The shapes of crystals

Naturally occurring crystalline minerals often show well-developed, flat faces, as described previously, so that much can be learned about the shape of a unit cell from an examination of the geometrical arrangement of the faces of a crystal. The size of a crystal is a function of the amount of material available and the time over which crystallization has been allowed to proceed. The interfacial angles, however, are a function of the internal atomic arrangement in the crystal, and of the shape and relative dimensions of the unit cell. The prominence of faces is governed by the population density of the atoms in particular planes therefore, those faces that contain the most closely spaced atoms occur most readily. 

Interfacial angles (angles between crystal faces) are characteristic of the. crystal form being studied and, in certain cases, may aid in identification of the material. When calcite crystals are broken, they form rhombohedra with interfacial angles of 75°. Haiiy discovered this when, to his chagrin, he dropped a valuable Iceland spar (calcite) crystal. He found that the particles into which it broke were always the same type of rhombohedra, with the same shape (but not necessarily the same size). This occurred no matter what external form the original crystal had. 

These observations led to the concept of a unit cell. Haiiy was able to build realistic models of calcite crystals by stacking rhombohedral building blocks of uniform size (each with interfacial angles of 75°). Clearly, the interfacial angles are important dimensions of the exteriors of crystals. The Law of the Constancy of Interfacial Angles was first proposed by Steno. It states that in all crystals of a givpn 

FIGURE 2,12. The indexing of crystal faces, (a) The derivation of the integers used to describe the crystal faces (Miller indices). The (213) face is marked as an example. It intersects the axes of the unit cell at a/2, 6/1, and c/3. (b) Several parallel planes can be represented by these Miller indices. They intersect at, for example, a, 26, 2c/3 3a/2, 36, c. (c) A crystal with its (100) face (stippled), intersecting the x tixis at x = a and the y and z axes at infinity (that is, parallel to them), (d) A crystal with a (213) face stippled, 

American or bar-/i, etc., for British practice). Planes with these indices cut the unit cell axes at a/h, -b/k, -c/l, respectively. If the values of h, k, and / are small for all observed crystal faces, then a reasonable unit cell has probably been chosen. For reasons of symmetry, planes in hexagonal crystals are conveniently described by four axes, three in a plane at 120° to each other. This leads to four indices, hkil, where i — - h + k), and equivalent crystal faces will have similar indices. 

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When asked to describe and to explain the properties of crystals, all the students immediately nsed ideas of particles, even though they had not been taught these ideas at school before. In contrast to the historical researchers, who set their focus on the shape of crystals, the students mainly described and explained properties snch as the hardness or the colour of crystals. The concept most used by students was the density of package of the particles. According to the students explanations, the lighter and the less stable a material was, the smaller was its density,. 

Part I (on identification) comprises four chapters. Chapter II is an introduction to the shapes of crystals and the relation between shape and structure, and Chapter III is an elementary account of crystal optics some knowledge of both subjects is essential, not only for the identification of crystals by microscopic methods, but also for the under-... 

Foreign substances even in minute amounts may have other kinds of effects on crystallization They may inhibit or accelerate growth rate or change the shape of crystals, say from rounded to needlelike, or otherwise. One of the problems sometimes encountered with translating laboratory experience to full scale operation is that the synthetic liquors used in the laboratory may not contain the actually occurring impurities, and thus give quite different performance. Substances that modify crystal formation are very important industrially and many such materials have been the subject of patents. 

In order to test the effect of the shape of crystals, a sodium chloride crystal with (111) and (-1-1-1) faces, together with 100 faces, has been examined. In the crystal 28 cations and 28 anions are included, and the (111) face consists of only sodium ions, and the (-1-1-1) face of chloride ions. No significant difference has in fact been found in the dissolution mechanism of crystals with different faces. 

The description of the shape of crystals (their morphology) is the subject of the science of crystallography. Crystals are divided into six systems, on the basis of their symmetry ... 

Shape in chemistry appears on many levels. The shapes of crystals, the shapes of titration curves, the shapes of spectral lines, the shapes of potential energy functions or the multidimensional shapes of potential energy hypersurfaces are some examples. However, few chemists would dispute that the most important shape problem of chemistry is that of the three-dimensional shapes of molecules. The study of molecular shape and molecular shape changes is fundamental to our understanding of chemical properties and reactions. 

The science of crystallography began, in the seventeenth century, with the stud of the shapes of crystals. It was observed that there is considerable variation in tb overall shape of crystals of a particular substance (or of crystals of one form if it polymorphic), but that however much a crystal departed from the ideal shap 42... 

