Question

The numbers of tetrahedral and octahedral holes in a CPP array of $ 100 $ atoms are respectively?
(A) $ 200 $ and $ 100 $
(B) $ 100 $ and $ 200 $
(C) $ 200 $ and $ 200 $
(D) $ 100 $ and $ 100 $

Answer 1

Hint: Unit cell: It is defined as the smallest repeating unit in the crystal lattice or you can say the building block of a crystal as like a cell is the unit cell of the body. They are of three types: (I) primitive cubic unit cell, (II) Body-centered cubic unit cell (III) Face-centered cubic unit cell.

Complete step by step solution:
Unit of crystal: It is defined as the repeating unit in the crystal. They are of three types: (I) primitive cubic unit cell, (II) Body-centered cubic unit cell (III) Face-centered cubic unit cell.
-Primitive cubic unit cell: In this unit cell, atoms are present only at the corners of the cube. We know that there are eight corners in a cube. So one atom is shared by right corners. So the contribution of an atom in one corner is $ \dfrac{1}{8} $ . The number of atoms present in a primitive cubic unit cell is one.
-Body-centered cubic unit cell: In this unit cell, atoms are present at the centre and corners of the cube. We know that there is only one centre in a cube. So one atom is completely present there. And one atom is contributing to the eight corners. So the total number of atoms present in a Body-centered cubic unit cell is $ 2 $ .
-Face-centered cubic unit cell: In this unit cell, atoms are present at corners and at the face centres of the cube. There are a total six faces in a cube. So one atom contributes to two face centres i.e. contribution is one atom is half. And one atom contributing to all eight corners of the cube. So there are four atoms present in the face-centered cubic unit cell.
Tetrahedral voids are unoccupied empty spaces present in substances having a tetrahedral (having four atoms) crystal system. Octahedral voids are unoccupied empty spaces present in substances having an octahedral (having eight atoms) crystal system.
In CPP (FCC) effective number of atoms is equal to octahedral voids and two times of tetrahedral voids.
Here we are given a CPP array of $ 100 $ atoms so the number of tetrahedral voids are $ 2 \times 100 = 200 $ and the number of octahedral voids are $ 100 $ . So, the numbers of tetrahedral and octahedral holes in a CPP array of $ 100 $ atoms are respectively $ 200 $ and $ 100 $ .
Hence option A is correct.

Note:
In HCP the number of tetrahedral voids and octahedral voids are just opposite to that in CPP. Hence the number of tetrahedral voids is equal to the number of atoms while the number of octahedral voids is twice to the number of atoms.