3 Bravais lattice & crystal structure

Prof. P. N. Kotru

TABLE OF CONTENTS

2.1 Symmetry Elements and Crystalline Solids

2.1.1 Centre of Symmetry.

2.1.2 Axis of Symmetry.

2.1.3 Plane of Symmetry.

2.1.4 Inversion Axes.

2.2 Bravais Lattice and Crystal Structure

2.2.1 Translation Vectors.

2.2.2 Unit cell.

LEARNING OBJECTIVES

• The symmetry elements exhibited by crystals are described.
• Centro – symmetric crystals are illustrated.
• Symmetry about an axis is described. It is explained that crystals exhibit two -, three-, four – and six – fold axis of rotation. Five fold symmetry is ruled out.
• Crystal form showing symmetry about a plane is explained
• Combination of rotation and inversion through a point give rise to a different set of symmetry operation.
• Translational periodicity as the most fundamental feature of the Bravais lattice is explained.
• The concept of the fundamental building block of a crystal is explained.

2.1 Symmetry Elements and Crystalline Solids

Symmetry is the most general law inherent in the structure and properties of a crystalline substance; it is sometimes said to be property of properties of solid crystals. Most of the important properties of crystalline solids are structure sensitive properties.

There are certain laws of symmetry according to which the faces of a crystal are arranged and this symmetry may be defined in relation to:

• A centre of symmetry
• An axis of symmetry
• A plane of symmetry

Let us take it up one by one.

2.1.1 Centre of Symmetry

A crystal is said to have a centre of symmetry if, for a point within it, faces occur in parallel pairs of equal dimensions on opposite sides of the point and equidistant from it. Crystals exhibit ing this symmetry are known as centrosymmetric crystals, example of which is shown in figure 2.1

Fig 2.1: schematic representation of a centrosymmetric crystal 2.1.2 Axis of Symmetry (Rotation Axes)

If rotation of a crystal about an axis through 2π/n brings about the identical appearance of the crystal, the axis is said to be n-fold rotation axis. n can take values 1,2,3,4 and 6. A rotation about an axis through 360⁰ will always result with identical appearance of a figure and therefore, the value of n = 1 is meaningless. If n = 2, the crystal has a twofold axis known as diad ; n = 3 , the crystal is said to have three fold axis known as triad ; n = 4 ,  the crystal is said to have four fold axis known as tetrad ; n = 6 , the crystal has six fold axis known as hexad. Figures that are examples of a diad, triad, tetrad and hexad axes are represented by diagrams of figure 2.2.

Conventionally, the symmetry elements are represented by written symbol and graphical symbol. For diad, triad, tetrad and hexad the written and graphical symbols are 2, ; 3,  ; 4,  ; 6 ,  respectively. n = 5 or 7 are not allowed in crystallography. Non-existence of symmetry axes other than 1 , 2 , 3 , 4 and 6-fold in a lattice is actually based on the fundamental requirement for an arrangement of points to define a lattice because the symmetry elements that are operational have to conform to their translation periodicities.

2.1.3 Plane of Symmetry (Mirror plane)

A plane in the crystal which divides it in a way that the halves on opposite sides of the plane are mirror images of each other. One of the crystal forms exhibiting mirror planes is shown in a schematic diagram of figure 2.3 .This symmetry element is represented by written symbol ‘m’ and graphical symbol ‘─’

Fig 2.3: Crystal form exhibiting mirror plane

2.1.4 Inversion Axes

It involves a combination of rotation and inversion through a central point. The written and graphical symbols of inversion axes are 1 , 0 ; 2 , none ( m is generally used because 2 is the same as reflection in a plane to which 2-fold axis is perpendicular and , therefore, 2 = m ) ; 3 ,  ; 4 ,  ; 6 ,  respectively. 1 is actually identical to a centre of symmetry described earlier and, therefore, the symbol is used for centre of symmetry. There are all the symmetry elements which may be exhibited by the external form of the crystal.

