Ideal imperfect crystal

These expressions containing F2 are valid only for ideally imperfect crystals, to which class most known crystals belong. It is a curious fact that really perfect crystals like certain diamonds, in which afi portions of the lattice are parallel to a high degree of precision, give an integrated intensity which is proportional directly to F (not to its square), and is thus smaller than that given by an imperfect crystal of the same substance. 

If the combined expressions are used, it is only necessary to consider each reference atom in turn the expressions take care of the rest of each group. The cosine term for each independent group is evaluated and then all the cosine terms are added together. Sine terms for all the independent groups are likewise added together. A2- -B2 is then = F2, which for an ideally imperfect crystal is proportional to the intensity the hlcl reflections would have if the atoms really were in the postulated positions. The combined expression for the contribution of a set of equivalent atoms, for each space-group, is to be found in International Tables (1952). 

Extinction. This effect, described in Sec. 4-12, is a reduction in diffracted intensity as a crystal becomes more nearly perfect. Equation (14-1) is derived for the ideally imperfect crystal, one in which extinction is absent. Samples for chemical analysis should therefore be free of extinction, and this can be adcom-plished, for powder samples, by grinding or filing. If a solid aggregate must be analyzed directly, the possibility of some extinction in the individual grains of the aggregate should be kept in mind. 

The simple model implies an infinite perfect crystal. The crystal specimen studied is necessarily finite (rarely more than 0.5 mm in size) and, if perfect by conventional criteria, would be utterly unsuitable for collection of intensity data of the kind required for ordinary structure determination and refinement. What is needed is an ideally imperfect crystal shot through with dislocations and intergrain boundaries so that it behaves, so far as diffraction is concerned, like a mosaic of perfect crystal blocks of the order of micrometers or tenths of micrometers in size, tilted with respect to one another by angles of the order of a few seconds of arc and scattering independently (incoherently) with respect to one another. The assumption of an ideally imperfect ... 

A quite nice illustration of such a type of study has been published in 1985 by Bachman et al. on a 6 pm CaF crystal This was the smallest single crystal ever used for X-ray diffraction experiment. For an ideally imperfect crystal, the scattering power S may be defined as ... 

The crystals available in practice thus lie somewhere between amorphous and ideally crystalline. but nearer the ideal end of the scale ( ideal imperfect crystals"). A crystalline specimen may be a single crystal (smallest dimension ca. 0.05 - 0.5 mm) or a polycrystalline material with crystallite size ca. 0.04 mm. 

Graphite is commonly produced by CVD and is often referred to as pyrolytic graphite. It is an aggregate of graphite crystallites, which have dimensions (L ) that may reach several hundred nm. It has a turbostratic structure, usually with many warped basal planes, lattice defects, and crystallite imperfections. Within the aggregate, the crystallites have various degrees of orientation. When they are essentially parallel to each other, the nature and the properties of the deposit closely match that of the ideal graphite crystal. 

In an ideal world, crystals would be perfect or stoichiometric with constant composition. But like people crystals are not exempt from imperfections or defects. Crystals with variable composition are termed non-stoichiometric crystals. The defect chemistry of oxides is enormously complex and is extremely vital to their properties. It has involved extensive research in many laboratories and is providing extraordinary insights into structural variations, the stability of structures and the formation of new structures. Here, we first define order-disorder phenomena that are commonly associated with oxides and describe our current understanding of them. The disorder or non-stoichiometry plays a crucial role in oxide applications including catalysis and it is therefore of paramount importance. 

Most actual crystals are imperfect different portions of the lattice are not quite parallel, and the crystal behaves as if it consisted of a number of blocks (of the order of 10 5 cm in diameter) whose orientation varies over several minutes or even in some cases up to half a degree. This imperfection is perhaps connected with the manner of growth in thin layers (see Chapter II and Plates I and II) each layer may be slightly wavy, and there may be cracks or impurities between the layers. Most crystals are imperfect in this way, and in structure determination it is usually safe to assume that the intensity of any reflection is proportional to the square of the structure amplitude. To make quite sure that a crystal is ideally imperfect , it may be dipped in liquid air the shock-cooling produces imperfections. 

