Numerals, structural notation

NUMERIC CODE NOTATION FOR THE MEMBERS OF LINEAR STRUCTURE SERIES... 

The numeric code notation for the intergrowth structures depends on the choice of the initial fragments In the parent structures. As a consequence of this choice, the staictures of a linear homogeneous series can be considered in two different ways. 

Both methods of numeric notation can be used for the classification of the linear homogeneous structure series. However, for the derivation of a numeric code notation for the stmctures of a linear inhomogeneous structure series we prefer the second... 

Only a numeric code notation is useful for the derivation of the symmetry of an intergrowth structure (see 3.3.). The other notation with the formula of the parent structures offers the possibility to study the crystal chemical relationships of the intergrowth structure with structures of compounds in chemically analogous systems. 

Let us construct a binary matrix A whose rows correspond to the equations and whose columns correspond to the variables. Let the element alV be 1 if variable j occurs in equation i, and let it be zero otherwise. Such a matrix is called an occurrence matrix. For the special case of Eq. (39) the occurrence matrix is symmetric. It reflects the structure of the underlying graph, since a0 = 1, if and only if the graph contains an edge i, j. If we now introduce the notation to denote the operation which assigns a value of one to a variable if its numerical value is nonzero, and a value of zero if otherwise, that is to say, for any variable x... 

Because there are numerous silicates whose structures are made up of repeating patterns based on the Si04 tetrahedron, a type of shorthand notation has been developed for drawing the structures. For example, the Si04 unit can be shown as follows. 

Tables C. 1-C.4 provide conversion factors from a.u. to SI units and a variety of practical (thermochemical, crystallographic, spectroscopic) non-SI units in common usage. Numerical values are quoted to six-digit precision (though many are known to higher accuracy) in an abbreviated exponential notation, whereby 6.022 14(23) means 6.022 14 x 1023. In this book we follow a current tendency of the quantum chemical literature by expressing relative energies in thermochemical units (kcal mol-1), structural parameters in crystallographic Angstrom units (A), vibrational frequencies in common spectroscopic units (cm-1), and so forth. These choices, although inconsistent according to SI orthodoxy, seem better able to serve effective communication between theoreticians and experimentalists.

In the development of the concepts of atomic structure much of the experimental evidence came from optical and x-ray spectroscopy, From this work certain notations have arisen that are now an accepted part of the language. For example, the n = I shell is sometimes known as the K-shell, the n 2 shell as the L-shell. the it = 3 shell as the JM-shell. etc., with consecutively following letters of the alphabet being used to designate those shells with successively higher principal quantum numbers. A Roman numeral subscript further subdivides the shells in accordance with the n, J, and j quantum numbers of the electrons, as shown in Table 4,... 

Another approach for representing 2D chemical structures is the linear notation. Linear notations are strings that represent the 2D structure as a more or less complex set of characters and symbols. Characters represent the atoms in a linear manner, whereas symbols are nsed to describe information about the connectivity [3]. The most commonly nsed notations are the Wiswesser line notation (WLN) and the simpUfled molecnlar inpnt line entry specihcation (SMILES) [2]. The WLN, invented by William J. Wiswesser in the 1949, was the hrst line notation capable of precisely describing complex molecnles [4]. It consists of a series of uppercase characters (A-Z), numerals (0-9), the ampersand ( ), the hyphen (-), the oblique stroke (/), and a blank space. 

Structure of the Cluster. Definition of variables is important in the following discussions we use the following notations c, molar concentration of the cluster p, distance from its center v, 0, 1, volume, cross-section, and length of one monomer. These values are obtained from crystallographic data. The chains are treated as ideal free jointed rods of monomers. For numerical applications, we generally use the case of polyethylene v = 50 A3, 0 = 20 X2, 1 = 2,5 I. (1 + a)v will be taken as the volume of one neutralized charged ionomer. In many applications we shall take, for simplicity, a = 0 for the dry state. If the cations are solvated by V water molecules for each of volume vQ (30 AJ), a simple additivity rule for the volumes give av = WQ. 

The technique of Raman scattering (RS) to study vibrational spectra in the numerous polytypes of SiC will be described. An explanation of the various notations used to describe the stacking sequences in these polytypes will then be given. Section C discusses the various optical phonons studied by RS and the concept of a common phonon spectrum for all polytypes will be introduced. Raman studies are also used to assess crystalline structure and quality of epitaxial layers of SiC on Si and SiC substrates. Section D outlines several other excitations of interest, e.g. polaritons, plasmons, and electronic RS, as well as impurity and defect recognition in irradiated and ion implanted material. 

Blaser and Worms (2) have proposed the use of the abbreviated notations that are shown in the 4th column of Table I. These notations are based on the linkages of phosphorus and oxygen that compose the framework of a given compound. A numeral attached to a phosphorus atom represents an oxidation number of the phosphorus atom. The reason that all lower oxo acids of phosphorus can be represented by means of abbreviated notations of this type is based on the fact that every phosphorus atom in the molecules of the lower oxo acids of phosphorus has the following structure V... 

As with the numeric data, the content of stmctural information in the secondary literature is fimited, and we therefore have a situation in which many specicdised systems have been developed to deal with stractural data. Files of chemical stractures, and means of accessing them, have been with us for a number of years, certainly from before computers. Traditionally, these have used notations, fragment codes, etc., as the means of recording structural information, and have allowed searches to be made for complete compounds or for all compounds containing certain specified substractures. More recently, topographical systems have been developed. In these the complete stractures are recorded in the form of "connection tables", which store full details of all the atoms and bonds in a molecule, and the precise arrangement in which they are connected. 

Briefly, notation systems attempt to record full, multidimensional structural descriptions in a linear form, by the use of more comprehensive symbols than atoms and bonds (e.g. symbols for particular chains, rings, functional groups). Thus, more information is recorded implicitly, in Uie rides of the notation, and less is recorded explicitly in the notations for individual compounds. The rules can therefore be quite complicated, in order to ensure the notations are unique and unambiguous. For the Wiswesser Line Notation, the rules are given in Smith, E. G. The Wis-wesser Line-Formula Chemical Notation. New York McGraw-Hill 1968. In this notation, for example, saturated carbon chains are simply indicated by an arable numeral equal to the number of carbons in the chain, branch-... 

An array is a multidimensional (not linear) data structure. An appointment schedule for the business week, hour by hour and day by day, is an example of an array. A mathematical function can generate data for an array structure. For example, the four-dimensional array shown in Table I (with entries in exponential notation) was obtained by substituting numeric values for x, y, w, and z in the expression ... 

The first nomenclature for inorganic structure types has been proposed by Ewald Hermann in 1931. They used letters to designate the kind of chemical compound and numerals to distinguish among compounds with the same general formula. The chemical elements are designated by the letter A, the binary compounds by the letter B, the AB2 compounds by C and so on. Examples are A1 for Cu, A10 for Hg, B1 for NaCI, B3 for ZnS (sphalerite), HI2 for Mg2Si04 (olivine), G1 for CaCOa (calcite). This notation has not received much acceptance, possibly due to its lack of self explanatory structural information. 

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