Question

The number of atoms contained in an fcc unit cell of a monatomic substance is:
A. 1
B. 2
C. 4
D. 6

Answer 1

Hint: We know that the smallest repeating units in a crystal is termed as unit cell. A unit cell can either be primitive cubic, body-centered cubic or face centered cubic.
Let’s discuss face centered cubic unit cell in detail.

Complete step by step answer:
Let’s discuss face centred unit cell.
A face centred cubic unit cell has atoms at centre of all faces and at all corners of the cube.
In a cubic unit cell, the sharing of each corner atom is in between 8 unit cells.
In an fcc unit cell, 8 atoms are present in the corner. So, the atoms contributed by corner atoms can be find out as follows:
8 corner atoms $ \times $ $\dfrac{1}{8}$ atom per unit cell= $8 \times \dfrac{1}{8} = 1$ atom
There are 6 face centred atoms present in the face centred unit cell and each face centred atom is shared in two unit cells. So, the atoms contributed by face centred atoms can be calculated as follows:
6 face-centred atoms $ \times $ $\dfrac{1}{2}$ atom per unit cell= $6 \times \dfrac{1}{2} = 3$ atom.
So, atoms of the fcc unit cell are known by adding the number of atoms contributed by corner atoms and face centred atoms.
Therefore, the total atoms of the face centered cubic cell are 4.

Hence, the correct option is C.

Note: Students might be confused about the number of atoms in body centred and primitive unit cell.
In primitive unit cell, 8 atoms present at the corner and sharing of every corner atom is in between 8 unit cells, so, the number of atoms contributed is equal to $8 \times \dfrac{1}{8} = 1$ atom
In a body centred cubic unit cell, at the corners 8 atoms are present and one atom is occupied at the centre of the body. We know that 8 corner atoms contribute one atom $\left( {8 \times \dfrac{1}{8} = 1} \right)$ and one body centred atom belongs to only one unit cell. So, the total number of atoms in a body centred unit cell is 2.