5 Graphical Representation of Point Groups & Space Groups

Prof. P. N. Kotru

4.1 Graphical Representation and Labelling of Point Groups.

Various schemes for labelling the crystallographic point groups, as has already been explained, have been suggested. However, Hermann-Mauguin is considered as modern and treated as international notation. The symbols used for representation of axes of symmetry are shown below in table 4.1:

Table 4.1 Graphical Representation of axes of symmetry

The most simple point groups are those which possess one axis of symmetry of order 1, 2, 3, 4, or 6. Inversion axis is symbolized by 1, 2, 3, 4 or 6. On the stereogram 1 and 1 is represented as shown in figure 4.1. In these representations points above and below the plane are indicated by solid (shown as ●) and hollow (shown as   ○) circles respectively.

Fig 4.2: Representation of 2 and 2 on the stereogram

2 is also represented as ‘ m ‘, suggesting a plane of symmetry ; vertical two-fold inversion axis being the same as a horizontal plane of symmetry. Likewise, 3 and 3 or 4 and 4 and 6 and 6 are indicated as is shown in figures 4.3

Fig 4.3: stereograms of rotation & Inversion axes

More point groups are created by adding ‘ ⁄ m’ ( plane of symmetry at right angles to the relevant axis of symmetry ) to the symbol of the particular point group. For example, 1/m , 2/m , 3/m , 4/m and 6/m. Out of these 1/m is the same as m and so can be symbolized as 2. Similarly, 3/m is equivalent to 6 .

Plane of symmetry may also be added vertically through the axes of symmetry. It is symbolically represented by adding ‘m’ to the number of axis of rotation.

Addition of plane to an axis say 4 m leads to new elements of symmetry. In this case twofold axes of symmetry appear. It is, therefore, indicated by including ‘2’ in the symbol and so is written as 4 2 m or 4 m 2.

New point groups can also be produced by adding a two-fold axis at right angles to the basic point groups 1 , 2 , 3 , 4 , and 6. In that case ‘2’ is added to the point group symbol, as for example, 3 2.

More than one plane of symmetry say a plane through the axis and a plane at right angles to the axis also leads to some new point groups. To symbolize this ‘/m’ is added to the relevant point group symbol, as for example, 3/mm. A total of 32 crystallographic point groups corresponding to each crystal system is shown in the stereograms of figure 4.4.

Fig 4.4: Thirty two Crystallographic Point Groups corresponding to each crystal system.

Schoenflies notation is given on the right and Hermann-Mauguin notation is given on the left for each point group.Equivalent points above and below the plane is indicated by hollow circle (○) & solid circle (●).

4.2 Space Groups

Point groups and the Bravais lattices have already been described. So far we have considered point group operations only. Point group operations combined with lattice translation leads to a space group which is actually required in order to describe the symmetry of a real crystal. We may recall the concept of screw rotation axes or the glide reflection planes of symmetry which have been summarily explained earlier. These two types of rotation do not occur in the point groups on account of the fact that the operations of a point group will have to leave the centre of the object in the place from the spot it started. New symmetry elements required for assignment of space groups are the screw axis and glide plane. It is a kind of rotation axis which is chosen parallel to a translation direction in the crystal. So, the operation is a combination of rotation (say by Ө) and translation T. A glide plane is a combination of reflection and a translation in a direction parallel to the reflection. There are two types of space groups, one being called as symmorphic space groups and the other as non-symmorphic or asymmorphic space groups. The former type of space groups are in all 73 whereas the latter types are 157 in number. In all there are 230 space groups.

Let us consider structures having the full symmetry of the cubic lattice i.e., the symmetry of the cubic point group m3m. The body-centred and face-centred cubic structures are assigned to the space groups I m 3 m and F m 3 m respectively, where point group symbol e.g., m 3 m is prefixed by symbol of body -centred ( b. c. c.) and face-centred (f.c.c.) to indicate the Bravais lattice ; I standing for b.c.c. and F for f.c.c.

Each of the 230 space groups is represented by a symbol for identification which is designed to reveal the main features of the group.

Symmorphic space group contains two parts. One part symbolizes Bravais lattice to which a particular structure belongs and the second part symbolizes the point group associated with the Bravais lattice. Thus , the face-centred cubic structure having point group m 3 m and the associated Bravais lattice is face-centred cubic denoted by F belongs to the space group F m 3 m .This way , the five cubic point groups corresponding to face-centred cubic ( F ) symbolized by 2 3, m 3 , 4 3 m , 4 3 2 and m 3 m will be assigned the space groups F 2 3 , F m 3 , F 4 3 m , F 4 3 2 and F m 3 m. Similarly, the five cubic point groups associated with the other two cubic lattices viz., primitive cubic lattice ( P ) and body -centred cubic lattice ( I ) are assigned to the space groups as

(i) P 2 3, P m 3 , P 4 3 m , P 4 3 2 and P m 3 m , and (ii) I 2 3 , I m 3 , I 4 3 m , I 4 3 2 and I m 3 m.

