Unit cells 266 INDEX

Figure 2.23 High resolution TOF powder diffraction pattern. Note (a) the range and resolution of the smaller d spacings is impressive, but (b) the largest d spacings are not present. The consequences of this are that excellent structural parameters can be derived from such data sets, especially for anisotropic displacement parameters, which require a wide range of d spacing for their precise determination, but that unit cell indexing may be a problem unless the poorer resolution data from low angle detector banks is exploited.

The lines on Figure 5.33(A) represent planes in a cubic crystal parallel to the c axis, which is normal to the plane of the page. The circles mark the comers of the cubic unit cell. Index planes (a)-(d). 

Here U(q/c) is the amplitude which is independent of the unit cell index /, Cl) is the frequency, and q is the wave vector which is a vector in reciprocal space and designates the modes of the system. Its components q, q,) have values... 

Figure 16.21. Theoretical variations of lattice parameter (a) scaled by chain length (/) as a function of bilayer volume fraction 9) for (a) Type 1 and (b) Type 2 bicontinuous mesophases. The curves are calculated for distinct mesophase topologies per unit cell, indexed by the crystallographic genus, y...

In describing a particular surface, the first important parameter is the Miller index that corresponds to the orientation of the sample. Miller indices are used to describe directions with respect to the tluee-dimensional bulk unit cell [2]. The Miller index indicating a particular surface orientation is the one that points m the direction of the surface nonual. For example, a Ni crystal cut perpendicular to the [100] direction would be labelled Ni(lOO). 

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-temiinated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (licp) and body-centred cubic (bcc) lattices (a) fee (lll)-(l X 1) (b)fcc(lO -(l X l) (c)fcc(110)-(l X 1) (d)hcp(0001)-(l x 1) (e) hcp(l0-10)-(l X 1), usually written as hcp(l010)-(l x 1) (f) bcc(l 10)-(1 x ]) (g) bcc(100)-(l x 1) and (li) bcc(l 11)-(1 x 1). The atomic spheres are drawn with radii that are smaller than touching-sphere radii, in order to give better depth views. The arrows are unit cell vectors. These figures were produced by the software program BALSAC [35]-...

Indexings and Lattice Parameter Determination. From a powder pattern of a single component it is possible to determine the indices of many reflections. From this information and the 20-values for the reflections, it is possible to determine the unit cell parameters. As with single crystals this information can then be used to identify the material by searching the NIST Crystal Data File (see "SmaU Molecule Single Stmcture Determination" above). 

All of the observed reflections could be indexed on the basis of a cubic unit cell with Oo = 11.82 A the estimated probable error is 0.01 A. The only systematic absences were hhl with l odd this is characteristic of the space group 0 -PmP>n, which also was reported by von Stackelberg from his single-crystal work on sulfur dioxide hydrate. For 46 H20 and 6 Cl2 in the unit cell the calculated density is 1.26 densities reported by various observers range from 1.23 to 1.29. 

The distance of each reflection from the center of the pattern is a function of the fiber-to-film distance, as well as the unit-cell dimensions. Therefore, by measuring the positions of the reflections, it is possible to determine the unit-cell dimensions and, subsequently, index (or assign Miller indices to) all the reflections. Their intensities are measured with a microdensitometer or digitized with a scanner and then processed.8-10 After applying appropriate geometrical corrections for Lorentz and polarization effects, the observed structure amplitudes are computed. This experimental X-ray data set is crucial for the determination and refinement of molecular and packing models, and also for the adjudication of alternatives. 

The second choice is a simpler solution. According to Sarko and Muggli,66 all 39 observed reflections in the Valonia X-ray pattern are indexable by a two-chain triclinic unit cell with a = 9.41, b =8.15 and c = 10.34 A, a = 90°, 3 = 57.5°, and y = 96.2°. Ramie cellulose, on the other hand, is completely consistent with the two-chain monoclinic unit cell. Also, there are significant differences between their high-resolution solid-state l3C NMR spectra, indicating that Valonia and ramie celluloses, the two most crystalline forms, reflect two distinct families of biosynthesis. On this basis, the Valonia triclinic and the ramie monoclinic forms are classified69 as Ia and Ip, respectively. It has been shown from a systematic analysis of the NMR spectra by these authors, and from electron-dif-... 

