Unit cells, in crystals

Amorphous stereotactic polymers can crystallise, in which condition neighbouring chains are parallel. Because of the unavoidable chain entanglement in the amorphous state, only modest alignment of amorphous polymer chains is usually feasible, and moreover complete crystallisation is impossible under most circumstances, and thus many polymers are semi-crystalline. It is this feature, semicrystallinity, which distinguished polymers most sharply from other kinds of materials. Crystallisation can be from solution or from the melt, to form spherulites, or alternatively (as in a rubber or in high-strength fibres) it can be induced by mechanical means. This last is another crucial difference between polymers and other materials. Unit cells in crystals are much smaller than polymer chain lengths, which leads to a unique structural feature which is further discussed below. 

Tetragonal unit cells. In crystals of tetragonal symmetry the unit cell is a rectangular box with two edges equal (a) and the third (c) different from the first two. The spacings of hkO planes—those parallel. to c—are in the same ratios as those of the hkO planes of cubic crystals, that is, in the ratios 1/Vl2 l/ /(l2+12) 1/V22 l/ /(22+l2), and so on. But the 001 spacing is not related in any simple way to a the ratio cfa may have any value and is different for every tetragonal crystal and... 

Defects Irregularities in the precise periodicity of the arrangement of unit cells in crystals, i.e., imperfections in the crystal lattice. 

Steric descriptors and/or -> size descriptors representing the volume of a molecule. The volume of a molecule can be derived from experimental observation such as the volume of the unit cell in crystals or the molar volume of a solution or from theoretical calculations. In fact, analytical and numerical approaches have been proposed for the calculation of molecular volume where the measure depends directly on the definition of - molecular surface-, -> van der Waals volume and -> solvent-excluded volume are two volume descriptors based on van der Waals surface and solvent-accessible surface, respectively. 

The structure of a crystal is characterized by the fact that it is formed by the indefinite repetition in three dimensions of the contents of a parallelo-piped, termed the unit cell if the contents of one unit cell is known, the structure of the whole crystal is given by stacking identical cells in parallel orientation in such a way that each corner is common to eight of these cells. (In an analogous way the pattern of a wallpaper is formed by the indefinite repetition in two dimensions of the design contained within a unit parallelogram.) Unit cells in crystals commonly have linear dimensions of the order of io A, and contain a small whole number of formula units. 

With the INS technique, wavelengths compare to chemical bonds or unit cells in crystals. Therefore, collective osciUations can be excited and, simultaneously, the... 

As a result of having two chiral centers, four stereoisomers of ascorbic acid are possible (Table 1) (Fig. 2). Besides L-ascorbic acid (Activity = 1), only D-araboascorbic acid (erythorbic acid (9)) shows vitamin C activity (Activity = 0.025-0.05). The L-ascorbic acid stmcture (1) in solution and the soHd state are almost identical. Ascorbic acid crystallizes in the space group P2 with four molecules in the unit cell. The crystal data are summarized in Table 2. 

Figure 18.1 A crystal is built up from many billions of small identical units, or unit cells. These unit cells are packed against each other in three dimensions much as identical boxes are packed and stored in a warehouse. The unit cell may contain one or more than one molecule. Although the number of molecules per unit cell is always the same for all the unit cells of a single crystal, it may vary between different crystal forms of the same protein. The diagram shows in two dimensions several identical unit cells, each containing two objects packed against each other. The two objects within each unit cell are related by twofold symmetry to illustrate that each unit cell in a protein cr) stal can contain several molecules that are related by symmetry to each other. (The pattern is adapted from a Japanese stencil of unknown origin from the nineteenth century.)...

The elementary building block of the zeolite crystal is a unit cell. The unit cell size (UCS) is the distance between the repeating cells in the zeolite structure. One unit cell in a typical fresh Y-zeolite lathee contains 192 framework atomic positions 55 atoms of aluminum and 1atoms of silicon. This corresponds to a silica (SiOj) to alumina (AI.O,) molal ratio (SAR) of 5. The UCS is an important parameter in characterizing the zeolite structure. 

Of special interest to intercalation studies are complex non-stoichiometric systems, such as the so-called misfit layer chalcogenides that were first synthesized in the 1960s [45]. Typically, the misfit compounds present an asymmetry along the c-axis, evidencing an inclination of the unit cell in this direction, due to lattice mismatch in, say, the -axis therefore these solids prefer to fold and/or adopt a hollow-fiber structure, crystallizing in either platelet form or as hollow whiskers. One of the first studied examples of such a misfit compound has been the kaolinite mineral. 

Boron is as unusual in its structures as it is in its chemical behavior. Sixteen boron modifications have been described, but most of them have not been well characterized. Many samples assumed to have consisted only of boron were possibly boron-rich borides (many of which are known, e.g. YB66). An established structure is that of rhombohedral a-B12 (the subscript number designates the number of atoms per unit cell). The crystal structures of three further forms are known, tetragonal -B50, rhombohedral J3-B105 and rhombohedral j3-B 320, but probably boron-rich borides were studied. a-B50 should be formulated B48X2. It consists of B12 icosahedra that are linked by tetrahedrally coordinated X atoms. These atoms are presumably C or N atoms (B, C and N can hardly be distinguished by X-ray diffraction). 

The insertion of the oxygen atoms widens the silicon lattice considerably. A relatively large void remains in each of the four vacant octants of the unit cell. In natural cristobalite they usually contain foreign ions (mainly alkali and alkaline earth metal ions) that probably stabilize the structure and allow the crystallization of this modification at temperatures far below the stability range of pure cristobalite. To conserve electrical neutrality, probably one Si atom per alkali metal ion is substituted by an A1 atom. The substitution of Si... 

Unlike crystals that are packed with identical unit cells in 3D space, aperiodic crystals lack such units. So far, aperiodic crystals include not only quasiperiodic crystals, but also crystals in which incommensurable modulations or intergrowth structures (or composites) occur [14], That is to say, quasiperiodicity is only one of the aperiodicities. So what is quasiperiodicity Simply speaking, a structure is classified to be quasiperiodic if it is aperiodic and exhibits self-similarity upon inflation and deflation by tau (x = 1.618, the golden mean). By this, one recognizes the fact that objects with perfect fivefold symmetry can exist in the 3D space however, no 3D space groups are available to build or to interpret such structures. 

The edge of the unit cell in a crystal is sometimes called the lattice or cell constant. The structure known as antifluorite is the structure of K20 and its cell constant is 644 pm. Determine the value for each of the following ... 

The mathematics necessary to understand the diffraction of X rays by a crystal will not be discussed in any detail here. Chapter 4 of reference 10 contains an excellent discussion. The arrangement of unit cells in a crystal in a periodic manner leads to the Laue diffraction conditions shown in equations 3.3 where vectors a, b, and c as well as lattice indices h, k, and l have been defined in Figure 3.5 and S is a vector quantity equal to the difference between the resultant vector s after diffraction and the incident X-ray beam wave vector So so that S = s - So-... 

Figure 3.22 A (101) twin plane in rutile, Ti02. The two parts of the crystal are related by mirror symmetry. The unit cells in the two parts are shaded.

Fig. 7.6. Addition of two unit cells in the theophylline monohydrate crystal phase [20].

A is the volume of the unit cell in the direct lattice of the crystal The range of integration is restricted to the first Brillouin zone of the crystal, and the volume of the zone is (27t)3/A. 

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