## crystal
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 crystalsMost 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 crystalsIf 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 classificationCrystals 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°.
- 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*
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 categories• GEOLOGY AND PLANETARY SCIENCE• PHYSICAL CHEMISTRY | |||||||||||

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