depicted in Figure 10.27, which shows two planes separated by a distance d. The incident beam is reflected, partly from the top row and partly from the second row. For constructive interference to occur, the path difference 8 must be a multiple of the wavelength 2, as expressed by Bragg s Law ... 

This type of thickener is based on modified ureas dissolved in N-methylpyrrolidone, which are insoluble in common coating solvents. Upon careful incorporation into coating systems, the controlled precipitation of additive forms very fine, needle-like microcrystals. The crystals form a three-dimensional lattice structure via hydrogen bonding, which results in thixotropic behavior of the system. 

Morphology. A crystal is highly organized, and constituent units, which can be atoms, molecules, or ions, are positioned in a three-dimensional periodic pattern called a space lattice. A characteristic crystal shape results from the regular internal stmcture of the soHd with crystal surfaces forming parallel to planes formed by the constituent units. The surfaces (faces) of a crystal may exhibit varying degrees of development, with a concomitant variation in macroscopic appearance. 

NbOF3 and TaOF3 are characterized by very low stability, are sensitive to moisture and decompose in air even at ambient temperature. Nevertheless, Kohler et al. [217] succeeded in investigating these compounds. Both compounds crystallize in a SnF4 type structure forming planes that share oxyfluoride octahedrons with each other, at four comers. These planes are stacked in a three-dimensional lattice via van der Waals interactions [217]. 

The crystal structure of tantalum and niobium dioxyfluorides, TaC F and Nb02F, consists of oxyfluoride octahedrons linked via their vertexes to form a three-dimensional lattice with a Re03 type structure, as demonstrated by Andersson and Astrom [233] and by Frevel and Rinn [234]. Fig. 39 shows the structure of NbC F. 

Ionic transport in solid electrolytes and electrodes may also be treated by the statistical process of successive jumps between the various accessible sites of the lattice. For random motion in a three-dimensional isotropic crystal, the diffusivity is related to the jump distance r and the jump frequency v by [3] ... 

As we stated previously, particles usually grow from a nucleus to which atoms are added in a regular manner to form a three-dimensional structure. Such crystals cease grovidng when the "nutrient" (the material which serves to form the peirticle) becomes depleted. Such particles are known as "ciystalUtes". Each will consist of several grains, having a differing orientation of the crystal lattice, within each individual particle, namely ... 

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