2 Elements of Crystallography

Prof. P. N. Kotru

1.1 ORIGIN

The science of crystallography started with the study of the external form and habit of crystals which grow in nature. The naturally occurring crystals bounded by multiple faces have attracted the fascination of man since long.

Figure 1.1: schematic diagram showing natural quartz crystal with multiple habit faces

Figure 1.1 is a schematic illustration of naturally occurring quartz crystal with multiple flat faces. The morphology, (i.e., the external regular arrangement of crystal faces) was taken as suggestive of internal regular arrangement of atoms. Nicolaus Steno (1969) established a law of interfacial angles. He observed that the interfacial angles of a crystalline solid are always the same irrespective of how and where they grew. The external form of crystals manifests internal regular arrangement of atoms that compose them. Crystals may be of several different shapes and sizes, say for example, small cubes, long thin needles, small platelets, but for any given chemical composition they are all built up out of atoms , molecules or ions which are arranged regularly in the crystal in the same way irrespective of the external shape of the crystal. This is illustrated through schematic diagrams of figure 1.2

Fig 1.2: Schematic diagrams showing crystals of different shapes & sizes in two dimensions

Considering for the sake of simplicity a two-dimensional example. Here, we take the example of four sets of solid circles (●), each set being of different shape but each of them has the solid circle arranged in the same pattern within the figure. Each figure consists of several squares, with solid circle at their corners , which are located side by side in the same order, and one of these solid circles is drawn in each of the cases shown in figures 1.2 ( a-d). The differences between the four figures are on account of the fact that the boundaries have been drawn in different ways. One could cut out two of these shapes from a large area containing solid circles arranged in this same regular order. It is not the shape and size of a crystal that holds so much importance. It is, however, an important fact that the angles between the habit faces are always the same on account of regularity in the arrangement of atoms of which the crystal is composed. The schematic diagrams of figure 1.2 (a-d) illustrate the fact that the crystals of any chemical composition may differ very widely in shape and size, their interfacial angles are the same for various different specimens of the chemical composition. A crystal may, therefore, be defined as a solid with plane naturally occurring faces and in which the composing atoms, ions or molecules are regularly arranged. The outward orderliness of crystal is only a manifestation of the internal orderly arrangement of the atoms or molecules of which the crystal is composed.

Generally speaking, the term solid is applied to rigid elastic substances which exhibit elastic behaviour when they are subjected to hydrostatic forces and also under tensile or shear stresses. Materials regarded as solids may, generally, be divided into two categories, amorphous and crystalline. In amorphous substances, the atoms or molecules may be bound very strongly to one another, but there is no geometrical regularity or periodicity in the arrangement of atoms composing these substances. Such substances are usually viscoelastic and may be regarded as super cooled liquids.

Fig 1.3: Schematic illustration of difference in the atomic arrangement between crystalline solid & an amorphous solid

Figures 1.3(a,b) are schematic representation in two dimensions showing the difference in atomic arrangement between a crystalline and an amorphous material.

The term crystalline state is used to describe substances which apparently may not look like crystals but which, nevertheless, possess an orderly internal arrangement of atoms or molecules. The best example is those of metals which apparently may not appear to be crystals on account of the fact that they are found in irregular lumps but they certainly belong to the crystalline state because the atoms in a given metal are arranged regularly. For example, one may offer the case of Aluminium which is made up of a very large number of the small cubes, as shown in figure 1.4 all packed together so as to fill space completely.

Fig 1.4: The unit cell (phase–centred cubic type) of aluminium metal

The solid circles in the figure are the aluminium ions. It is a face–centred cubic structure, in which the unit cells or building blocks represented by say LMNOPQRS are cubes with an atom at each of the corners and an atom at the centre of each of the six faces of the cube. Several metals, for example, gold, cobalt, copper, silver and several others, exhibit this structure. It only points out the fact that though some solid substances may not necessarily look like crystals, yet they in, in reality, are actually crystalline.

Guglielmini (1688) observed that some solids exhibit the property of cleavage under an application of small stress. Calcite crystals exhibit easy fracture along certain crystallographic planes, each of two surfaces being smooth and plane. Mica, diamond and several other crystals show the property of easy fracture along certain crystallographic planes. The crystallographic plane along which cleavage takes place is a characteristic property of the crystal. It was Guglielmini who studied the constancy of cleavage direction in a given crystalline solid. Abbe Hauy carried the study further in 1784.Study on cleavage of solids led him to conclude that the smallest unit of which the crystal is built up was independent of the external shape (morphology).His work laid the foundation of crystallography and,as such, is credited with the title of father of crystallography. It is now known and established beyond doubt that a certain fundamental building block, each block may be in the form of a group of atoms, ions or molecules etc., which repeats at regular intervals exhibiting periodicity in all the three dimensions in a crystalline solid. The fundamental building block is termed as unit cell of the crystal.

Solid substances may exist in the form of single crystals or as polycrystalline. In case of single crystals the periodicity exists throughout the material and all planes have the same orientation throughout the crystal. That is not so in case of polycrystalline materials. In such materials the periodicity is limited and does not exist throughout. The periodicity breaks down at grain boundaries. There are very tiny crystallites randomly oriented with respect to each other and as such there are several grain boundaries in the mass of polycrystalline material.

1.2. Symmetry Elements

Basic understanding of symmetry is that if a geometrical figure is folded along a line so that the two halves match exactly, then the figure is said to have a line of symmetry as shown in figure 1.5

Fig 1.5e: Hexagonal figure having multiple lines of symmetry

Any regular figure exhibits one or the other type of symmetry. Element of symmetry refers to an operation which when performed may bring about congruency of the figure. Crystals show external symmetry which suggests that most of them exhibit geometric symmetry; the symmetry may be about a point, line or a plane or any of these combinations. Symmetry is characteristic of any regular geometrical figure. This is what applies to solid crystalline materials which exhibit regular arrangement of atoms, ions or molecules that compose them.

