4 Crystallographic point groups

Prof. P. N. Kotru

3.1     Lattice & Crystal Structure

The concept of unit cell has already been explained. We only need to describe one cell and its contents in order to be able to define the whole crystal structure. Identical unit cells are packed together so as to fill all of space to make the whole crystal.

It is important to understand the difference between a Bravais lattice or space lattice and a crystal structure. Bravais lattice is a lattice of points in space which is formed by repeating fundamental units with atoms replaced by points. There are 14 ways of arranging points in space lattices such that all the lattice points have exactly the same surroundings. These 14 point lattices are called Bravais lattices. So, a Bravais lattice is a discrete set of points in space which looks exactly identical when looked at from any one of its points as when looked at from any other lattice point.

A crystal structure can be built up by placing basis on each point of a Bravais lattice. A basis may be a single atom or ion, a group of atoms or a molecule. At each lattice point there has to be an identical arrangement and orientation of the basis. In this manner by repeating the basis in space in accordance with the requirements of the applicable Bravais or space lattice the whole crystal is built up.

3.2     Crystallographic Point Groups

By now we are familiar with symmetry operations as applied to crystallography. A point group is a collection of symmetry operations that apply at a point in a space. These are the operations which leave one point fixed such as rotations about an axis. Such symmetry operations may apply to a molecule, to a crystal of a particular external form or to the region surrounding a point in a given lattice structure. In this kind of operation translation is not involved because there is no question of movement of a point. There are 32 crystallographic point groups which consist of combinations of these elements.

The 14 Bravais lattices, as would be described further in the text are grouped into seven crystal systems. Each of these is defined by their characteristic symmetry elements. The symmetry elements that specify the crystal systems may be put as follows:

1. Rotation axis: If rotation about an axis through an angle 2π/n radians leaves the lattice unchanged, it is said to have n-fold rotation axis. n takes the values 1 ,2 , 3 , 4 and 6. n cannot take the value 5, because five-fold rotational symmetry in a crystal is not possible.
2. Plane of symmetry: If one half of the crystal on reflection in such a plane, passing through a lattice point, reproduces the other half, it is said to have plane of symmetry.
3. Inversion Centre: If a lattice point about which the operation r → – r, where r is a vector to any other lattice point, keeps the lattice identical, it is said to be an inversion centre.
4. Rotation inversion axis: It is a symmetry operation which involves rotation and inversion about an axis. If rotation about an axis by 2π/n radians (where n = 1, 2, 3, 4 and 6) followed by inversion about a lattice point through which the rotation axis passes maintains the lattice identical, it is said to be rotation-inversion axis.

There is a possibility of other symmetry operations but they are equivalent to linear combination of the above four symmetry operations described above.

Table 3.1 provides a list of seven crystal systems alongwith their characteristic symmetry elements and the characteristics of unit cell.

Table 3.1Conventional unit cells for the 14 Bravais lattices arranged according to seven crystal systems

Fig 3.1a: Conventional unit cells for the 14 Bravais lattices and seven crystal systems

Fig 3.1b: Notation for angles & dimensions of unit cell

The various identifying features, axial angles and lengths of these 14 Bravais lattices are shown in figure 3.1a. The notation for angles & dimensions of unit cell used in figure a is indicated in figure 3.1b.All the 14 Bravais lattices fall under seven systems depending upon the type and number of rotation axes present. Rotation symmetry is the most important in classifying crystals according to the seven different systems.

