Patterns external shape

Conventional theory of powder diffraction assumes completely random distribution of the orientations among the infinite amount of crystallites in a specimen used to produce a powder diffraction pattern. In other words, precisely the same fraction of the specimen volume should be in the reflecting position for each and every Bragg reflection. Strictly speaking this is possible only when the specimen contains an infinite number of crystallites. In practice it can be only achieved when the number of crystallites is very large (usually in excess of 10 to lO particles). Nonetheless, even when the number of crystallites approaches infinity, this does not necessarily mean that their orientations will be completely random. The external shape of the crystallites plays an important role in achieving randomness of their orientations in addition to their number. 

Without a doubt, crystals such as diamonds, emeralds and rubies, whose beauty has been exposed by jewelry-makers for centuries, are enjoyable for everybody through their perfect shapes and astonishing range of colors. Far fewer people take pleasure in the internal harmony - atomic structure -which defines shape and other properties of crystals but remains invisible to the naked eye. Ordered atomic structures are present in a variety of common materials, e.g. metals, sand, rocks or ice, in addition to the easily recognizable precious stones. The former usually consist of many tiny crystals and therefore, are called polycrystals, for example metals and ice, or powders, such as sand and snow. Besides external shapes and internal structures, the beauty of crystals can be appreciated from an infinite number of distinct diffraction patterns they form upon interaction with certain types of waves, e.g. x-rays. Similarly, the beauty of the sea is largely defined by a continuously changing but distinctive patterns formed by waves on the water s surface. 

Amorphous substances (e.g., Ga, Se, Ge, Si02, C) are mostly derived from face-centered cubic crystals or hexagonal ones. The law of Donnay and Marker [4] allows one to connect the diffraction pattern to the external shape of a crystal. The more developed face of a crystal is the one where the atomic density is maximum (i.e., when the atoms are in contact). Correspondingly, electrostatic... 

Crystals are characterized by the orderly, cohesive arrangement of their atoms, ions, or molecules (Table 1.1, page 5). As a result of (his internal orderliness, a crystal assumes a recognizable external shape. The angles at which its sur ces (faces) meet each other are a characteristic and reproducible property of the crystal. These geometrical patterns have been of interest to man since earliest civilization, and more... 

So far we have discussed the macroscopic symmetry elements that are manifested by the external shape of the three-dimensional patterns, that is, crystals. They can be studied by investigating the symmetry present in the faces of the crystals. In addition to these symmetry elements there are two more symmetry elements that are related to the detailed arrangements of motifs (atoms or molecules in actual crystals). These symmetry elements are known as microscopic symmetry elements, as they can only be identified by the study of internal arrangement of the motifs. As X-ray or electron diffraction can reveal the internal structures, these symmetry arrangements can only be identified by X-ray, Electro or Neutron diffraction. Obviously, they are not revealed in the external shape of the pattern. These symmetry elements are classified as microscopic symmetry elements. There are two such types of synunetry elements (i) glide plane of symmetry and (ii) screw axis of synunetry. 

Figure 16-27 compares the various constant pattern solutions for R = 0.5. The curves are of a similar shape. The solution for reaction kinetics is perfectly symmetrical. The cui ves for the axial dispersion fluid-phase concentration profile and the linear driving force approximation are identical except that the latter occurs one transfer unit further down the bed. The cui ve for external mass transfer is exactly that for the linear driving force approximation turned upside down [i.e., rotated 180° about cf= nf = 0.5, N — Ti) = 0]. The hnear driving force approximation provides a good approximation for both pore diffusion and surface diffusion. 

The shape of an object is a descriptor of the outline of its external surface only. Thus the shape of an object is a property that reflects the recognized pattern of relationships among all the points that constitute its external surface. The difference between the shapes of two objects arises from the differences between the patterns of relationships among these point coordinates corresponding to the two shapes. While the size of an object, for example a material particle, is an indicator of the quantity of matter contained in it, its shape is concerned with the pattern according to which this quantity of matter is assembled together. Shape is an intrinsic rather than an extrinsic characteristic in that it is not additive. 

Surface-active contaminants play an important role in damping out internal circulation in deformed bubbles and drops, as in spherical fluid particles (see Chapters 3 and 5). No systematic visualization of internal motion in ellipsoidal bubbles and drops has been reported. However, there are indications that deformations tend to decrease internal circulation velocities significantly (MI2), while shape oscillations tend to disrupt the internal circulation pattern of droplets and promote rapid mixing (R3). No secondary vortex of opposite sense to the prime internal vortex has been observed, even when the external boundary layer was found to separate (Sll). 

The actual shape and size of the molecules depend on the substitution pattern of the arylene units. Although pure para-substituted cyclic phenylene-arylenes have been investigated, e.g. 1, most PAMs contain meta- or ortho-substituted aromatics at their corners and para-substituted aromatics for size expansion, respectively (2, Fig. 6.2) [11], The position where the side-groups are attached to the backbone determines if they point to the inside (intra-annular substituents, I, I ), to the outside (extra-annular substituents, E, E ) or if they can change their orientation according to an external parameter (adaptable substituents, A, A ). 

Thus, no sharp emission pattern can be expected with the overall emission spectrum. Nevertheless, assuming the Lambertian shape of the emission from microcavity structures may lead to an overestimate as large as 30% [571]. An attempt to compare the measured full spectrum external emission as a function of the emitter thickness (Alq3) with theoretical description of microcavity modes has shown substantial disagreement, the theoretical estimates lead to the emission output much below the experimental data, differing by a factor of 2 for a 40nm-thick emitter [567]. The reason for... 

---