Calculations Involving Unit Cell Parameters

The seven crystal systems

Cubic- In this type of unit cell lengths a, b and c all are equal in length and the angles between them alpha, beta and gamma are also equal to 90 degrees bravais lattices in this type are 3 that is primitive, face centred and body centred examples of cubic unit cell are NaCl, zinc blende and copper. 

Tetragonal – In this type of unit cell length a and b are equal but a and b are not equal to c this type of unit cell has two equal lengths but the angles between them are equal that is alpha, beta and gamma is equal to 90 degrees bravais lattices are primitive, body centred that it is of 2 types. Examples- white tin. 

Orthorhombic – (rhombic) In this type of cell lengths a, b and c are not equal a is not equal to b is not equal to c but the angles between them are equal alpha, beta and gamma is equal to 90 degrees. Bravais lattices are of 4 types in this type of cell that is primitive, face centred, body centred and end centred. Example – rhombic sulphur, match box, duster. 

Rhombohedral – ( trigonal) In this type of cell length a, b and c all are equal in length but the angles present between these sides are not equal; alpha is equal to beta but beta is not equal to gamma. The bravais lattices present in this type of unit cell are of one type that are primitive examples- calcite, cinnabar. 

Hexagonal- In this type of unit cell all the lengths are not equal to each other, that is a is equal to b but b is not equal to c and the angle between them is alpha and beta is equal to 90 degrees and the gamma is equal to 120 degrees. Bravais lattices in these types of cells are of one type that is primitive. Examples- graphite, zinc oxide. 

Monoclinic- In this type of cell all the lengths are not equal, that is a is not equal to b is not equal to c but the angle present between them is alpha and beta is equal to 90 degrees and gamma is equal not to 90 degrees. Bravais lattices in this type of unit cell are of two types that are primitive, end centred. Examples- monoclinic sulphur. 

Triclinic- In this type of unit cell a is not equal to b is not equal to c all the lengths are different and same with the angles alpha is not to beta is not equal to gamma is not equal to 90 degrees. Bravais lattices in these types of unit cells are of one type that is primitive. Examples- potassium chromate oxide. 

Classification of unit cell ( as per bravais) 

Unit cells are further classified as primitive unit cells and centered unit cells. In primitive unit cells the same type of particles are present at the corners only. As on the other side of the centred unit cell the same type of particles are present besides corners. It is further classified as of three types: face centred, body centred and end centred. In the end, the same types of particles are present at corners and any opposite face centres. End centred type of bravais lattice is present only in orthorhombic and monoclinic type unit cells. 

Types of cubic unit cell

The distance between successive lattice planes of the same type is called spacing of planes or interplanar distance between the planes. On the basis of this aspect, the lattices may be divided in following cases-

• Simple / primitive/ basic unit cell 

A unit cell having lattices point only at corners is called a primitive or simple unit cell. In this case there is one atom at each of the eight corners of the unit cell considering an atom at one corner as the centre, it will be found that this atom is surrounded by six equidistant neighbours and thus the coordination number will be six. If a is the side of the unit cell, then the distance between the nearest neighbours shall be equal to a. Relationship between edge length ‘a’ and atomic radius ‘r’ is a= 2r that is r = a/2. Number of atoms present in a unit cell is in this case one atom lies at each corner. Hence a simple cubic unit cell contains a total of 1/8 ×8 = 1 atom / unit cell. Packing efficiency of this unit cell is calculated in the following manner: packing efficiency is equal to volume occupied by atoms present in unit cell/ volume of unit cell . 

In a simple cubic unit cell, 52.4 % of total volume is occupied by atoms. % of void space is 47.6.

• Body centred cubic unit cell

A unit cell having lattices at the point of body center in addition to the lattice point at every corner is called a body centered unit cell, where the body diagonal particles are touching particles. Here the central atom is surrounded by eight equidistant atoms and hence the co – ordination number is eight. The nearest distance between two atoms will be a√3 /2 . Relationship between edge length and atomic radius is as follows , In BCC , along the cube diagonal all atoms touch each other and the length of cube diagonal is √3a. So √3a=4r that is r =√3a/4. Number of atoms present in this type of unit cell is 2 atoms per unit cell in that 1 is of corners and one is of body centre. In this case one atom lies at each corner of the cube. Hence the total number of atoms per unit cell is 1 +1 = 2 atoms. Packing efficiency in these types of cells are 68% of total volume occupied by atoms. % void space is 32.

• Face centred cubic unit cell

A unit cell having a lattice point at every face centre in addition to the lattice at every corner is called a face centred unit cell. In this case there are eight atoms at the eight corners of the unit cell and six atoms at the center of six faces. The coordination number will be 12 and the distance between the two nearest atoms will be a√2/2. Relationship between edge length and atomic radius is 4r =√2a that is r =√2a/4 that is a/2√2 that is a/2√2. R = a/2√2. Number of atoms per unit cell is 4 atoms/unit cell in this case one atom lies at the each corner of the cube and one atom lies at the centre of each face of the cube. It may be noted that ½ of each face sphere lies within the unit cell and there are six such faces. The total contribution of 8 corners is 1 , while that of 6 face centred atoms is  3 unit cells. Hence the total number of atoms per unit cell is 1+3 =4 atoms. Packing efficiency in the center is 74% of total volume occupied by atoms. % void space is 26.

Conclusion

Coordination number of nearest neighbouring particles around a specific particle in a given crystalline substance is called the coordination number of that crystalline substance. In a cube number of corners are 8 , number of faces are 6 , number of edges are 12, number of body centre is 1 , number of body diagonals is 4 , number of face diagonals is 12