Question

The ratio of number of atoms present in a simple cubic, body centered cubic and face centered cubic structure, respectively, are :

• A. $8:1:6$
• B. $1:2:4$
• C. $4:2:1$
• D. $4:2:3$

Answer 1

The correct option is B $1:2:4$

Number of atoms present in Simple Cubic Cell:
Here, $8$ atoms are present in every corner of a simple cubic unit cell. Every single atom present at the corner contributes only $\frac{1}{8}$ to the unit cell.
$\therefore$ , total contribution = total no. of atoms $\times$ contribution by one atom
= $8\times \frac{1}{8}=1$ atom per unit cell

Number of atoms present in Body Centered Cubic cell:
Here, $8$ atoms are present in every corner of the unit cell and 1 atom is present at the body center of the unit cell. Every single atom present at the corner contributes only $\frac{1}{8}$ to the unit cell while atom at the body center contributes as a complete $1$ atom to the unit cell.
$\therefore$Total contribution = (total no. of atoms at corner $\times$ contribution by one atom at corner) + (total no. of atoms at body center $\times$ contribution by atom at body center)
= $8\times \frac{1}{8}+1\times 1$
= $2$ atoms per unit cell

No. of atoms present in Face Centered Cubic cell:
Here, $8$ atoms are present in every corner of the unit cell and $6$ atoms are present at the face center of the unit cell. Every single atom present at the corner contributes only $\frac{1}{8}$ to the unit cell and every atom present at the face contributes $\frac{1}{2}$ to the total unit cell.
Total contribution = (total no. of atoms at corner $\times$ contribution by one atom at corner) + (total no. of atoms at face center $\times$ contribution by atom at face center)
= $8\times \frac{1}{8}+6\times \frac{1}{2}$
= $1+3$
= $4$ atoms per unit cell

⇒ Ratio of atoms in simple cubic : body center: face center = $1:2:4$