Crystal unit cell, size

Experiments were performed using a Phillips PW-3010 automated powder diffractometer with CuKa radiation (40mA, 40kV). Measurements of the crystal unit cell size (u.s.c.) [19] and hence the framework Si/Al ratio ((Si/Al)jy) [20] as well as the crystallinity of the zeolites were obtained, and are given in Table 1, for the parent and hydrothermally treated zeolites, and in Table 3 for the AHFS treated zeolites. As expected, both the u.c.s (framework Si/Al ratio) and crystallinity reflect the degree of steam treatment in the hydrothermally treated... 

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. 

Certain factors are likely to influence future analyses of more complex viruses. Crystal stability is governed by packing interactions and, as can be seen from Fig. 16.4, is, to a first approximation, inversely proportional to the square of the virus radius, presumably underl)dng the problems with crystal stability for analyses such as that of PRDl. Even assuming that well-ordered, stable crystals can be formed, technical considerations will place an upper limit on the unit cell size from which useful data can be collected. Nevertheless, with some improvements in beam and detector technology, we expect that data collection from cells up to 2000 A should be feasible for even a primitive unit cell. 

X-ray Diffraction Structural characterization of single crystal samples can reveal both the atomic arrangement and the composition of a sample. The unit cell size and symmetry alone can quickly establish whether a crystal is a known material or a new phase. A detailed discussion of X-ray diffraction studies of the known superconductors is the topic of another chapter in this book. 

Before interface energy was understood, the concept was explained by Friedel [2] in terms of a compound or twin lattice. This may be explained as follows (see Fig. 7.2). If the lattice of one individual crystal is extended to superpose that of the other, where both are projected onto the same plane, a new lattice consisting of common lattice points results. This lattice is called a twin lattice or compound lattice, and the twin index is defined by the number of multiples of the unit cell size of a single crystal. The smaller the twin index, the higher the probability that twinning will occur. 

The corresponding relation between the host and guest crystals when evaluating the misfit ratio may be a one-to-one lattice relation in the same direction (a X b to a xb axes), or in different axial directions (aX b axes versus aX <110> axes), or on the basis of one unit cell versus a few unit cell sizes (see Fig. 7.13). Royer s misfit ratio is generally a two-dimensional correspondence, but Hartman [13] extended this relation to the misfit ratio in PBCs (see Section 4.2), which is a one-dimensional correspondence. Royer s epitaxial relations correspond to a relation between the F faces of the host and guest crystals containing more than two PBCs, and an epitaxial relation is not allowed between S faces or K faces. In Hartman s analysis, rela-... 

The lamella size - 100-500A A multiple of the lamella thickness, say 1 fi The crystal unit cell. 

Techniques of transmission electron microscopy have proved valuable in many areas of solid state science. Use of electron diffraction permits identification of crystal types, determination of unit cell sizes and characterization of crystal defects in the phases. Measurement of Energy Dispersive X-ray (EDS) line intensity allows calculation of the elemental composition of the phases. It is difficult to overestimate the value of such applications to metallic alloys, ceramic materials and electron-device alloys (T-4V Applications to coal and other fuels are far fewer, but the studies also show promise, both in characterization of mineral phases and in determination of organic constituents (5-9. This paper reports measurements on a particular feature of coal, the spatial variation of the organic sulfur concentration. 

Stephens et al. determined the lattice structure of powdered TDAE-C60 to be monoclinic C2/m with one formula unit per unit cell [84]. However, a structural analysis performed on single crystals [85] showed the room temperature structure to be monoclinic with unit cell dimensions a=15.858(2) A, b=12.998(2) A, c=l9.987(2) A, (3=93 37° and four formula units per unit cell. The correct space group was found to be C2/c and not C2/m as originally reported from the powder data. The unit cell in fact consists of two subcells, which are stacked along the c-direction so that the unit cell size in the c-direction is doubled (Fig. 13). In one of the subcells the TDAE ion is shifted by about 0.02 A along... 

The discovery of x-rays provided crystallographers a powerful tool for the thorough determination of crystal structures and unit cell sizes [20-26], X-rays have wavelengths between 0.2 and 10 nm. As x-rays possess dimensions comparable to the interplanar distances in crystals, x-ray crystallography is an ideal nondestructive method for material characterization, since nanometer parameters as well as macroscopic properties of the tested samples can be determined from x-ray diffraction data. 

If two heavy-atom derivatives can be crystallized which preserve the space group and unit cell size of a large protein, then the structure can be solved directly this method of multiple isomorphous replacement was used by Perutz152 and Kendrew153 to solve the first two protein structures by laborious, decade-long film methods hemoglobin and myoglobin. 

The next stage of characterization focuses upon the different phases present within the catalyst particle and their nature. Bulk, component structural information is determined principally by x-ray powder diffraction (XRD). In FCC catalysts, for example, XRD is used to determine the unit cell size of the zeolite component within the catalyst particle. The zeolite unit cell size is a function of the number of aluminum atoms in the framework and has been related to the coke selectivity and octane performance of the catalyst in commercial operations. Scanning electron microscopy (SEM) can provide information about the distribution of crystalline and chemical phases greater than lOOnm within the catalyst particle. Differential thermal analysis (DTA) and thermogravimetric analysis (TGA) can be used to obtain information on crystal transformations, decomposition, or chemical reactions within the particles. Cotterman, et al describe how the generation of this information can be used to understand an FCC catalyst system. 

The technique of low energy electron diffraction (LEED) has been the most widely used tool in the study of surface structure. LEED experiments involve the scattering of monoenergetic and collimated electrons from a crystal surface and detection of elastically diffracted electrons in a backscattering geometry (Figure 2). The characteristic diffraction pattern in LEED arises from constructive interference of electrons when scattered from ordered atomic positions. The diffraction pattern represents a reciprocal map of surface periodicities and allows access to surface unit cell size and orientation. Changes in the diffraction pattern from that of a clean surface can be indicative of surface reconstruction or adsorbed overlayers. 

Figure H.4. The crystals are manipulated by scooping them up with a small loop of nylon that is glued to the end of a pin. Surface tension firom the liquid will hold the crystal in the loop, but the crystal can also be held by using a loop that is smaller in size than the crystal of interest. This technique will work particularly well with fragile crystals, thin plates for example, that would normally fall apart in a capillary mount. Once the crystal is frozen, it is placed on an axis in line of both an X-ray source and a stream of nitrogen set to about 100,000to keep the crystal frozen. The crystal is rotated in increments during the data collection procedure to collect a full data set (typically one or two degrees per frame, depending on the resolution limits, mosaicity of the crystal, unit cell lengths, etc.).

This vast number of possibilities calls for a systematic procedure to identify a subset of the most likely interface matchings of the parent crystals. This subset will then be the starting point for atomistic modeling. The question about unit cell size and shape is relatively simple to address. Many related procedures based on linear elasticity theory and lattice strain estimates may be adopted. The basic situation is sketched in Fig. 4 an overlayer unit cell A needs to be matched together with a substrate unit cell B. Matching pairs of unit cells are, in general, multiples of primitive cells in the interface plane for the metal and ceramic, respectively. 

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