Let us consider fundamental forms of symmetry operation that can be executed to periodically repeat the basic building block of atoms, ions or molecules, also known as motif, in space.

1.2.1 Translational Symmetry

To illustrate translational symmetry we may consider numerical figure ‘7’ here for the sake of convenience and understanding as shown in figure no 1.6.If we move numerical figure ‘7’ by a distance , say ‘a’ ,  to the right, the same numerical figure ‘7’ would be located at P . This is called translational symmetry operation and, obviously, can be repeated any number of times. There can also be translational symmetry operations in other directions, up the page, down the page, to the right or left of the page, and there can be compound translations such as ‘2a’ to the right and ‘1a’ up the page taking ‘O’ to ‘R’. The situation is very similar to how chess players move knights while playing the game of chess.

Fig 1.6: Illustration of translational symmetry

1.2.2. Rotational Symmetry

One can achieve congruency of a figure by rotation about an axis through a particular angle say ‘Ө’ , as for example , in the case of numerical figure ‘7’ shown in figure 1.7. If the congruency of the numerical figure about an axis is achieved by a rotation of 2 π/n, where n can be 1, 2, 3, 4, and 6, we can say it has axis of symmetry. For n = 1, 2, 3, 4 and 6 congruency is achieved by rotation about an axis through 360⁰ (possible in all cases), 180⁰, 120 ⁰, 90⁰ and 60⁰ respectively. If n=2 , the axis is named as diad ; n=3 , the axis is triad ; n = 4 , the axis is tetrad ; n = 6 , the axis is hexad.

Fig 1.7: Illustration of rotational symmetry

1.2.3 Reflection Symmetry

One can achieve congruency of a figure by reflection from a line in the two-dimensional case or from a plane as shown in figure 1.8(a,b).If we keep a mirror along BD , BCD is the mirror image of ABD in figure (b) or the top numerical figure ‘7’ is the mirror image of the lower numerical figure ‘7’ in figure (a).

Fig 1.8: Illustration of Reflection symmetry: (a) about a line, (b) about a plane.

1.2.4 Centre of Symmetry

Symmetry about a point is possible by inversion through the point, as shown in figure1.9 for the numerical figure ‘7’.

Fig 1.9: Illustration of symmetry by inversion through a point ‘o’.

1.2.5. Mixture of Symmetry Operations.

Symmetry elements can be of more complex type. To understand this we may consider a situation shown in a schematic diagram of figure 1.10.

Fig 1.10: Illustration of mixed symmetry operations

In figure (a) we have a translational symmetry as explained above. Repetition of numerical figure ‘7’ occurs if we move it by a distance ‘a’. However, the situation is different in case of an arrangement shown in figure (b). Unlike the arrangement of figure (a) one does not find repetition of numerical figure ‘7’ occurring after every movement by a distance a. One can achieve repetition of numerical figure ‘7’ by moving it a distance a (say digit ‘7’ at M) to the right but also to be reflected in the line XY so as to make it appear to be the same as before. On performing this compound operation the numerical figure ‘7’ at M moves to position N and turns it upside-down. In the same way it moves O to position P followed by reflection in the line XY, moves it to Q, turning it upside down at the same time. It is a composite operation which is known as glide-reflection operation of symmetry. One of the many examples of solids which exhibit this kind of symmetry element is diamond that possesses glide planes of symmetry. The glide planes of symmetry are a mixture of a translation operation and reflection operation. Figure 1.11(a, b) is another schematic representation which shows a glide in which reflection is combined with translation parallel to the plane of mirror plane.

Fig 1.11: schematic representation of a glide in which reflection is combined with a translation

Symmetry operations which are a mixture of translation operation and rotation operation are known as “screw axes of symmetry”. The screw rotation operation is a combination of a rotation operation and translation operation in the direction of the screw axis. The structure assumed by diamond has a fourfold screw rotation axis of symmetry.

REFERENCES

1.Kittel, C.    “ Introduction to Solid State Physics “, Wiley , New York , 1971.

2. Azaroff, L.V. “ Introduction of Solids”, McGraw Hill, New York , 1960.

3. Cracknell, A.P. “ Crystals and their Structures “ , Pergamon Press, New York , 1969.

4. McKelvey, John P. “ Solid State and Semiconductor Physics “, Harper & Row, New York,1966.

5. Verma,A.R. &Srivastava, O.N. “Crystallography for Solid State Physics”, Wiley Eastern Ltd., N.Delhi 1982.

6. Philips, F.C. “ An introduction to Crystallography”, Wiley, New York , 1963.

7.Buerger, M.J. “ Elementary Crystallography”, John Wiley & Sons, New York 1956.

8. Omar, M.A. “ Elementary Solid State Physics”, Addison Wiley, Reading ,1975.

9. Dekker, A.J. “ Solid State Physics” Macmillan ,London (1958).

Interesting & Informative reading on crystals & symmetry.

1. Bunn,C. “ Crystals : Their Role in Nature and in Science”, Academic Press, New York.
2. Ehrenfeucht,A. “ The cube made interesting” ,Pergamon Press, Oxford.
3. Fejeshtoth,L . “ Regular Figures” Pergamon, Oxford.
4. Shubnikov, A. V. and Koptsik, V.A. “ Symmetry in Science and Art “ Plenum 1974.
5. Woolfson ,M.M. “ An Introduction to X-ray Crystallography”, Cambridge Vikas.