It is important in defining the monodromy matrix, which quantifies changes in the unit cell in Figs. 4 and 5, to specify the lengths of the unit cell sides that define the basis. The monodromy theorem—that the monodromy index is equal to the number of pinch points on the pinched torus [40]—applies in a basis in which the cell sides represent unit changes in the relevant quantum number. 

The SAD patterns were Indexed to a [101] zone axis, as described In reference ( ). The unit cell parameters of the mineral and synthetic aurlchalclte are given In Table I together with the Cu/Zn ratios. 

The group-subgroup relation of the symmetry reduction from diamond to zinc blende is shown in Fig. 18.3. Some comments concerning the terminology have been included. In both structures the atoms have identical coordinates and site symmetries. The unit cell of diamond contains eight C atoms in symmetry-equivalent positions (Wyckoff position 8a). With the symmetry reduction the atomic positions split to two independent positions (4a and 4c) which are occupied in zinc blende by zinc and sulfur atoms. The space groups are translationengleiche the dimensions of the unit cells correspond to each other. The index of the symmetry reduction is 2 exactly half of all symmetry operations is lost. This includes the inversion centers which in diamond are present in the centers of the C-C bonds. 

VTO, and that of the G structure (Ia3d symmetry, Fig. 4) index to /6 /8 /l4 VT6 V O V"22 /7A. Finally the measurements of the surface area inside the cubic cell can yield partial information about a topology of the structure. One can expect a small unit cell and the small surface area per side of... 

Figure 14. The phase diagram of the gradient copolymer melt with the distribution functions g(x) = l — tanh(ciit(x —fo)) shown in the insert of this figure for ci = 3,/o = 0.5 (solid line), and/o — 0.3 (dashed line), x gives the position of ith monomer from the end of the chain in the units of the linear chain length. % is the Flory-Huggins interaction parameter, N is a polymerization index, and/ is the composition (/ = J0 g(x) dx). The Euler characteristic of the isotropic phase (I) is zero, and that of the hexagonal phase (H) is zero. For the bcc phase (B), XEuier = 4 per unit cell for the double gyroid phase (G), XEuier = -16 per unit cell and for the lamellar phases (LAM), XEuier = 0.

The orthogonal projection of the epitaxial poly(DMDA) could not be indexed using the unit cell data for the bulk polymerized crystal (8). However, poly(DMDA) cannot usually be polymerized to completion or to high crystallinity in the bulk due to cross-linking. The use of an epitaxial substrate may have controlled the polymerization process that led to oriented single crystals. 

When stored under increasing relative humidities (RH), cromolyn sodium absorbed water, resulting in a continuous series of interstitial solid solutions [11]. The amount of water absorbed was proportional to the relative humidity of the environment and could be up to about nine molecules of water per molecule of drug. Such an unusual system was characterized by combining XPD with single-crystal x-ray studies. The unit cell parameters of cromolyn sodium were obtained from single-crystal x-ray studies, and this permitted the authors to index the powder pattern. The b axis spacing was found to increase dramatically as a function of the relative humidity up to 20% RH (Table 2). Above 40% RH, the unit cell dimensions were nearly constant. 

The unit cell of cellulose from Chaetomorpha melagonium is monoclinic, with a = 16.43 A (1.643 nm), b(fiber axis) = 10.33 A (1.033 nm), c = 15.70 A (1.570 nm), and /3 = 96.97°. In base-plane projection, each of the Meyer-Misch subcells that make up the super-lattice are identical. All equatorial reflections can be indexed by using a one-chain unit-cell, meaning that every single chain has... 

The Miller indices of planes in crystals with a hexagonal unit cell can be ambiguous. In order to eliminate this ambiguity, four indices, (hkil), are often used. These are called Miller-Bravais indices and are only used in the hexagonal system. The index i is given by... 

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