The intensities of crystal reflections are in some circumstances reduced by effects known as primary and secondary extinction. If the crystal is not ideally imperfect but consists of rather large lattice blocks, the intensities of the reflections are proportional to a power of F between 1 and 2 this is primary extinction . Secondary extinction affects only the strongest reflections and is due to the fact that the top layer of a crystal (the part nearest the primary beam) reflects away an appreciable proportion of the primary beam, thus in effect partially shielding the lower layers of the crystal the strongest reflections are therefore experimentally less strong than they should be in comparison with the weaker reflections. The relation between the actual intensity p and the intensity p which would be obtained if there were no secondary extinction is, for reflection at a large face,... 

Major limitations of the Hildebrand equation occur when co-crystallization occurs between the fractions of TAGs (Humphrey et al. 2003). Also, deviations from ideality occur at a high solvent/solute ratios. Inclusion of solvent in the crystal may also cause deviation from ideality as imperfect crystals have a higher solubility than perfect crystals. These major limitations, which have been observed in our SSS/OOO ideal system, have lead authors to explore other methods to quantify and predict solubility. 

Actually, adsorption of lattice ions is far from ideal (Section 8-5), owing to the nonequivalence of perfect and imperfect crystal surfaces and the interference of nonlattice ions that may also be potential-determining. 

This is the lattice parameter (unit cell edge length) of an ideal single crystal of naturally occurring Si free of impurities and imperfections, and is deduced from measurements on extremely pure and nearly perfect single crystals of Si by correcting for the effects of impurities. 

Some authors have observed that the growth rate at very small supersatirration is greater than predicted by the nucleation models. This can be explained by the so-called BCF model (Bruton et al. 1951). The authors assume that the presence of spiral dislocations which end somewhere on the crystal surface creates steps, which are thus a continuous soirrce of favorable integration sites. The soirrce of such screw dislocations is a lattice imperfection which prevents an ideally smooth crystal surface. The steps of these spiral dislocations are remote from the centers and considered to be parallel and the same distance apart from each other. The linear displacement rate of a face is controlled by surface diffusion. With the siuface diffusion coefficient the growth rate Vg p according to Burton, Cabrera, and Frank is... 

In the same way that many properties of the crystalline state are viewed as being determined by imperfections in an ideal, perfect, crystal, it is useful to view the properties of passes as influenced by imperfections in an ideal. 

The 2500°C heat-treatment causes the reordering of the structure. The basal planes coalesce and become more parallel and closer together. The various crystallite imperfections such as vacancies, stacking faults, dislocations, and rotational disorders, tend to heal and disappear the crystallite size (Lg) increases the 002 line narrows considerably and becomes close to the position of the ideal graphite line as the interlayer spacing (d) decreases to approach that of the ideal graphite crystal (0.3354 nm). This observed reduction of the interlayer spacing is attributed in part to the removal of interstitial elements, mostly carbon.P l... 

Imperfect crystals are close to the asymmetric stage of matter but being variant in the characteristics of symmetrical state, they demonstrate some important properties that an ideally perfect symmetrical state fail to give. Therefore, attention is then shifted from single crystal state to polycrystalline state and some of their characteristic properties. 

Two-dimensional nucleation tequites ideally smooth crystal surfaces which exist in reality only under exceptional circumstances. In reality, imperfections of the crystal surface play the predominating role for nucleation in electrolytic crystal growth and dissolution. The presence of dislocations on the surface enhances the formation of nuclei for growth and dissolution drastically. The real process consists, therefore, of an alternating combination of layer growth and nucleation. The relation between these two processes depends very much on components of the solution and can be widely modified by the presence of adsorbates. The same situation is foimd in electrolytic dissolution of crystals. 

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