Crystallographic notation

To a good approximation this structure can be regarded as f.c.c. based with (002) planes composed alternatively of Tl and Al the cubic crystallographic notation Is. therefore, commonly used. If we denote as c and a the lattice parameters In the [001) and 11001/1010) directions, respectively, the c/a ratio is 1.02 at the stoichiometric composition and Increases to 1.03 at Al-rich compositions (Duwez and Taylor 1952 Huang, et al. 1987). 

The selective oxidation of ra-butane to give maleic anhydride (MA) catalyzed by vanadium phosphorus oxides is an important commercial process (99). MA is subsequently used in catalytic processes to make tetrahydrofurans and agricultural chemicals. The active phase in the selective butane oxidation catalyst is identified as vanadyl pyrophosphate, (V0)2P207, referred to as VPO. The three-dimensional structure of orthorhombic VPO, consisting of vanadyl octahedra and phosphate tetrahedra, is shown in Fig. 17, with a= 1.6594 nm, b = 0.776 nm, and c = 0.958 nm (100), with (010) as the active plane (99). Conventional crystallographic notations of round brackets (), and triangular point brackets (), are used to denote a crystal plane and crystallographic directions in the VPO structure, respectively. The latter refers to symmetrically equivalent directions present in a crystal. 

The 1 Operation This entails no net rotation but inverts all the coordinates. In short, it converts a point a, y, z to the points, y, z. Thus 1 is simply the crystallographers equivalent of L The crystallographic notation employs 1, not /. 

In crystallographic notation it is customary to place a bar over lattice index numbers to indicate negative direction relative to the origin of the axes.)... 

Crystallographic Plane Notation. We may now proceed to deal with the individual sets of atom planes. In order to describe and define the position in space of these sets of planes we shall need to use a special form of crystallographic notation. This notation is used to define a plane, or set of planes, in terms of the number of parts into which the set of planes divides the edges of the unit of the lattice under consideration. Thus, Fig. 15 represents the unit of the fundamental cubic lattice it might be emphasised that such a simple unit is not found in actual metal crystals. The cube of side a is intersected by a number of planes, portions of which are shown in the diagram, in such a way that the side OA is divided into three parts, the side OB into two parts and the side OC into one part the... 

Degree symbol, 143, 163, 165 Derivative works and copyright, 345, 347 Descriptors, See Chemical descriptors Determinants, 150-152 Deuterium, 258, 259 Dictionaries, desk references of ACS editorial staff, 67, 75 Different from in comparisons, 10 Diffraction lines, crystallographic notation, 260-261... 

Zero before decimal point, 148 Zones, crystallographic notation, 260-261... 

As a complement to this section on crystallography, some remarks on the old notations are given. In the nineteenth-century treatises, the crystallographic notations are varied and difficult to understand. However C. Friedel (1893) recorded all notations used from Haiy onward. For all the simple forms, he compared five different modes (Levy, Weiss, Naumann, Dana, Miller) which could be reduced substantially to two different schemes. Some crystallographers assigned a symbol to a simple form (e.g., p is cube in Levy, O is octahedron in Naumann), whereas others referred to the intercepts of the faces on axes and assigned some numbers to each face in comparison with a reference face, with direct (Weiss) or inverse (Miller) proportionality. The following table, which reports the various symbols for some simple forms of the cubic system, clarifies these differences. 

Unit Cell. Various workers (2,5-6) have determined similar dimensions for the ramie unit cell. As shown In Table I, however, the and dimensions have ranges of about 0.07 A. (The a and dimensions of the MGW (Mann, Gonzalez and Wellard) and RMF cells have been Interchanged to conform to standard crystallographic notation (7 ) as discussed below.)... 

We describe each crystal system and include crystallographic notation to help distinguish the characteristics of each type of crystal system. Figure 8-4 shows a diagram with a box that represents the basic unit of a crystal... 

Diamond is the perfect example of an atomic crystal or giant molecule, where there is complete electron pair covalent bonding, which links all atoms in all directions in space. Diamond can occur in several crystal forms and these are classified using the crystallographic notation for the simple planes of a cubic crystal (Figure 2.5). Diamond has three major crystal forms cubic (100 plane), dodecahedral (110 plane) and the octahedral form (111 plane), which are shown in Figure 2.6. Both cubic and octahedral forms occur in high pressure synthetic diamond and CVD diamond. 

To some degree problems of notation did arise with the naming of columnar meso-phases. Originally they were called discotic liquid crystals, and indeed they also acquired a crystallographic notation. Both of these notations have, however, fallen out of favor and the naming of the state has been redefined. As research in disc-like systems remains relatively active, it is to be expected that further phases will be discovered, and as our understanding of the structures of these phases increases changes may be made to our current notation. 

Whether the constituent molecules are chiral or whether we make the medium chiral by dissolving chiral molecules to a certain concentration c>0, in both cases the only symmetry element left would be the twofold rotation axis along the y-direction, and a polarization along that direction is thus allowed. The symmetry of the medium is now C2, which is lower than the symmetry of the property. The polarization P, like the electric field, is a polar vector, hence with the symmetry (or <=om in the crystallographic notation). 

It is often said that group 432 is too symmetric to allow piezoelectricity, in spite of the fact that it lacks a center of inversion. It is instructive to see how this comes about. In 1934 Neumann s principle was complemented by a powerful theorem proven by Hermann (1898-1961), an outstanding theoretical physicist with a passionate interest for symmetry, whose name is today mostly connected with the Hermann-Mau-guin crystallographic notation, internationally adopted since 1930. In the special issue on liquid crystals by ZeitschriftfUr Kristal-lographie in 1931 he also derived the 18 symmetrically different possible states for liquid crystals, which could exist between three-dimensional crystals and isotropic liquids [100]. His theorem from 1934 states [101] that if there is a rotation axis C (of order n), then every tensor of rank rcubic crystals, this means that second rank tensors like the thermal expansion coefficient a, the electrical conductivity Gjj, or the dielectric constant e,y, will be isotropic perpendicular to all four space diagonals that have threefold symme-... 

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