Wherever we look, we seem to see matter demonstrating order, whether it is the regularity of a leaf on a tree, our own ten fingers and toes, or the shape of crystals. It is not surprising that the study of matter has also been the search for order, both in... 

We have said nothing so far about the shape of crystals, preferring to concentrate instead on their interior structure. However, the shape of crystals is, to the layman, perhaps their most characteristic property, and nearly everyone is familiar with the beautifully developed flat faces exhibited by natural minerals or crystals artificially grown from a supersaturated salt solution. In fact, it was with a study of these faces and the angles between them that the science of crystallography began. 

Nevertheless, the shape of crystals is really a secondary characteristic, since it depends on, and is a consequence of, the interior arrangement of atoms. Sometimes the external shape of a crystal is rather obviously related to its smallest building block, the unit cell, as in the httle cubical grains of ordinary table salt (NaCl has a cubic lattice) or the six-sided prisms of natural quartz crystals (hexagonal lattice). In many other cases, however, the crystal and its unit cell have quite different shapes gold, for example, has a cubic lattice, but natural gold crystals are octahedral in form, i.e., bounded by eight planes of the form 111. ... 

In practice, optical and scanning electron microscopes are commonly used to view the shapes of crystals. Measurement can be performed offline by taking samples from the crystallizer. Therefore, care should be taken to avoid the complications associated with sampling. Apparatus for in-situ viewing of slurry without sampling is also available commercially. However, the resolution of such devices is limited to tens of microns due to the limitation of the optical focal depth. 

Such gratings are put into our hands by nature, as von Laue (1912) has shown, in the shape of crystals, in which the lattice distances are just of this order of magnitude. If a beam of X-rays is passed through a crystal, we do in fact obtain... 

Figure 63 The shape of crystals depends on the growth parameter a.

There has been a series of systematic studies of the effect of impurities and tailor-made additives on the shape of crystals grown from solution (Addadi et al. 1982 Berkovitch-Yellin 1985 Weissbuch et al. 1999.) They used amino acids and amino acid derivatives as model solutes. By combining habit observations with crystal structure and molecular orientation, they were able to rationalize a number of effects, including resolution of steroisomers. Hopefully, further studies in this area will provide a rational basis for the many impurity and additive effects found during crystallization of biochemicals. 

The shape of the crystals that are formed by a particular compound is determined by the symmetry of the unit cell of the compound and by conditions such as ion concentrations, temperature, pressure, and the space available for the formation of the crystals. The shapes of crystals are described by their forms, such as cube, tetrahedron, etc., and by the symmetry of the crystals. In the sections that follow, we will first develop a system of designating the relationship of the faces of a crystal to the crystallographic axes and then use this system to develop the general idea of a crystal form. 

SEM photographs of produced crystals are shown in Photos 1,2 and 3, where the Mg/P molar ratio is varying from 1 to 4. Also Xray diffraction analysis of each crystal are shown in Figs. 5,6 and 7. At Mg/P molar ratio 1, the shape of crystals was observed to be one of typical MAP. However the crystals were ag omerat at Mg/P molar ratio 2. When Mg/P molar ratio became 4, the shape of crystals became needlelike and in addition, fine crystals were observed. These crystals were found to be MAP by Xray diffraction analysis, although the shape of the crystals was different. But the results of the composition analysis shown in Table 3, indicate the amount of ammonium ions in the case of Mg/P molar ratio 2 and 4, was higher than in the case of Mg/P molar ratio 1. The reason for the excess uptake of ammonium ion was not clear, but this phenomenon is important when considering the ammonium ion removal. 

Solid-state chemistry is based on the study of atoms that combine to build solid structures, or crystals. In Chapter 18, you learn how solid-state chemists describe the shape of crystal structures and how this determines the size and shape of the unit cell, which is then used to characterize the many different forms that solid structures take. For example ... 

Nyvlt, J. (1973) The effect of periodic change on the shape of crystals. In Particle Growth in Suspensions, A.L. Smith (ed.). Academic Press, London, 131-136. 

While the exponent n characterizes the shape of crystallization curves, in equation (6.22), the rate parameter k characterizes the position on the... 

In this specific case, the condition for the equilibrium is met when dF=0, that is, dA = 0 this condition is fulfilled when the droplet shape becomes a sphere. Nevertheless, the equilibrium shape of solids is not determined merely by the condition dA = 0 (as in the case of liquids), because the surface energy y varies as the crystallographic orientation changes. Curie [28] demonstrated that the equilibrium condition, dP—0,as established by Gibbs [13], is satisfied when the shape of crystals changes in a way that the capillarity constant of each crystal surface becomes identical. To have a better idea on the capillarity constant concept, it is helpful to follow the argument proposed by Curie. 

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