Orientational symbols

1 A) DA symbols. The first symbolic description is from Dekeyser and Amelinckx (1953), who used a set of vectors and numerical symbols to indicate the complete stagger of the layer, defined as the (001) projection of the vector connecting two (OH/F) sites on the two sides of the octahedral sheet. Six characters n = 1,2,3,1,2,3 represent the stagger of the layer with respect to a space-fixed reference (Fig. 8). These symbols apply to the homo-octahedral approximation only and therefore cannot correctly describe polytypes containing M2 layers. 

These symbols can be considered the predecessors of OD and Z symbols (cf Fig. 4). With respect to the original figure in Dekeyser and Amelinckx (1953), the b and c axes have been taken in the opposite direction b left instead of right, and c coming out from the plane instead of into) in accordance with the conventions of the International Tables for Crystallography. 

IB) Z symbols. Zvyagin (1962) introduced a numerical/vectorial description giving the stacking of half-layers, as defined by the interlayer cations and the origin of the O sheet. This choice made Zvyagin s symbols more general than the symbols introduced previously and also suitable for some other phyllosilicates. However, 

The fundamental merit of Z symbols is that they can describe also meso-octahedral polytypes. Their shortcoming is that the symbols describing homo-octahedral mica polytypes are identical with those describing meso-octahedral polytypes consisting of Ml layers, and additional information must be given also. Moreover, in their present form, they cannot handle hetero-octahedral polytypes. 

1C) OD symbols. The OD school, inspired by Z symbols, derived the most general symbols to describe mica polytypes (Durovic and Domberger-Schiff 1979 Dornberger-Schiff et al. 1982 Backhaus and Durovic 1984 Durovic et al. 1984 Weiss and Wiewiora 1986). These symbols consist of a sequence of characters referring to one period, placed between vertical bars two lines of characters are used the first line indicates the packet orientations, and the second line the packet-to-packet displacements. A dot separating the orientational characters for packets P2y and q2y+i indicates the position of Oc layer. The OD symbols are thus expressed  

---

Not only a symmetry plane but also other symmetry elements may serve as antisymmetry elements. We have already seen the contour of the oriental symbol Yin/Yang representing twofold rotational... 

Fig. 47. Idealized versions of the graphs shown in Fig. 3a and h (a and c, respectively) can be produced from the (3.6.3.6) graph by elimination of one sixth of its white vertices and aU edges incident upon them. Idealized black-and-white graph of the [(U02)3(Se04)5]" sheets observed in a- (d) and P-Mg2[(U02)3(Se04)5](H20)i6 (f), and [(1102)3(1 004)5] sheet observed in (NH3(CH2)3NH3)(H30)2[(U02)3(Mo04)5] (e) together with the tetrahedral orientation symbols written next to the white vertices tabular forms of orientation symbols for these sheets (g, h and i, respectively). See text for details.

Subfamily B. These polytypes are described by orientational symbols with characters of alternating parity, i.e. by all-odd characters in the RTW symbol. Successive layers are related by (2 +l)x60° rotations. Only polytypes with an even number of layers appear in this subfamily. In addition, because layers with different orientational parity have an opposite x component of the stacking vector, Ev is ( ), (+) or -) and it is not possible to have a Class a polytype. 

Mixed-rotation polytypes. These polytypes correspond to orientational symbols with... 

Polytypes are usefully classified not only within the same family, but also between different families. On the basis of the number of layers and of the parity of the corresponding characters in the orientational symbols, several meso-octahedral polytypes can be related to one homo-octahedral polytype similarly, taking into account the chirality of the packets, several hetero-octahedral polytypes can be related to one meso-octahedral polytype. In mathematics, a 1 relation is a homomorphism, of which the 1 1 relation (isomorphism) is a special case the 1 relation of polytypes of different families is hence termed relations of homomorphy. 

The contribution by molecule (rs) is a function of the position and orientation (symbolized by R and A) of this molecule. Substituting (4.29) into (4.26) we note that we shall obtain nonzero contributions only when I k ) is either one of (Ik), i.e., the central molecule or (rs), the contributing molecule. Therefore, the double sum over I k and over rs reduces to a single sum with two terms ... 

The data from Table 2 show that the algorithm developed in allows sizing of different cracks with complex cross-sections and unknown shapes for orientation angles not exceeding 45°. It is seen that the width 2a and the parameter c (or the surface density of charge m=4 r // e at the crack walls) are determined with 100% accuracy for all of the Case Symbols studied. The errors in the computation of the depths dj and di are less than 4% while the errors in the computation of d, dj, d, and d are less than 20% independent of the shape of the investigated crack and its orientation angle O <45°. 

For a binaiy system, r = Otg = L/Ot g. The symbol r applies primarily to the process, while Ot is oriented toward interactions between pairs of solute species. For each binaiy pair, fij = C( ji = l/Otiy. ... 

Maximum information is obtained by making Raman measurements on oriented, transparent single crystals. The essentials of the experiment are sketched in Figure 3. The crystal is aligned with the crystallographic axes parallel to a laboratory coordinate system defined by the directions of the laser beam and the scattered beam. A useful shorthand for describing the orientational relations (the Porto notation) is illustrated in Figure 3 as z(xz) y. The first symbol is the direction of the laser beam the second symbol is the polarization direction of the laser beam the third symbol is the polarization direction of the scattered beam and the fourth symbol is the direction of the scattered beam, all with respect to the laboratory coordinate system. 

The Schrodinger equation can be solved approximately for atoms with two or more electrons. There are many solutions for the wave function, ij/, each associated with a set of numbers called quantum numbers. Three such numbers are given the symbols n, , and mi. A wave function corresponding to a particular set of three quantum numbers (e.g., n = 2, = 1, mi = 0) is associated with an electron occupying an atomic orbital. From the expression for ij/y we can deduce the relative energy of that orbital, its shape, and its orientation in space. 

Fig. 13 a, b. Possible stacking of macromolecular bilayers in the P form crystals of s-PS. The regular succession of bilayers ABAB. .. gives rise to the ordered P" modification (a) defects, corresponding to pairs of bilayers of the kind AA or BB, would characterize the disordered P modification an AA defect is reported in (b). The symbols (/) and ( ) indicate the orientation of the lines connecting the adjacent phenyl rings of each chain inside the macromolecular bilayers A and B, respectively [29]... 

Fig. 9. Plot of amorphous orientation average fa, obtained from birefringence, etc., versus f, obtained from im etc- Triangles, draw temperature 80 °C circles draw temperature 85 °C open symbols, single-stage draw full symbols, two-stage draw. Reproduced from Polymer by permission of the publishers, Butterworth Co (Publishers) Ltd. (C)...

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

A Hermann-Mauguin point-group symbol consists of a listing of the symmetry elements that are present according to certain rules in such a way that their relative orientations can... 

The region within which k is considered (—n/a first Brillouin zone. In the coordinate system of k space it is a polyhedron. The faces of the first Brillouin zone are oriented perpendicular to the directions from one atom to the equivalent atoms in the adjacent unit cells. The distance of a face from the origin of the k coordinate system is n/s, s being the distance between the atoms. The first Brillouin zone for a cubic-primitive crystal lattice is shown in Fig. 10.11 the symbols commonly given to certain points of the Brillouin zone are labeled. The Brillouin zone consists of a very large number of small cells, one for each electronic state. 

---