Relations between crystal structures

Our aim in the first part of this book is to set out the basic geometry and topology which is necessary for an understanding of the 3-dimensional systems of atoms which constitute molecules and crystals and to enable us to describe the more important structures in the simplest possible way. In view of the extraordinary variety of atomic arrangements found in crystals it is important to look for any principles which will simplify our task. We therefore note here some relations between crystal structures which will be found helpful. 

We shall illustrate these relations by dealing in some detail with some of the simpler structures in Chapter 6. It will also be shown how the Cdl2 layer (Fig. 6.14) is used in building the structures of a number of basic salts , some with layer and others with 3D structures. 

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W. Bronger, Ternary sulfides a model case of the relation between crystal structure and magnetism. Angew. Chem. Int. Ed. 20 (1981) 52. 

Fig. 7 Relation between crystal structure and mechanical properties...

FIG. 9.18. Relation between crystal structure and conhguration of molecule in the vapour state (a) molecules AX, AiXi, and A4X4 derivable from the NaQ structure, (b) molecules AX2 and AX 3 derived from halides with layer structures. 

Another important area of correlation is the relation between crystal structure, band shape, and bonding. In Table 1 all of group A carbides with the exception of Cr3C2 b ive the NaCl-type structure. However, as shown in Table 3, NbCo,5and TaCj- shave sufficiently high... 

Munch, W., Kreuer, K. D., Adams, S., Seifert, G., and Maier, J. (1999). The relation between crystal structure and the formation and mobility of protonic charge carriers in perovskite-type oxides a case study of Y-doped BaCeOa and SrCeOs. Phase Transitions 68 367-386. 

Ito, Y., Idemoto, Y., Tsimoda, Y., Koura, N. 2003. Relation between crystal structures, electronic structures, and electrode performances of LiMn2 j.Mj.O4 (M = Ni, Zn) as a cathode active material for 4V secondary Li batteries,... 

When water freezes the crystalline form adopted depends upon the detailed conditions employed. At least nine structurally distinct forms of ice are known and the phase relations between them are summarized in Fig. 14.9. Thus, when liquid or gaseous water crystallizes at atmospheric pressure normal hexagonal ice If, forms, but at very low temperatures (—120° to — 140°) the vapour condenses to the cubic form, ice Ic. The relation between these structures is the same as that between the tridymite and cristobalite forms of SiOa (p. 342), though in both forms of ice the protons are disordered. 

As clearly pointed out, for instance, by Barnighausen (1980), one of the main objectives of crystal chemistry is to order the profusion of structure types and to show the general principles involved. To this end relations between cognate structures evidently play an important role. 

The often mentioned relations between the structure types of cryolite and perovskite (page 41) may be explained best with the example of the elpasoHte type. The elpasoUte structure is really a superstructure of the perovskite-lattice, generated by substituting two divalent Me-ions in KMeFs by two others of valency 1 (Na) and 3 (Me) resp. The resulting compound K (Nao.5Meo.5)F3 crystallizes with an ordered distribution of Na+ and Me + because of the differences in size and charge of the ions. Thus to describe the unit cell the lattice constant of the perovskite ( 4 A) has to be doubled to yield that of the elpasohte structure ( 8 A). 

One of the more notable features of antimony(V) halide chemistry is the tendency to achieve a CN of six, thus resulting in the facile formation of complex anions, particularly with halide donors (Table 21) the d(Sb—F) depends on the nature of the cation. Their structures are related to the F—Sb- -F interactions between crystal structure units, which is dependent upon the potential field of the cation. 

Of the morphological phenomena mentioned in the last few paragraphs, that of twinning is likely to be of most frequent value in identification problems, but all the phenomena are significant from the point of view of crystal structure and the relation between internal structure and growth characteristics. The subject of crystal morphology in relation to internal structure will not, however, be pursued further at present it will be taken up again in Chapters VII and VIII. For the present, we shall continue our consideration of the problem of the identification of microscopic crystals we pass oij to discuss crystal optics, the relation between optical properties and crystal shape and symmetry, and the determination of refractive indices and other optical characteristics under the microscope. 

It is not essential to measure the absolute intensities experimentally Hargreaves (see Lipson and Cochran, 1953) pointed out that by making use of the knowledge that the difference between the absolute structure amplitudes of corresponding reflections of the two crystals must be constant (and equal to the difference between the diffracting powers of the two ions), two sets of relative structure amplitudes can be put on the same scale. We know that the relations between absolute structure amplitudes must be as on the left-hand side of Fig. 210, where the... 

A different type of bridging occurs in hydrolysis complexes of tho-rium(IV) (219) and uranium(IV) (130). Here a distinct peak at 3.94(2) A in the hydrolyzed solutions can be ascribed to the metal-metal distances in the hydrolysis complexes. Discrete dinuclear complexes with a very similar metal-metal distance, 3.988(2) A, in which the metal atoms are joined by double hydroxo bridges have been found in crystals ofTh2(OH2)(N03)6(H20)8 (229). The same type of bridging, therefore, must occur in solution. When hydrolysis is increased, however, the number of metal-metal distances per metal atom increases beyond a value of 0.5, valid for a dinuclear complex, and larger hydrolysis complexes are obviously formed. These structures are unknown but an extensive X-ray investigation of highly hydrolyzed thorium(IV) solutions has shown that there is probably no close relation between the structures of the hydrolysis complexes in solution and the structure of thorium dioxide, which is the ultimate product of the hydrolysis process (230). 

The faces and angles of natural crystals result from the orderly arrangements of the atoms and molecules that make up a crystal. The relation between crystal shape and microscopic structure was suggested in the seventeenth century by Robert Hooke and Christian Huygens. It was confirmed in the twentieth century with the development of x-ray diffraction, a technique that uses x rays to examine the atomic structures of materials. 

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