Unit cell content

Fig. 10. (continued)—(b) A projection of the unit cell contents along the c-axis, with a down and b across the page. A water molecule (crossed circle) per trisaccharide bridges three surrounding helices. 

Fig. II. (continued)—(b) A view of the unit-cell contents down the c-axis and remaining four helices, which surround a cluster of 6 water molecules (crossed circles) per disaccharide in the middle.

Fig. 18.—Antiparallel packing arrangement of 2-fold poly(ManA) (15) helices, (a) Stereo view of two unit cells roughly normal to the hoplane. The helix at the center (filled bonds) is antiparallel to the two in the back (open bonds). Intrachain hydrogen bonds stabilize each helix. Association of helices through direct hydrogen bonds involve the carboxylate groups for parallel chains, but involve the axial hydroxyl groups for antiparallel chains, (b) A view of the unit-cell contents down the t-axis highlights the interactions between the helices.

Fig. 21.—Structure of the 6-fold anhydrous curdlan III (19) helix, (a) Stereo view of a full turn of the parallel triple helix. The three strands are distinguished by thin bonds, open bonds, and filled bonds, respectively. In addition to intrachain hydrogen bonds, the triplex shows a triad of 2-OH - 0-2 interchain hydrogen bonds around the helix axis (vertical line) at intervals of 2.94 A. (b) A c-axis projection of the unit cell contents illustrates how the 6-0H - 0-4 hydrogen bonds between triple helices stabilize the crystalline lattice.

Two helices are packed antiparallel in the orthorhombic unit cell. Association of the helices occurs through a series of periodic carboxylate potassium water - carboxylate interactions. An axial projection of the unit-cell contents (Fig. 23b) shows that the helices and guest molecules are closely packed. This is the first crystal structure of a polysaccharide in which all the guest molecules in the unit cell, consistent with the measured fiber density, have been experimentally located from difference electron-density maps. The final / -value is 0.26 for 54 reflections, of which 43 are observed, and it is based on normal scattering factors.15... 

Fig. 24. (continued)—(b) An axial projection of the unit cell contents. The double helix at each corner can be either up- or down-pointing," in terms of the X-ray data. All are, however, up in this diagram so that a calcium ion (crossed circle) is connected to the sulfate groups in three surrounding... 

In general, spinel and other chalcogen-based thiospinels with Z=8 leads to the total unit cell content of 8 and 16 (8 a and 16 d sites) cations and 32 anions in the 32 e site corresponding to the formula A8B15O32. In spinels, the AX4 tet-rahedra share comers with BXg octahedra and the octahedra are Hnked together by sharing edges (Fig. 15.1). 

Structural characteristics. Both natural and synthetic mordenite have an orthorhombic structure that consists of parallel, 12-membered ring channels in the c-direction, having an eliptical cross-section with dimensions of 6,7 x 7.0 A (Figure 9). Smaller 8-membered ring channels with dimensions of 2.9 x 5.7 A, which are perpendicular to the main channels, are too small to allow the movement of molecules from one main channel to another. Mordenite has been synthesized in a "large -port" and "small-port" form that have different sorption properties. A typical unit cell content is Na0[(A10o)o(Si0o)/rJ. 24 HO. 8 28 2 40... 

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... 

Commercially significant zeolites include the synthetic zeolites type A (LTA), X (FAU), Y (FAU), L (LTL), mordenite (MOR), ZSM-5 (MFI), beta ( BEA/BEC), MCM-22 (MTW), zeolites E (EDI) andW (MER) and the natural zeolites mordenite (MOR), chabazite (CHA), erionite (ERl) and clinoptiloUte (HEU). Details of the structures of some of these are given in this section. Tables in each section lists the type material (the common name for the material for which the three letter code was established), the chemical formula representative of the unit cell contents for the type material, the space group and lattice parameters, the pore structure and known mineral and synthetic forms. 

