Question

The number of carbon atoms in the unit cell of diamond is____
A. 6
B. 1
C. 4
D. 8

Answer 1

We need to know about the unit cell, their types and how are atoms arranged in the unit cell of a diamond. Diamond is a crystalline solid with a regular and repeating pattern of constituent particles in a three-dimensional arrangement which is called a crystal lattice. In this crystal lattice, the smallest portion which when repeated in different directions produces the lattice is called the unit cell of that crystalline solid.

Complete step by step answer:
We know that the unit cells are divided into two categories – primitive and centred unit cells. When the constituent particles are present only on the corner positions then it is called a primitive unit cell. We must understand that when one or more constituent particles are present in other positions in addition to those at corners are called centred unit cells. Centred unit cells are of three types:
Body-centred Unit Cell: Such a unit cell contains one constituent particle at its body-centre besides the one present at the corners.
Face- centred Unit Cell: Such a unit cell has one constituent particle present at the centre of each face, besides the ones that are at its corners.
End-centred unit cell: Such a unit cell has one constituent particle at the centre of any two opposite faces besides the ones present at its corners.
There in total seven crystal systems and primitive unit cells are a part of all the seven systems. However, the three centred unit cells are a variation of a few of them. We shall discuss only one of the systems associated with diamonds which is face-centred cubic unit cells, also known as fcc unit cell.
An fcc unit cell contains atoms at all the corners and the centre of all the faces of the cube and each atom located at the face-centre is shared between two adjacent cells and only half of each atom belongs to one unit cell.
The 8 corners of the cubic unit cell contribute only 1/8 atom per unit cell.
In addition, there are 6 face-centred atoms in each unit cell, each contributing ½ atom to one unit cell.
In diamond, every C is tetrahedrally bonded to 4 other C atoms and hence when the unit cells are closely packed, one layer of unit cells above another creates a tetrahedral void which is occupied by another carbon atom which contributes only ½ of its atom to one unit cell. Hence there are 8 voids created by the closed packing arrangement.
Keeping the above arrangement of atoms in a diamond’s unit cells, we can calculate the number of atoms as follows:
Number of atoms contributed by the corners of the unit cell = $8 \times \dfrac{1}{8} = 1$atom
Number of atoms contributed by the face-centred atoms of the unit cell = $6 \times \dfrac{1}{2} = 3$atoms
Number of atoms contributed by the tetrahedral voids of the unit cell = $8 \times \dfrac{1}{2} = 4$atoms
Therefore, the total number of atoms in the unit cell of a diamond = $1 + 3 + 4 = 8$ atoms.

Therefore, option D is correct.

Note: It must be noted that this arrangement of atoms in a diamond unit cell is not the same for all the cubic unit cells. The corner and face-centred atoms may be the same but the contribution of voids may vary depending on the close packing. Note that the arrangement of atoms in diamond makes it one of the hardest substances on earth.