Unit cells description

Figure 5.88 is an illustration of two-dimensional defects in the form of surfaces and grain boundaries. They either terminate a crystal or separate it from the three-dimensional defects. In polymer crystals, these surfaces and grain boundaries are rarely clean terminations of single-crystalline domains, as one would expect from the unit cell descriptions in Sect. 5.1. The surfaces may contain folds or chain ends and may be traversed by tie molecules to other crystals and cilia and loose loops that enter the amorphous areas, as is illustrated in Figs. 5.87 and 2.98. The properties of a polycrystalline sample are largely determined by the cohesion achieved across such surfaces and the mechanical properties of the interlamellar material, the amorphous defects. 

Retrieval systems should allow access to any item entered as keyboard input and stored as an independent descriptor alone and/or in combination with others by Boolean operators If this makes sense. Because, e.g., of the variety of the unit cell descriptions used, it is not useful to search for a specific cell dimension. On the other hand, a query can be composed of the atomic symbols and the number of different elements present (to avoid compounds containing more elements than wanted). The answer will not always be an exact one but visual inspection of the list of answers can be more fruitful than the expenditure for an exact query. 

Structure Type Lattice Unit Cell Description Examples... 

The description of the crystallographic unit cell is followed by probably the most important seetion of the file - the Cartesian eoovdinates of the atoms. 

The structure requires 160 valence electrons per unit cell computed as follows internal bonding within the 4 icosahedra (4 x 26 = 104) external bonds for the 4 icosahedra (4x12 = 48) bonds shared by the atoms in 2(b) positions (2x4 = 8). However, 50 B atoms have only 150 valence electrons and even with the maximum possible excess of boron in the unit cell (0.75 B) this rises to only 152 electrons. The required extra 8 or 10 electrons are now supplied by 2C or 2N though the detailed description of the bonding is more intricate than this simple numerology implies. 

In the weak-coupling limit unit cell a (, 0 7a for fra/u-polyacetylene) and the Peierls gap has a strong effect only on the electron states close to the Fermi energy eF-0, i.e., stales with wave vectors close to . The interaction of these electronic states with the lattice may then be described by a continuum, model [5, 6]. In this description, the electron Hamiltonian (Eq. (3.3)) takes the form ... 

The unit cell of the hydrates crystallizing in Structure II is rather complicated, and for a detailed description the reader is referred to the original publications.6 48 Its composition is characterized by ... 

The unit cell content. To complete the description of the crystal structure, the list of the atoms contained in the unit cell and their coordinates (fractional coordinates related to the adopted system and unit cell edges) are then reported. These are usually presented in a format such as M El in n x, y, z. In the MoSi2 structure, also reported in Table 3.2, and in Fig. 3.7, for instance, four silicon atoms... 

Notice that in the description of the content of the unit cell, the atomic coordinates are given for the first cell from the chosen origin, and are formulated modulo 1 thus, for instance, —x, —y, —z is written (x,y, z) rather than 1 — x, 1 — y, 1 — z and the following equivalences could be considered as typical examples ... 

Common crystal-chemical formulae. Unit cell volumes and interatomic distances. In the analysis and description of a structure some calculation may be useful. A few common formulae are collected here. 

Examples of alternative descriptions of the unit cell In a number of cases, in order for instance, to compare a given atomic distribution and arrangement with several others, it may be useful to use different descriptions of the same structure (to refer to different, but obviously equivalent, unit cells). The transformations (of the unit cell constants and, consequently, of the coordinates of the atomic positions) are described, for the general case, for instance, in the International Tables (Hahn 2002). A few, frequently used, transformation formulae of the unit cell constants are reported here. 

Description of a rhombohedral unit cell in terms of the equivalent, triple-primitive, hexagonal cell (see Fig. 3.9). 

A detailed example of the alternative descriptions of a given compound, both in terms of its hexagonal unit cell and of the corresponding rhombohedral primitive cell is presented in Chapter 4 the rhombohedral compound Mo6PbSx (the prototype of the family of the so-called Chevrel phases) is described and unit cell constants and atomic positions are listed for its conventional hexagonal cell and for the rhombohedral primitive cell. 

Description of a hexagonal unit cell (ah, ch cell edges) in terms of an ortho-hexagonal cell (equivalent orthorhombic cell a0, b0, c0) (see Fig. 3.10). 

Description of a cubic (primitive, body centred or face centred) unit cell (ac) in terms of the equivalent, primitive rhombohedral, (a,-, a) and triple-primitive hexagonal, cells (ah, ch). See Fig. 3.11. 

All the requisite structural information for a solid phase is contained (either explicitly or implicitly) in the description of its unit cell. A number of features, however, which are especially relevant for chemical-physical considerations, such as local coordination geometries, the existence of clusters of atoms or chains or layers, etc., are not self-evident in the aforementioned structural descriptions and can be deduced only by means of a more or less complicated series of calculations. 

Figure 3.20. A lateral view of different stacking sequences of triangular nets. They correspond to some typical close-packed structures. The first layer sequence shown corresponds to a superimposition according to the scheme ABABAB... (equivalent to BCBCBC... or CACACA... descriptions) characteristic of the hexagonal close-packed, Mg-type, structure. With reference to the usual description of its unit cell, the full stacking symbol indicating the element, the relative position of the superimposed layers and their distance is Mg Mg. The other sequences correspond to the schemes ABC.ABC. (Cu, cubic), ABAC.ABAC. (La, hexagonal), ACACBCBAB. (Sm, hexagonal). For Cu the constant ch of the (equivalent, non-conventional) hexagonal cell is shown which may be obtained by a convenient re-description of the standard cubic cell (see 3.6.1.3). ch = cV 3, body diagonal of the cubic cell.

Finally, as another simple example of description (and symbolic representation) of structures in terms of layer stacking sequence, we now examine structures which can be considered as generated by layer networks containing squares. A typical case will be that of structures containing 44 nets of atoms (Square net S net). The description of the structures will be made in terms of the separation of the different nets, along the direction perpendicular to their plane, and of the origin and orientation of the unit cell. 

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