In asymorphic space group the reflection planes and rotation axes get turned into glide planes and screw axes. If one of the axes in the point group is replaced by a screw axis, suffix 1 , 2 , 3 , 4 or 6 is added to the figure representing the axis and if one of the planes is replaced by a glide plane , the letter m in the point group symbol is replaced by one of the letters a , b , c ,d or n, as may be applicable in case of the nature of the glide plane. F d 3 m , P n 3 m , I a 3 d and P 41 3 2 are all asymmorphic cubic space groups. Same procedure of assigning space groups is applied to any crystal system.

Space group is important where the internal arrangement of atoms within a crystal is under consideration. They are most important and very much informative in the study of the microscopic properties of crystals. Assignment of space group to a crystal structure provides the following information:

1. The symmetry of the structure is specified.
2. The type of Bravais lattice and location of the point group and other symmetry elements in the unit cell is revealed, and
3. The positions of equivalent points within the unit cell are determined.

The space groups for large number of crystals are given in the literature (Reference:

1. Handbook of Chemistry and Physics, Chemical Rubber Publication Co.,Cleveland, 1960 and
2. M.J.Buerger, Crystal Structure Analysis, John Wiley and Sons, New York, 1960.)

Space groups of some crystals are given in table 4.2.

Table 4.2 Space groups of some crystals in Hermann-Mauguin & Schoenflies notations

*a, b ,c are used for axial glide planes whereas n & d are used for diagonal glide and diamond glide respectively

One can easily identify the crystal class or point group symmetry of a crystal from the space group notation. The simple procedure is to drop the capital letter of the space group symbol, and then change all other letters to m and finally drop all subscripts. Let us take up an example of some well known crystal say diamond. Its space group is F 41/d 3 2/m. Now let us proceed step-wise as follows:

(i) Drop F

(ii) Change letters d and m to m,

(iii) Drop all subscripts

Following the above procedure we get the result regarding crystal class or point group symmetry of the substance.

These steps may be summarily indicated as follows:

F 41/d 3 2/m→ 4/m 3 2/m = m 3 m

Therefore, the point group symmetry or class to which diamond belongs is m 3 m.It is important to know that as far as the external appearance of a crystal is concerned it is only the point group and not the space group which is distinguishable. Space group is important only when one deals with the internal arrangement of the atoms that compose the crystal. The 32 point groups also known as crystal classes have a physical relevance .Each and every crystalline solid, be it organic or inorganic crystals, metals, alloys or minerals show symmetries corresponding to each of them. A few examples may be given in the table 4.3

TABLE 4.3SOME COMMON STRUCTURE TYPES AMONG CRYSTAL CLASSES

References.

1. Brown, F.C.: “ The Physics of Solids “.,W.A.Benjamin, Inc.,N.Y., 1967.
2. Cracknell, A.P.: “ Crystals & Their Structures”, Pergamon Press Ltd.,U.K., 1969.
3. Phillips , F.C.: “ An Introduction to Crystallography”, Longmans, London.
4. Woolfson ,M.M. “ X-ray Crystallography”, Cambridge Vikas.
5. McKelvey,J.P. “ Solid State & Semiconductor Physics”, Haroer& Row, N.Y., 1966.
6. Verma ,A.R.&Srivastava, O.N.” Crystallography For Solid State Physics”, Wiley Eastern Ltd.,1982.

General Information & Informative Reading On Space Groups Of Crystals.

1. Koster, G.F. : “ Solid State Physics : Space Groups & Their Representations”: Vol. 5,Academic Press, N.Y.,1957.
2. Hand Book Of Chemistry & Physics, Chemical Rubber Co., Cleveland ,1960.
3. Buerger, M.J.: “Crystal Structure Analysis” John Wiley & Sons, N.Y. 1960.
4. Seitz, F. “ Modern Theory Of Solids”, MacGraw, N.Y., 1940.
5. Buerger,M.J. “ Elementary Crystallography”, Wiley, N.Y.
6. International Tables For X-Ray Crystallography” by the International Union of Crystallography ( Kynoch Press, Birmingham) Vol.I (1952) & Vol. II ( 1959).