Question

How many atoms are in the following cubic unit cell -
(a) Simple cubic unit cell.
(b) Body centered cubic unit cell.
(c) Face centered cubic unit cell.​

Answer 1

Explanation:

(a). In each cubic unit cell, there are 8 atoms at the corners. Therefore, the total number of atoms in one unit cell is

8 × 1/8 = 1 atom.

(b). In a BCC cell, we have:

8 corners × 1/8 per corner atom = 8 × 1/8 = 1 atom

1 body centre atom = 1 × 1 = 1 atom

Therefore, the total number of atoms present per unit cell = 2 atoms.

(c). In a face-centered cubic unit cell, we have:

8 corners × 1/8 per corner atom = 8 × 1/8 = 1 atom

6 face-centered atoms × 1/2 atom per unit cell = 3 atoms

Therefore, the total number of atoms in a unit cell = 4 atoms.

Answer 2

To find:

Total number of atoms in:

• Simple cubic unit cell

• Body centred unit cell

• Face centred unit cell

Calculation:

SIMPLE CUBIC UNIT CELL:

1/8th contribution from 8 vertices.

Total number of atoms

$= 8 \times \dfrac{1}{8}$

$= 1 \: atom$

BODY CENTRED UNIT CELL:

1/8th contribution from 8 vertices and 1 central contribution.

Total number of atoms

$=( 8 \times \dfrac{1}{8} ) + 1$

$= 1 + 1$

$=2 \: atoms$

FACE CENTRED UNIT CELL:

1/2 contribution from 6 faces and 1/8th contribution from 8 vertices.

Total number of atoms

$= (6 \times \dfrac{1}{2} ) + (8 \times \dfrac{1}{8} )$

$= 3 + 1$

$= 4 \: atoms$

Refer to attached diagram.