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Quartz crystals
Quartz forms long hexagonal shaped crystals with pointed ends. The line diagram alongside shows the ideal geometrical shape which a perfect crystal would assume.
A crystal is any substance that has solidified in a definite geometrical form. The visible orderly arrangement of crystal faces reflects a similar orderly arrangement of atoms or molecules, making up the substance, at a very tiny scale.

Most solid substances, when pure, will form crystals. Solids that don't are said to be amorphous. Crystals are classified according either to the structure of their lattices (e.g., cubic or hexagonal), or the type of chemical bond that holds them together, i.e., ionic crystals, covalent crystals, or metallic crystals. There are also such things as liquid 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 invisible to the unaided eye but can be distinguished in a microscope.

Crystals have always fascinated people, and scientists have studied them in detail for several centuries. One of the earliest scientific investigators of crystals was the Danish geologist and anatomist Nicolaus Steno (1638–1686), who carried out studies of quartz in 1669. A century later the French mineralogist Romé de l'Isle (1736–1790), and the Abbé Hauy (1743–1842) made comprehensive measurements of the geometry of a wide variety of crystals.

Good specimens of crystals have well-formed faces that occur in characteristic angular positions. In most crystals the faces are very unevenly developed, making it hard to distinguish the underlying symmetry. The sizes of the individual faces depend on chance circumstances which determine the way the crystal has grown in different directions, but when idealized the crystals 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. Measurements of the angular inclinations of the various faces of a particular crystal species reveal that they occur at specific angles. The situation is reminiscent of that seen in a well-planned orchard of fruit trees, where we notice that only at a few special viewpoints do the trees fall into lines. (The author carried out crystallographic studies using neutron diffraction at the Harwell Atomic Energy Establishment near Oxford, England, in the late 1970s. His supervisor at the time, Professor George Bacon of Sheffield University, was fond of pointing out that the atoms of crystals aligned much as did the trees in the orchards nearby the laboratory.)

Symmetry in crystals

If we look in more detail at a collection of crystal models we find that it includes representatives of a number of different types of symmetry. For example, there's the highly developed symmetry of the cubic system; crystals of this class are often metals or simple ionic compounds, such as sodium chloride (table salt). A much lower type of symmetry is expressed by crystals belonging to, say, the monoclinic system, which cannot be described in terms of three mutually perpendicular coordinate axes. The monoclinic system has many representatives among organic materials. Comparing the cubic and monoclinic systems, as represented by metals and organic molecules respectively, illustrates an important concept. When identical spherical atoms or atoms of similar size are packed together a highly symmetrical structure is likely to arise. On the hand, when anisotropic shapes, such as the flat molecules which often occur among organic compounds, are packed together then the resulting symmetry is likely to be much lower. The difference in the symmetries of the solids which occur in these two cases can be compared 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

representation of the distribution and orientation of crystal faces
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:
  1. the cubic system, which has three equal axes ar right angles
  2. the tetragonal system, with three axes at right angles, two of them equal and the third one unequal
  3. the orthorhombic system, with three axes at right angles, all unequal
  4. the rhombohedral system, with three equal axes, which are equally inclined but not at right angles
  5. 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
  6. the monoclinic system with three unequal axes, one of which is perpendicular to the plane of the other two
  7. the triclinic system with three unequal axes, all inclined at unequal angles which are different from 90°.
unit cell of a triclinic crystal
The unit cell of a triclinic crystal, conventionally drawn to show the crystallographic axes a, b, c and the angles α, β, γ between them
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.

the seven crystal systems

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