FAU type zeolites exchanged with many different cations (Na, K, Ba, Cu, Ni, Li, Rb, Sr, Cs, etc.) have been extensively studied. The unit cell contents of hydrated FAU type zeolite can be represented as M,j(H20)y [A Sii92 0384] -FAU, where x is the number of A1 atoms per unit cell and M is a monovalent cation (or one-half of a divalent cation, etc.). The number of A1 atoms per cell can vary from 96 to less than 4 (Si/Al ratios of 1 to more than 50). Zeolite X refers to zeolites with between 96 and 77 A1 atoms per cell (Si/Al ratios between 1 and 1.5) and Zeolite Y refers to zeolites with less than 76 A1 atoms per cell (Si/Al ratios higher than 1.5). 

Fig. 1 a and b. The unit cell contents of (a) monoclinic y-sulfur (a horizontal, c vertical) and (b) monoclinic a-selenium (c horizontal, b vertical) as completed to show full molecules. Atomic coordinates of monoclinic y-sulfur are according to Wata-nabe and those of monoclinic a-selenium according to Cherin and Unger... 

Fig. 2. The unit cell contents of Se Si2 n as completed to show full molecules (c horizontal, b vertical). Atomic coordinates are according to Weiss and Bachtler f Of the four independent atomic sites two (2 and 4) are occupied by sulfur only while the other two (1 and 3) are occupied by both sulfur and selenium. The occupation factor of selenium in these sites is 25 %...

If one assumes that the anomalous component is small relative to the normal scattering from the unit cell contents, then... 

The typical unit cell content of zeolite L is (K,Na)gAi9Si27072.nH20 and its Si/AI ratio varies in the range of 2.6 - 3.5 [1-4]. Takaishi recently determined the distribution of Al atoms in the framework of zeolite L ly analyzing Si-MAS-NMR spectra. He thereby deduced five kinds of extra-framework cation sites as shown in Fig. 1., and estimated the relative strengths of their cation affinities [5]. 

Drawing structures in three-dimensions is not easy and so crystal structures are often represented by two-dimensional plans or projections of the unit cell contents—in much the same way as an architect makes building plans. These projections are called packing diagrams because they are particularly useful in molecular structures... 

The theoretical density of a crystal can be obtained from the volume of the unit cell and the mass of the unit cell contents. The results of an X-ray diffraction structure determination gives both of these data, as the unit cell dimensions are accurately measured and the type and number of formula units in the unit cell are also determined. An example of this type of calculation for FeO follows ... 

If the mass of the unit cell contents is Mand the unit cell volume is 7then the density, p is given by... 

FIGURE 32. Unit cell content of 53 (space group Ibam)... 

FIGURE 58. Unit-cell contents of crystalline 119 showing the separated anthracene radical anions and [Mg2Cl3(THF)6] cations... 

FIGURE 2. Unit cell contents of 4 in the space group 14... 

FIGURE 4. Unit cell contents of 6 atoms outside the cell are omitted for clarity... 

If the unit-cell contents are symmetric, then the reciprocal lattice is also symmetric and certain sets of reflections are equivalent. In theory, only one member of each set of equivalent reflections need be measured, so awareness of unit-cell symmetry can greatly reduce the magnitude of data collection. In practice, modest redundancy of measurements improves accuracy, so when more than one equivalent reflection is observed (measured), or when the same reflection is observed more than once, the average of these multiple observations is considered more accurate than any single observation. 

In brief, obtaining a detailed molecular model of the unit-cell contents entails calculating p(x,y,z) from Eq. (6.7) using measured intensities from the native... 

The crude molecular image seen in the F0 map, which is obtained from the original indexed intensity data (IFobsI) and the first phase estimates (a calc), serves now as a model of the desired structure. A crude electron density function is devised to describe the unit-cell contents as well as they can be observed in the first map. Then the function is modified to make it more realistic in the light of known properties of proteins and water in crystals. This process is called, depending on the exact details of procedure, density modification, solvent leveling, or solvent flattening. 

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