The unit cell content

2 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, indeed, that a space group is a group of symmetry elements. If an atom is placed in a quite general position in the unit cell it is multiplied by the symmetry elements and thus other atoms, exactly equivalent to the first, are found at other positions precisely related to those of the first. Each space group has its own characteristic number of equivalent general positions. 

This is the highest multiplicity Mmax of the given space group and corresponds to the lowest site symmetry (each point is mapped onto itself only by the identity operation ). In this general position the coordinate triplets of the Mmax sites include the reference triplet indicated as x, y, z (having three variable parameters, to be experimentally determined). In a given space group, moreover, it is possible to have several special positions. In this case, points (atoms) are considered which 

In the International Tables of Crystallography, for each of the 230 space groups the list of all the Wyckoff positions is reported. For each of the positions (the general and the special ones) the coordinate triplets of the equivalent points are also given. The different positions are coded by means of the Wyckoff letter, a, b, c, etc., starting with a for the position with the lowest multiplicity and continuing in alphabetical order up to the general position. 

A similar presentation has been used for the Mg2Ge structure description. 

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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... 

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... 

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... 

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. 

Materials. The zeolites studied are summarized in Table I. The unit cell contents of the dehydrated forms of the starting materials were ... 

The relationship between the crystal lattice, the unit cell contents, and the crystal structure is shown in two dimensions in Figure 2.16. This raises the concept of convolution, which is the equivalent of folding one function into another. It is a method for converting a unit of a pattern into many identical copies arranged on a lattice all that need be defined is the unit of a pattern and the lattice the convolution gives the rest. 

Systematically absent Bragg reflections Bragg reflections that have no intensity, because of translational components of any symmetry in the unit-cell contents and which have h, k, and I values that are systematic in terms of oddness or evenness. These absences depend only upon symmetry in the atomic arrangement in the crystal, and they can be used to derive the space group. For example, all reflections for which h + k is odd may be absent, showing that the unit cell is C-face centered. 

The structure factor F hkl) is the Fourier transform of the unit cell contents sampled at reciprocal lattice points, hkl. The structure factor amplitude (magnitude) F is the ratio of the amplitude of the radiation scattered in a particular direction by the contents of one unit cell to that scattered by a single electron at the origin of the unit cell under the same conditions (see Chapter 3). The first report of the structure factor expression was given by Arnold Sommerfeld at a Solvay Conference. The structure factor F has both a magnitude F(hkl) and a phase rel-... 

The weight of the unit cell contents can be calculated if the unit cell dimensions and the density of the crystal are known. The results ma indicate the presence of solvent of crystallization in the crystal it may also indicate that the crystal cannot possibly contain what it was expected to. Therefore, crystal density measurements are essential in the early stages of a crystal structure analysis. 

FIGURE 11.15. (a) Stereoview of the unit cell contents of CuS04-5H20 (Refs. 8 and 9), and (b) identification of atoms on (a). The fifth water molecule is drawn in (b) with thick bonds, while the other four are simply labelled H2O. 

Measure the density of flotation in a mixture of liquids. Check the weight of the unit-cell contents. 

When the angles are all 90° (so that their cosines are 0), this formula reduces to the simple result V = abc for the volume of a rectangular box. If the mass of the unit cell contents is known, the theoretical cell density can be computed. This density must come close to the measured density of the crystal, a quantity that can be... 

In addition to the influence brought about by the instrumental parameters, there are two kinds of crystallographic (structural) parameters, which essentially define the structure of every powder diffraction pattern. These are the unit cell dimensions and the atomic structure (both the unit cell content and spatial distributions of atoms in the unit cell). Thus, a powder diffraction pattern can be constructed (or simulated) as follows ... 

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