2.2 Bravais Lattice and Crystal Structure

A Bravais lattice is simply a lattice of points in space formed by repeating units. A Bravais lattice is just a discrete set of points in space which appears to be the same when one looks at it from any one of its points or when looked at from any other lattice point.

A crystal structure is built up by placing basis down on each point of Bravais lattice. Basis actually is a single atom or ion, or a group of atoms, or a molecule. At each lattice point one finds an identical arrangement and orientation of the basis. The crystal then develops by repeating the basis in space in accordance with the requirements of a particular Bravais or space lattice involved.

2.2.1 Translation Vectors

Translational symmetry or translational periodicity is the most fundamental feature of a Bravais lattice. Let us consider Bravais lattice of figure 2.4. There are three vectors a, b and c which correspond to elementary translations along three different directions. The elementary translation vectors a, b and c are as shown in figure 2.4

Fig 2.4: Bravais lattice with elementary translation vectors

Here, a represents shortest period in the lattice, b represents shortest period not parallel to a whereas c represents the shortest period not coplanar to a and b.Translational symmetry or periodicity can be understood as follows: Consider r to be any position in the lattice. There would be location of some point at the position r/ which is derivable from r, using the below given equation:

The lattice arrangement has to be the same at r∕ as at r as per the translational symmetry requirements. The translation operation is then obtained as:

and then the same is added to the vector r . Equation 2.1 can be expressed in terms of an operator as :

The totality of operations for all values of integers n1, n2 and n3 is called translation group for the crystal. The translation group is a part of the space group that includes both translation and rotation operations . The translational periodicity described above is in terms of the vectors a, b and c which define the edges of the primitive cell. The primitive cell of a Bravais lattice is a unit cell which contains at most one lattice point.

2.2.2 Unit cell

A unit cell is a small unit of the crystal which on repeating in space forms the entire crystal pattern. It is the fundamental smallest unit of which the entire crystal is built up. Let us consider a lattice in two dimensions as shown in figure 2.5

Fig 2.5: Three primitive cells 1, 2, 3.Cell marked 4 is non-primitive

Notice that four different cells marked 1, 2, and 3 are primitive whereas cell marked 4 is non-primitive. The primitive cells are those which have atoms at their corners only. Though the primitive cells appear to have different shapes, they enclose the same area. Figure 2.2 represents the lattice of a two-dimensional crystal. The parallelogram (say marked 1) is determined by the basis vectors a and b. All translations of the parallelogram (say 1) by integral multiples of the vectors a and b, along the a and b directions , will lead to translating it to a region of the crystal which would be just like the original one. This way the whole crystal may be reproduced simply by translating the area marked 1 along a and b directions by all possible combinations of multiples of the basis vectors a and b .As such every lattice point in the crystal could be described by a vector r where

In the same way one can extend the argument to define unit cells and basis vectors for three-dimensional crystal lattices. One is now in a position to define unit cell as a region of the crystal which is defined by three vectors a , b and c so that when the same is translated by any integral multiple of those vectors, leads to a similar region of the crystal. A set of linearly independent vectors a, b, c is used to define a unit cell. The unit cell enclosing smallest volume which is used to define a given lattice is called primitive unit cell whereas the primitive basis vectors are a set of linearly independent vectors which defines a primitive unit cell.

Every lattice point in a three-dimensional crystal lattice is described by a vector r defined as:

SUMMARY

1. The crystals exhibit symmetry about a point, a line and a plane.
2. The translational symmetry is the most characteristic feature of the Bravais lattice.
3. Crystals showing symmetry about a point are known as centro-symmetric crystals.
4. Crystals show symmetry about an axis. Congruencies is achieved on rotation about an axis through an angle of 180°, 120°, 90° and 60° and are designated as diad, triad, tetrad and hexad respectively. The possibility of achieving congruence by rotation about an axis through 72° is ruled out.
5. Crystals show plane of symmetry, also known as mirror planes.
6. The concept of unit cell as the fundamental building block of the crystal is described and discussed.