Isomorphic composition series

In Section 5.4, we define composition series of schemes having finite valency. The main result will be a generalization of a group theoretic theorem due to Camille Jordan and Otto Holder. This generalization says that any two composition series of a closed subset of a scheme of finite valency are isomorphic. 

Theorem 5.4.2 Let T be a closed subset of S. Then any two composition series ofT are isomorphic. 

On the other hand, as above, the fact that V is a composition series of T yields Uv = UuV n UVUVV. Therefore, Uu C Uvv would lead to Uu n UvV C Uu, and that contradicts the above given isomorphism. Thus, Uu % U"v. 

First, a series of incompatible systems is discussed, including blends of different elastomers, two-component fibers and films, blends having paperlike characteristics, two-component membranes having highly ordered structures, and wood. Next, some aspects of the flow behavior of blends are considered, with emphasis on the effects of flow on morphology. Finally, the behavior of compatible blends, including isomorphic composition, is described. 

There is a series of arsenatophosphates corresponding with each of the three forms of condensed potassium arsenate and phosphate, the transition points and melting point of which vary systematically with composition (see Fig. 10). A chain structure for the a-form of potassium arsenate and arsenophosphate may be inferred from their isomorphism with (KP03)xB. The 7-forms also contain high-molecular anionic chains for, when they are hydrolyzed, monoarsenate and polyphosphates with chain length up to n = 6 are formed, depending on their phosphorus content. No metaphosphate is produced, however. 

In mineralogy, a linear relation between hardness and chemical composition is common, applying specifically to all series of isomorphous minerals, such as Au-Ag (Fig. 5.16a), plagioclases, ferberite-wolframite-hibnerite,... 

In glauconites, octahedral A1 ranges from 0.00 to 1.28 and Fe3+ from 0.35 to 1.77 (11 out of 82 samples have more octahedral A1 than Fe3+). Within these limits, there is a continuous isomorphous series between A1 and Fe3+. The composition of the basic unit cell of many glauconites is within the range of composition of the nontronite clays however, on the average, glauconites have more Fe2 + and Mg and less tetrahedral A1 than the average nontronite. The most variable components within the layer are the Al3+ and Fe3+ ions in the octahedral sheet. Fe3+ increases as A1 decreases. 

An empirical set of effective ionic radii in oxides and fluorides, taking into account the electronic spin state and coordination of both the cation and anion, have been calculated (114). For six-coordinate Bk(III), the radii values are 0.096 nm, based on a six-coordinate oxide ion radius of 0.140 nm, and 0.110 nm, based on a six-coordinate fluoride ion radius of 0.119 nm. For eight-coordinate Bk(IV), the corresponding values are 0.093 and 0.107 nm, based on the same anion radii (114). Other self-consistent sets of trivalent and tetravalent lanthanide and actinide ionic radii, based on isomorphous series of oxides (145, 157) and fluorides (148, 157), have been published. Based on a crystal radius for Cf(III), the ionic radius of isoelectronic Bk(II) was calculated to be 0.114 nm (158). It is important to note, however, that meaningful comparisons of ionic radii can be made only if the values compared are calculated in like fashion from the same type of compound, both with respect to composition and crystal structure. 

Iridium sulphate unites with the sulphates of the alkali metals to yield a series of well-defined crystalline salts known as alums. These are isomorphous with the better known alums of aluminium, chromium, and iron. Application of Mitscherlich s Law, therefore, points to their having a composition represented by the general formula ... 

Isomorphism Similarity of crystal shape, unit cell dimensions, and structure between substances of similar chemical composition. Generally only the identity of one atom in the chemical formula is changed. Ideally, the substances are so closely similar that they may form a continuous series of solid solutions. 

Diadochy. This refers to the substitution of a chemical element by another one in a crystal lattice, leaving the basic structure unchanged, except for relatively minor variations in the unit-cell parameters. If the substitution occurs over a wide compositional range, such a range is commonly referred to as an isomorphous series or a solid-solution series. 

That solids with different compositions can adopt identical crystal shapes was documented in 1819 by Mitscherlich, who called the phenomenon isomorphism (Mitscherlich 1819, Melhado 1980). Isomorphism can describe phases with similar atomic architectures but unlike constituents, such as NaCl and PbS, and it also can refer to members of a continuous solid solution series, such as the olivine group with formula (Mg,Fe)2Si04. Three years later, Mitscherlich documented the complementary property of polymorphism, whereby phases with identical compositions occur as different structures (Mitscherlich 1822). Although mineralogists of the nineteenth century recognized the important inter-relationship between crystal structure and composition, the crystallographic probes available for structure determination did not keep pace with advances in wet chemical analysis. Consequently, understanding the effects that chemical modifications exert on crystal structures could be revealed only by careful measurements of subtle variations in habit. 

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