Crystal standardized description

Aside from the conventions mentioned for the cell choice, further rules have been developed to achieve standardized descriptions of crystal structures [36], They should be followed to assure a systematic and comparable documentation of the data and to facilitate the inclusion in databases. However, contraventions of the standards are rather frequent, not only from negligence or ignorance of the rules, but often for compelling reasons, for example when the relationships between different structures are to be pointed out. 

Shortened version of the standard description structural types Several intermetallic phases are known which have the same (or a similar) stoichiometry and crystallize in the same crystal system and space group with the same occupied point positions. Such compounds are considered as belonging to the same structure type. About 30000 intermetallic phases have been described. These, however, may be grouped in about 2800 types. 

The series 6 - 12/21/2 + 8/31/2 - 6/41/2 + 24/51/2 -. .. eventually becomes convergent and gives the value for the Madelung constant for the sodium chloride lattice (the standard description of lattices which have the same form as that adopted by sodium fluoride). The values of Madelung constants for some common crystal lattices are given in Table 7.5. 

In 1912 Bom and von Karman [1, 2] proposed a model for the lattice dynamics of crystals which has become the standard description of vibrations in crystals. In it the atoms are depicted as bound together by harmonic springs, and their motion is treated collectively through traveling displacement waves, or lattice vibrations, rather than by individual displacements from their equilibrium lattice sites [3]. Each wave is characterized by its frequency, wavelength (or wavevector), amplitude and polarization. 

According to Parthe et al. (1993), a standardization procedure is necessary in the presentation of the relevant data characteristic of a crystal structure (see also Parthe and Gelato 1984). A convenient description of the structure types is then possible using the Wyckoff sequence (the letters of the occupied Wyckoff sites). This allows a finer classification of structure types and offers suggestions not only for recognizing isotypic structures but also possible structural relationships like substitution, formation of vacancy or filled-in structure variants. 

The results in sections 2 and 3 describe the adsorption isotherms and diffusivities of Xe in A1P04-31 based on atomistic descriptions of the adsorbates and pores. The final step in our modeling effort is to combine these results with the macroscopic formulation of the steady state flux through an A1P04-31 crystal, Eq. (1). We make the standard assumption that the pore concentrations at the crystal s boundaries are in equilibrium with the bulk gas phase [2-4]. This assumption cannot be exactly correct when there is a net flux through the membrane [18], but no accurate models exist for the barriers to mass transfer at the crystal boundaries. We are currently developing techniques to account for these so-called surface barriers using atomistic simulations. 

For a mathematical description of crystal faces, take any three non-parallel faces (chosen to be mutually orthogonal, if possible) and take their intersections as reference axes, which are labeled OA, OB, and OC with the origin at O, as shown in Fig. 9.1.2(a). Let another face (the standard face or parametral face A B C ) meet these axes at A, B, and C, making intercepts OA = a, OB = b and OC = c, respectively. The ratios a b c are called the axial ratios. 

Part of the difficulty encountered in searching and interpreting the literature on polymorphic behaviour of materials is due to the inconsistent labelling of polymorphs. In many cases, the inconsistency arises from lack of an accepted standard notation. However, often, and perhaps more important, it is due to the lack of various authors awareness of previous work or lack of attempts to reconcile their own work with earlier studies (see, for instance. Bar and Bernstein 1985). While many polymorphic minerals and inorganic compounds acmally have different names (e.g. calcite, aragonite and vaterite for calcium carbonate or rutile, brookite, and anatase for titanium dioxide) this has not been the practice for molecular crystals, which have been labelled with Arabic (1, 2, 3,...) or Roman (I, II, III,...) numerals, lower or upper case Latin (a, b, c,... or A, B, C,...) or lower case Greek a, P,y, ) letters, or by names descriptive of properties (red form, low-temperature polymorph, metastable modification, etc.). 

It is well known that one of the standard approaches to Eq. (2.1) is the linear combination of atomic orbitals (LCAO) method it consists in expanding the states of the solid in linear combination of atomic (or molecular) orbitals of the composing atoms (or molecules). This method, when not applied in oversimplified form, provides an accurate description of core and valence bands in any type of crystal (metals, semiconductors, and insulators). Applied with some caution, the method also provides precious information on lowest lying conduction States, replacing whenever necessary atomic orbitals with appropriate localized orbitals. ... 

The complete manufacture of trinitrotoluene involves the several processes of nitration, separation, washing, crystallization and possibly purification. The experimental stage has been passed in every one of these various divisions in the manufacture of this product, until the modern plant runs as smoothly as a well-oiled machine. The apparatus necessary to carry on any one or all of the steps in the manufacture of TNT is now well standardized, and many excellent machines are on the market for accomplishing the end toward which every manufacturer works—a pure product. A detailed description of the necessary apparatus will not be gone into, but a brief outline of the requirements to be fulfilled by the various machines will be given in their respective places. 

In this monograph, a number of notations, units, and abbreviations will be used, and they are summarized in Appendix A. It contains lists of notations for crystal orientations, process parameters for CVD, analytical techniques, CVD reactors, crystal growth, and carbon materials in addition to a description of standard diamond film characterizations, i.e. Raman spectroscopy and cathodoluminescence (CL). The readers are recommended to just quickly read through Appendix A at this point. 

If then we abandon the standard three-dimensional Euclidean perspective and adopt this non-Euclidean two-dimensional view, it can be seen that stable polymorphs are characterised by a global geometric constraint surface density 2"1, and a local constraint Gaussian curvature, . We shall see in Chapter 4 that this description is identical to one that accounts for the mesophase behaviour of lyotropic liquid crystals in amphiphile-water mixtures. 

The formalism introduced in the previous subsections is able to incorporate the effect of these influences in the crystallization kinetics, thus providing a more realistic modeling of the process, which is mandatoiy for practical and industrial purposes. Due to the strong foundations of our mesoscopic formalism in the roots of standard non-equilibrium thermodynamics, it is easy to incorporate the influence of other transport processes (like heat conduction or diffusion) into the description of crystallization. In addition, our framework naturally accounts for the couplings between all these different influences. 

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