PHYSICAL CHEMISTRY A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z                    HOME ABOUT CATEGORIES SITE MAP COPYRIGHT ADVERTISE CONTACT entire Web this site crystal A substance solidified in a definite geometrical form. Most solid substances, when pure, are obtainable in a definite crystal form. Solids that do not form crystals are said to be amorphous. Crystals are classified according to the structure of their lattices, or according to the type of chemical bond that holds them together, i.e., ionic crystals, covalent crystals, or metallic crystals. Large scale appearance of crystals Most solids are crystalline to a large degree, and although many solids may appear very imperfect on a macroscopic scale, and quite different from the large crystals seen in mineralogical collections of museums, nevertheless they are almost invariably built up of crystalline regions. In an ordinary piece of metal, for example, these crystalline regions may be too small to be seen with the unaided eye, but they can be distinguished in a microscope. Crystals have always fascinated people and scientists have studied them in detail for several centuries. For example, Steno, a Danish geologist, was studying quartz in 1669 and a century later Romé de l'Isle and the abbé Hauy (1784) were making comprehensive measurements of the geometry of a wide variety of crystals. Good specimens of crystals have highly perfect faces, and these faces occur in characteristic angular positions. usually the faces are very unevenly developed, so that it is difficult at first sight to visualize the underlying symmetry of the crystal form. The actual sizes of the individual faces will depend on chance circumstances which determine the way in which the crystal has grown in different directions, but when idealized the crystals will have regular shapes. These shapes may be a combination of simpler individual forms, such as a pyramid with a vertical prism, or several different prisms and pyramids. If we measure the angular inclinations of the various faces which occur for a particular species, then we'll find that they occur at very special angles. The situation is reminiscent of what we see in a well-planned orchard of fruit trees, where we notice that only at a few special viewpoints do the trees fall into lines. Symmetry in crystals If we look in more detail at a collection of crystal models we'll find that they include representatives of many different types of symmetry. For example, there is the highly developed symmetry of the cubic system; crystals of this class are often metals or simple ionic compounds. A much lower type of symmetry is shown by crystals belonging to, say, the monoclinic system, which cannot be described in terms of three mutually perpendicular coordinate axes. The monoclinc system has many representatives among organic materials. The comparison of the cubic and monoclinc systems, as represented by metals and organic molecules respectively, illustrates an important concept. When we are packing together identical spherical atoms or atoms of similar size we are likely to get a highly symmetrical structure. On the hand, when packing together anisotropic shapes, such as the flat molecules which occur rather generally among organic compounds, then the resulting symmetry is likely to be very much lower. We may compare the difference in the symmetries of the solids which occur in these two cases with the difference, on a much larger scale, between the regular packing which exists in a pile of ball-bearings and the stacked arrangement which is produced when we try to pack together a collection of cards. Crystallographic classification As a result of extensive study of the external forms of crystals, crystallographers developed methods of formal classification. Crystals may be idealized by representing their faces by a system of normals drawn perpendicular to the faces as shown in figure (i), where the normals a, b, c, d are drawn perpendicular to the faces A, B, C, D, respectively. Then, as a further refinement these normals are represented by points such as a, b, c, d in figure (ii) at which they intersect the surface of a sphere. The problem then is to study the arrangement of such collections of points since the symmetry of this arrangement represents the symmetry of the crystal. In the case of the example shown, the crystal symmetry comprises a fourfold tetragonal axis, which is vertical. If we had also included in the diagram points to represent the underneath faces of the crystal, which have been omitted for clarity, we should see that there is also a horizontal plane of symmetry. The fourfold axis and the plane of symmetry are examples of symmetry elements. These elements are such that when they operate on the collection of points then the collection returns to its initial arrangement. Thus the operation represented by a fourfold axis is a rotation through 90° and it will be seen that when this operation is carried out by a vertical axis on the set of points shown in figure (ii) then a set of identical positions is produced. Investigation shows that there can be only a limited number of combinations of symmetry elements grouped together in this way and passing through a point, indicated as O, and it is a purely mathematical problem to work out this total number. In fact there are found to be 32 possible combinations of symmetry elements passing through a point and they are called the 32 point groups. These 32 arrangements can be divided into 7 symmetry systems, the division being made according to the systems of coordinates to which they can be related. These systems are as follows: the cubic system, which has three equal axes ar right angles the tetragonal system, with three axes at right angles, two of them equal and the third one unequal the orthorhombic system, with three axes at right angles, all unequal the rhombohedral system, with three equal axes, which are equally inclined but not at right angles the hexagonal system with three equal co-planar axes inclined at 120° to one another and a fourth unequal axis which is perpendicular to the plane the monoclinic system with three unequal axes, one of which is perpendicular to the plane of the other two the triclinic system with three unequal axes, all inclined at unequal angles which are different from 90°. We can therefore list the parameters which are necessary in order to give a full description of a crystal belonging to each of these systems as follows: for the cubic system the length a for the tetragonal system lengths a and c for the orthorhombic system lengths a, b, and c for the rhombohedral system length a and angle α for the hexagonal system length a and length c for the monoclinic system lengths a, b, and c and angle β for the triclinic system lengths a, b, and c and three angles α, β, and γ In order to examine the way in which these 32 crystal classes are distributed among the 7 systems of crystal symmetry it is convenient to use a method of representing direction which is known as the stereographic projection. In figure (ii) each crystal face, or normal direction, was represented by a point such as a, b, c, or d on the surface of a sphere. For each of these points, such as c or d, we draw a line to the south pole of the sphere as shown in figure (ii). The points, such as c', d' where these lines intersect the equatorial plane of the sphere, are the stereographic representation of the crystal face. This figure (iii) is the stereogram for the crystal which was first shown in figure (i). Points a, b on the circumference of the circle correspond to faces A, B which are vertical in the original crystal. For the crystal faces which are on the underneath of the crystal we shall have to join the ends of their normals, such as e, f in figure (i) to the north pole of the sphere. The existence of these underneath normals is indicated by circles which are included in figure (iii), overlapping the dots in the stereogram which correspond to the upper faces of the crystal. Related category    • PHYSICAL CHEMISTRY Also on this site: Encyclopedia of Alternative Energy & Sustainable Living Encyclopedia of History Transport Concepts & Designs (partner site) BACK TO TOP