Question

How many atoms constitute one unit cell of a face-centered cubic crystal?

Answer 1

Hint: A unit cell that contains atoms at all the corners of the crystal lattice along with at the centre of all the faces of the cube is known as the face-centered cubic crystal. The number of atoms in one unit cell of a face-centered cubic crystal is twice that present in a body-centered cubic crystal.

Complete step by step answer:
The unit cell of face-centered cubic crystal has atoms at all the corners of the cube along with atoms at each face of the cube.
We have to calculate the number of atoms that constitute one unit cell of a face-centered cubic crystal.
We know that a cube has eight corners. One atom at the corner is shared with eight other cubes. Thus, the contribution of one atom at the corner of one unit cell of a face-centered cubic crystal is $\dfrac{{\text{1}}}{{\text{8}}}$.
The cube has eight corners. Thus, the eight atoms at the corners contribute,
$\dfrac{{\text{1}}}{{\text{8}}} \times 8 = 1$
Thus, the contribution of atoms from the corners is 1.
We know that a cube has six faces. One atom at each face is shared with one other cube. Thus, the contribution of one atom at the face of one unit cell of a face-centered cubic crystal is $\dfrac{{\text{1}}}{{\text{2}}}$.
The cube has six faces. Thus, the six atoms at the faces contribute,
$\dfrac{{\text{1}}}{{\text{2}}} \times 6 = 3$
Thus, the contribution of atoms from the faces is 3.
The total number of atoms that constitute one unit cell of a face-centered cubic crystal is the sum of the contribution of the atoms from the corners and the contribution of atoms from the faces. Thus,
Total atoms in face-centered cubic crystal = 1 + 3
Total atoms in face-centered cubic crystal = 4

Thus, the number of atoms that constitute one unit cell of a face-centered cubic crystal are 4.

Note: Note that the face-centered cubic crystal has atoms at the centre of each face of the cube. No atom is centrally located or diagonally located inside the body of the cube. In the simple cubic crystal only one atom is present in the unit cell and in body centered total two atoms are present.