Characteristic parameters and seven classes of unit cell

Unit Cell

Crystals are built up of a regular arrangement of atoms or ions in three dimensions. The smallest structure of which the crystalline solid (or crystal) is built by its repetition in three dimensions is called as unit cell. A unit cell may be considered as the brick of a wall depends upon the shape of brick, the shape of crystal also depends upon the shape of unit cell. Therefore, a unit cell is the fundamental elementary pattern of a crystalline solid. The characterization of the crystal involves the identification of its unit cell.

Characteristic parameters of unit cell

1.    Crystallographic axes: The lines drawn parallel to the lines of intersection of any three faces of the unit cell which do not lie in the same plane are called crystallographic axes.

2.    Interfacial angles: The angles between the three crystallographic axes are known as interfacial angles.

3.     Primitives: The three sides a, b and c (as shown in figure ) of a unit cell are known as primitives or characteristic intercepts.

Crystallographic axes : OX, OY, OZ Interfacial angles : a, b, g Primitives (distances) : a, b, c

The unit cell is characterized by the distances a, b and c and angles a, b & g.

The size (edge length) of a unit cell depends on the size of the atoms or ions and their arrangement. Because a unit cell is representative of the entire structure, the ratio of ions in the unit cell is the same as the ratio in the overall structure.

There  are  seven  classes  of   unit  cells. 

1.  Cubic     , 

2.  Triclinic,

3. Monoclinic,

4. Orthorhombic,

5. Tetragonal, 

6.Hexagonal and

7. Rhombohedral.

Packing arrangement variations exist in each of the classes, yet here we will only explore the cubic system because it is the simplest.

Cubic unit cells

Three types of cubic unit cells are discussed in this chapter. Each unit cell is defined by one type of atom. Consequently, whichever atom you choose when defining that unit cell is the only atom used in defining the unit cell. (Ignore all other types.)

Simple cubic

The most simple unit cell is known as a simple cubic unit cell. This is where one atom occupies each of the eight corners of a cube. The distance from atom to atom along the lattice is the same in every direction, and the angle between each of axes is 90 o .

Body-centred cubic

The next unit cell is known as the body-centred cubic. In this form of crystal, there is an atom at each corner of the unit cell and also there is an additional atom in the center of the cube. This packing can fit more atoms into less space than the simple cubic unit cell.

Face-centred cubic

A slightly more tightly packed unit cell is the face-centred cubic unit cell. In this form of crystal, there is an atom at each corner of the unit cell. And there is an atom at the center of each of the six faces of the cubic unit cell. This crystal packing form has an even higher density than the body-centred cubic unit cell.

Atoms or ions are shared between adjacent unit cells. The lattice position of the atom or ion determines the number of unit cells involved in the share. There are four different lattice positions an atom or ion can occupy.

Body: Not shared

Face: Shared by two unit cells

Edge: Shared by four unit cells

Corner: Shared by eight unit cells.

Sodium chloride crystal

Space lattice of sodium chloride is known to consist of a face-centered cubic lattice of Na⁺ ions interlocked with a similar lattice of Cl^-ions. A unit cell of this combined lattice is shown in figure. This unit cell repeats itself in three dimensions throughout the entire crystal. The yellow spheres indicate chloride ions and red spheres represent sodium ions. The lattices are constituted entirely by ions are known as ionic lattices. All electrovalent compounds show such lattices.

There are four units of NaCl in each unit cube with atoms in the positions

Cl: 0 0 0;  ½½0; ½0½; 0½½;

Na:  ½½½; 00½; o½0; ½00;

As will be seen figure, the unit cell of sodium chloride consists of 14 chloride ions and 13 sodium ions. Each chloride ion is surrounded by 6 sodium ions and similarly, each sodium ion is surrounded by 6 chloride ions.

Notice that the particles at corners, edges and faces do not lie wholly within the unit cell. Instead these particles are shared by other unit cells. A particle at a corner is shared by eight unit cells, one at the centre of face is shared by two and one at the edge is shared by four. The unit cell of sodium chloride has 4 sodium ions and 4 chloride ions as shown below;

No. of Sodium ions

12 (At edge centers) x (1/4) + 1 ( At body center) x 1

 =Ncc / 4 + Nb / 1 + Nf / 2 + Nc / 8 = =      12 x ¼ + 1x 1 =3 + 1 =4

No. of Chloride ions

8 (At corners) x (1/8) + 6 ( At face centre) x (1/2 )

8 x (1/8)  + 6 x (1/2) =1+ 3 = 4

Thus, number of NaCl units per unit cell is 4.

The sodium chloride structure is also called rock-salt structure.

Representative crystals having the NaCl arrangements includes: LiH, NaI, KCl, RbF, RbI, PbS etc.

Cesium Chloride structure

 The cesium chloride, CsCl, structure has body-centred cubic system and is shown in figure. The body-centred cubic arrangement of atoms is not a close packed structure. There is one molecule per primitive cell, with atoms at the corners (000) and body-centred positions 1/2 1/2 1/2 of the simple cubic space lattice.

1. The Cl- ions are at the corners of a cube where as Cs+ ion is at the centre of the cube or vice versa

2. Each Cs+ ion is connected eight Cl- ions and each Cl- ion is connected eight Cs+ ions i.e., 8:8 coordination. Thus each atom is at the center of a cube of atoms of the opposite kind, so that the coordination number is eight.

The unit cell of cesium chloride has one Cs+ ion and one Cl- ion as shown below

No. of Cl- ions

8(At corners) x  1/8 (common to eight unit cell)

8 x 1/8  = 1

No. of Cs+ ion = 1 (At the body center) x 1

= 1 x 1 = 1

Thus number of CsCl units per unit cell is 1.

Representative crystals having the CsCl arrangements include: CsBr, CsI, TlBr, TlI, NH4Cl etc.