Question

What is a primitive unit cell?
How many unit cells are possible with a crystal system having $\alpha =\beta =\gamma ={{90}^{\circ }},a\ne b\ne c$ parameters?

Answer 1

By the primitive we means, the unit cells only occur at the corners and to find out the unit cells with a crystal system having $\alpha =\beta =\gamma ={{90}^{\circ }},a\ne b\ne c$ parameters, we should know about the seven crystals and then , we can easily answer the given statements.

Complete Solution :
- First of let’s discuss what is a primitive unit cell. By the primitive unit cell, we mean that unit cell which has lattice points only at the corners. It is also known as the simple cell.
- In all, there are seven types of the simple primitive unit cells among crystals. These are called the seven crystal systems or the crystal habits.
These seven crystal systems are as follows:

System	Possible variations	Axial distances	Axial angles	Examples
Cubic	Primitive Body- centeredFace-centered	$a=b=c$	$\alpha =\beta =\gamma ={{90}^{\circ }}$	Sodium chloride, zinc blende etc.
Tetragonal	Primitive Body- centered	$a=b\ne c$	$\alpha =\beta =\gamma ={{90}^{\circ }}$	White tin, calcium sulphate etc.
Orthorhombic	Primitive Body- centeredFace-centeredEnd-centered	$a\ne b\ne c$	$\alpha =\beta =\gamma ={{90}^{\circ }}$	Rhombic sulphur, barium sulphate etc.
Monoclinic	Primitive End-centered	$a\ne b\ne c$	$\alpha =\gamma ={{90}^{\circ }},\beta \ne {{90}^{\circ }}$	Monoclinic sulphur , sodium sulfate etc.
Hexagonal	Primitive	$a=b\ne c$	$\alpha =\beta ={{90}^{\circ }},\gamma ={{120}^{\circ }}$	Graphite , zinc oxide etc.
Trigonal	Primitive	$a=b=c$	$\alpha =\beta =\gamma \ne {{90}^{\circ }}$	Calcite, quartz etc.
Triclinic	Primitive	$a\ne b\ne c$	$\alpha \ne \beta \ne \gamma \ne {{90}^{\circ }}$	Hydrated copper sulphate, boric acid etc.

Therefore , from the above table we can see that the crystal system having $\alpha =\beta =\gamma ={{90}^{\circ }},a\ne b\ne c$ parameters is orthorhombic and there are four types of unit cells possible i.e.
1. Primitive
2. Body- centered
3. Face-centered
4. End-centered

Note: The smallest repeating unit in the space lattice which when repeated over and over again is called as the unit cell and unit cell is characterized by six parameters; edge length (a, b and c) and angles ($\alpha ,\beta \text{ }and\text{ }\gamma$).