Crystals shape determination

Since the crystal shape, or habit, can be determined by kinetic and other nonequilibrium effects, an actud crystal may have faces that differ from those of the Wulff construction. For example, if a (100) plane is a stable or singular plane but by processing one produces a plane at a small angle to this, describable as an (xOO) plane, where x is a large number, the surface may decompose into a set of (100) steps and (010) risers [39]. 

Crystalline Structures. Crystal shape of amino acids varies widely, for example, monoclinic prisms in glycine and orthorhombic needles in L-alanine. X-ray crystallographic analyses of 23 amino acids have been described (31). L-Glutamic acid crystallizes in two polymorphic forms (a and P) (32), and the a-form is mote facdely handled in industrial processes. The crystal stmeture has been determined (33) and is shown in Figure 1. 

The morphology (including crystal shape or habit), size distribution, and purity of crystalline materials can determine the success in fulfilling the function of a crystallization operation. 

In the particular case cited, a 20 per cent reduction in median crystal size is obtained. The process of attrition and methods for its determination were considered in detail in Chapters 4-6. Similar changes may also occur to crystal shape and form. 

Prabu-Jeyabalan M, Nalivaika E, Schiffer CA (2002) Substrate shape determines specificity of recognition for HIV-1 protease analysis of crystal structures of six substrate complexes. Structure 10 369-381... 

Freely suspended liquid droplets are characterized by their shape determined by surface tension leading to ideally spherical shape and smooth surface at the subnanometer scale. These properties suggest liquid droplets as optical resonators with extremely high quality factors, limited by material absorption. Liquid microdroplets have found a wide range of applications for cavity-enhanced spectroscopy and in analytical chemistry, where small volumes and a container-free environment is required for example for protein crystallization investigations. This chapter reviews the basic physics and technical implementations of light-matter interactions in liquid-droplet optical cavities. 

These conclusions concerning the structure of CIO 2 were recently confirmed by Edwards and Sills (SSa) who carried out a crystal structure determination for C102 Sb2Fii . They found the 0102 ion to be V-shaped, with an O-Cl-0 angle of 122° and a mean Cl-0 bond length of 1.31 A. 

From various observations it has been inferred that most AC and AEC complexes formed by the ligands of type 6—45 are 1 1 inclusion complexes, cryptates 34), in which the cation is held in the central cavity of the ligand molecule 34, 61, 106). This has been amply confirmed by several crystal structure determinations which also provided fundamental information about the shape of the ligand in the complex. 

The combination of SEM with EDS has also been applied to atmospheric particles (e.g., Posfai et al., 1995 Anderson et al., 1996 McMurry et al., 1996 Ganor et al., 1998). For example, individual sea salt particles were analyzed using TEM combined with EDS as well as selected-area electron diffraction (SAED) by Posfai et al. (1995) and Anderson et al. (1996). The crystal shapes correlated well with the chemical composition determined using EDS and SAED. For example, cubic crystals of NaCl were observed. Sulfate occurred in either rod-shaped crystals, which had significant concentrations of (Mg + K + Ca) compared to Na, or tubular crystals, with much smaller concentrations of these three metals. In the latter case, the EDS showed... 

Of the morphological phenomena mentioned in the last few paragraphs, that of twinning is likely to be of most frequent value in identification problems, but all the phenomena are significant from the point of view of crystal structure and the relation between internal structure and growth characteristics. The subject of crystal morphology in relation to internal structure will not, however, be pursued further at present it will be taken up again in Chapters VII and VIII. For the present, we shall continue our consideration of the problem of the identification of microscopic crystals we pass oij to discuss crystal optics, the relation between optical properties and crystal shape and symmetry, and the determination of refractive indices and other optical characteristics under the microscope. 

Crushing has been recommended as a primary method because it is safe and will lead to the determination of the principal refractive indices of any crystalline substance, provided a sufficient number of randomly oriented fragments is observed it is a beginner s method. But the more experienced worker may often dispense with it, when the crystals being examined have a well-defined polyhedral shape. If the relation between crystal shape and optical properties is properly understood, it is possible to determine the principal indices by a limited number of observations on crystals selected because they lie in such positions that they necessarily show their principal indices. 

In molecular crystals held together by ionic forces (for instance, salts of organic acids) or polar forces such as hydrogen bonds (for instance, alcohols and amides), the two influences, shape and distribution of forces, may not co-operate, and it is difficult to form any definite conclusions on the structure from crystal shape and cleavage, though it is well to keep these properties in mind during structure determination, for any suggested structure should account for them. 

The elastic energy of inhomogeneous, anisotropic, ellipsoidal inclusions can be studied using Eshelby s equivalent-inclusion method. Chang and Allen studied coherent ellipsoidal inclusions in cubic crystals and determined energyminimizing shapes under a variety of conditions, including the presence of applied uniaxial stresses [11]. 

The means to determine the minimum-energy shape for a crystal of fixed volume was developed by Wulff (38), who showed that the equilibrium shape can be determined if the surface tension, y, at all crystallographic orientations is known. As illustrated in Fig. 2, on a polar y plot of the surface tension as a function of orientation, the inner envelope of the planes drawn perpendicular to and at the ends of the radius vectors gives the equilibrium shape of a crystal of constant volume. Faceting in the equilibrium crystal shape is due to cusps in the polar y plot. 

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