Question

The number of atoms contained in one face-centred cubic unit cell of monatomic substance is
A.1
B.2
C.4
D.3

Answer 1

Hint: In fcc unit cell, there are 8 atoms present at the corners (each one contributes one-eighth to the unit cell) and six atoms at the centre of the faces of the cube (each one contributes one-half to the unit cell).

Complete step by step answer:
Face-centred cubic unit cell is also known as a cubic close packed arrangement. It has atoms at all the corners as well as at the centre of each of the faces. A cube has 8 corners and 6 faces. There is one atom at each of the eight corners in this arrangement and each of them contributes one-eighth to the unit cell, and one at each centre of the faces of the cube and each of them contributes one-half to the unit cell.
The number of atoms present at corners per unit cell \[ = 8 \times \dfrac{1}{8} = 1\]
The number of atoms present at faces per unit cell \[ = 6 \times \dfrac{1}{2} = 3\]
∴ total number of atoms in ccp or fcc arrangement = 1 +3 = 4
So, the number of atoms contained in one face-centred cubic unit cell of monatomic substance is 4.
Therefore, the correct answer is option (C).

Note: Alternative method, the number of atoms in a unit cell may be calculated by the given formula,
 $Z = \dfrac{{{n_c}}}{8} + \dfrac{{{n_b}}}{1} + \dfrac{{{n_f}}}{1} + \dfrac{{{n_e}}}{4}$
Where,
 ${n_c}$ = number of atoms at the corner
  ${n_b}$ = number of atoms at body centre
 ${n_f}$ = number of atoms at face centre
 ${n_e}$ = number of atoms at edge centre
An fcc crystal contains \[ = \dfrac{8}{2} + \dfrac{6}{2}\] = 1 + 3 = 4
∴ total number of atoms in ccp or fcc arrangement = 1 +3 = 4