Lattice geometry

The value of d, the distance between adjacent planes in the set hkl), may be found from the following equations. 

The following equations give the volume V of the unit cell. 

The angle (j) between the plane (h kili), of spacing d, and the plane (/i2 2 2)j of spacing d2, may be found from the following equations. (F is the volume of the unit cell.) 

The relationship between directions and planes depends upon the symmetry of the crystal. In cubic crystals, (and only cubic crystals), the direction [hlcl] is normal to the plane (hid). 

The most important metrical properties of lattices and crystals for everyday crystallography are given below. These are expressed most compactly using vector notation, but are given here in longhand , without derivation, as a set of useful tools. 

Sn = b2c2 sin2 a S22 = a2c2 sin2 / S33 = a2b2 sin2 y 

512 = abc2 cosacos/1 — cosy) S23 = a2bc( cos/ cosy — cosa) 

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To ensure quality control material suppliers and developers routinely measure such complex properties as molecular weight and its distribution, crystallinity and crystalline lattice geometry, and detailed fracture characteristics (Chapter 6). They use complex, specialized tests such as gel permeation chromatography (2, 3), wide- and narrow-angle X-ray diffraction, scanning electron microscopy, and high-temperature pressurized solvent reaction tests to develop new polymers and plastics applications. 

Let us consider a structural limiting model, in which the polymer molecules, presenting a periodic conformation, are packed in a crystal lattice with a perfect three-dimensional order. Besides this limiting ordered model, it is possible to consider models of disordered structures having a substantially identical lattice geometry. 

Crystalline forms corresponding to limiting ordered or disordered models, with equal lattice geometry, can be obtained with different procedures and can present dif-... 

The growing cell structure algorithm is a variant of a Kohonen network, so the GCS displays several similarities with the SOM. The most distinctive feature of the GCS is that the topology is self-adaptive, adjusting as the algorithm learns about classes in the data. So, unlike the SOM, in which the layout of nodes is regular and predefined, the GCS is not constrained in advance to a particular size of network or a certain lattice geometry. 

Molecular and crystal lattice geometry vary with the crystal modification of a pigment. Important details are discussed in connection with the corresponding classes of pigments. 

The earliest attempts at model analysis of polysaccharides -typified by the x-ray crystal structure analysis of amylose triacetate - were usually conducted in three steps ( L). In the first step, a model of the chain was established which was in agreement with the fiber repeat and the lattice geometry, as obtained from diffraction data. In the second step, the invariant chain model was packed into the unit cell, subject to constraints imposed by nonbonded contacts. This was followed, in the third step, by efforts to reconcile calculated and observed structure factor amplitudes. It was quickly realized that helical models of polysaccharide chains could be easily generated and varied using the virtual bond method. Figure 1 illustrates the generation of a two-fold helical model of a (l- U)-linked polysaccharide chain. 

As has been discussed in this article, C60 fullerene has shown its rich cohesive properties in various environments. It can form a van der Waals solid in the pristine phase and in other compound materials with various molecules. In fullerides, i.e., the compounds with metallic elements, valence electrons of metal atoms transfer to C60 partly or almost completely, depending on the lattice geometries and electronic properties of the metallic elements. So fullerides are ionic solids. Interestingly, these ionic fullerides often possess metallic electronic structure and show superconductivity. The importance of the superconductivity of C60 fullerides is not only in its relatively high Tc values but also in its wide... 

Figure 30. STM image of a platinum raft on a graphite powder particle. An organometallic route was used to deposit the metal at low temperatures. The discrimination of metal and graphite with the same lattice geometry is possible by the different lattice parameters...

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