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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?

Last updated date: 13th Jun 2024
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Hint: 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:

SystemPossible variationsAxial distancesAxial anglesExamples
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 $).