3.3 The thirty two Crystallographic Point Groups and their notations

Let us know what we mean by point group. Point group refers to a collection of symmetry operations viz., rotations, reflections and inversions which involve atleast one fixed point of the crystal while all other points remaining invariant. Construction of a finite number of collections of symmetry elements consistent with each other lead to groups which are referred to as crystallographic point groups or simply as point groups. They are also known as crystal classes. There are 32 point groups in all. Table 3.2 provides a list of point groups and the labels used to identify them. Two notations are given here one of which is the Hermann- Mauguin or international notation and the other one is an older notation known as Schoenflies notation. The former is more modern form of labelling point groups and is, therefore, followed by most of the crystallographers. These notations may be explained as follows:

In Schoenflies notation, the operations involved or the individual elements of the group are indicated. As for example, Cn refers to a single n-fold rotation axis. n stands for the number of elements or order of the group. Suppose n = 6, it means the symmetry operations refer to rotations through 2π/6 or 60⁰. The group contains the elements or operations

C₆ , ( C₆ )2 = C₃ , ( C₆ )³ = C₂ , ( C₆ )⁴ = ( C₃ )^─1 , ( C₆ )⁵ = ( C₆ )^─1 and ( C₆ )⁶= E, where E is known as the identity element.

Schoenflies indicates point groups corresponding to one symmetry axis by the letter C which is followed by an appropriate numerical subscript. So point group corresponding to a two-fold symmetry axis is denoted by C₂ , three-fold axis by C₃ and so on.The letter C is taken from the word ‘ Cyklisch’ which means circular and so implies axis of rotation . Groups Cni stands for rotation-inversion axes, Cnv stands for a number of symmetry related vertical mirror planes σv besides Cn rotation axis. Groups Cnh have a horizontal mirror plane σh . So, the addition of mirror plane is denoted by a letter subscript next to the figure. If the plane is added normal to the symmetry axis, the letter ‘h’ is used (h suggesting horizontal). If, however, the plane is parallel to the axis, the letter ‘v’ is used (v suggesting vertical). The letter ‘D’ denotes point groups containing symmetry axes only. Groups Dn ,Dnd and Dnh have n number of 2-fold axes perpendicular to a Cn axis. The subscript ‘d’ represents diagonal reflection planes which bisect the angles between the 2-fold axes. The subscript ‘h’ represents horizontal mirror planes σh.Point groups with rotoreflection axes are denoted by the letter ‘S’ ( S taken from the word ‘spiegelaxe i.e., a mirror axis ).

As for example, S6 = 6 = 3 . Cubic systems are highly symmetric and their groups are represented either by ‘T’ (i.e., tetrahedral), or ‘O’ (i.e., octahedral). The group represented by ‘T’ has 12 rotational operations whereas group represented by ‘O’ has 24 rotational operations.

The 32 crystallographic point groups corresponding to the type of crystal system in Herman─Mauguin and Schoenflies notation are listed in table 3.2

Table 3.2

that one is below and the other above instead of being together as for example C3h ≡ C3h & D3d ≡ D3d There is a practice that sometimes the two subscripts associated with various letters are written such and so on.

References

1. Brown, F.C. : “ The Physics of Solids” I W.A. Benjamin, N.Y., ( 1967 ).
2. Verma,A.R. &Srivastava, O.N.: “ Crystallography For Solid State Physics “ , Wiley Eastern Ltd.,
3. Cracknell, A.P.: “ Crystals & Their Structures “, Pergamon Press “, N.Y. 1966.
4. Phillips, F.C. ; “ An Introduction to Crystallography”,Longmans,London1963
5. McKelvey, J.P.: “ Solid State & Semiconductor Physics “, Harper & Row, N.Y.
6. Buerger, M.J.: “ Elementary Crystallography”, Wiley, N.Y.

Suggested Reading

1.Schoenflies, A. : Krystallsystene Und Krystallstrnktur Leipzig, 1891

2. International Tables for X-ray Crystallography ,Kynock Press, Birmingham, 1952 (Vol.1) &1959 ( Vol. 2 ).

3. Hermann, C.: Z.Krist, 68 ,P .257 (1928) ; 76, P .559.( 1931 ).

4. Mauguin,C.: Z. Krist. 76, P 542 ( 1931).

5. Azaroff, L. :” Introduction to Solids “. (McGraw Hill,N.Y. ) 1960.

6. De Jong, W.F.: “ General Crystallography “, W.H. Freeman & Co, Jan Francisco,1959.

7. Seitz, F. : “ Modern Theory of Solids “, McGraw-Hill Book Co.,Inc. ,N.Y. (1940).