A cubic lattice has $A$ atoms at the body centre, $B$ atoms at the corners, $C$ atoms at half of the face…
A cubic lattice has $A$ atoms at the body centre, $B$ atoms at the corners, $C$ atoms at half of the face centres. The formula of the lattice is
$A B C_2$
$A B_2 C_4$
$A_2 B_2 C_3$
$A B C_3$
Solution
$A$ atoms are present at body centre.
$\therefore$ Number of atoms $=1$
Contribution of body centre $=1$
$\therefore$ Effective number of atoms $A=1 \times 1=1$
$B$ atoms are present at cornres.
$\therefore$ Number of atoms at cornres $=8$
Contribution of corner $=\frac{1}{8}$
$\therefore$ Effective number of $B=8 \times \frac{1}{8}=1$
$C$ atoms are present at half of face centres.
$\therefore$ Number of atoms $=\frac{6}{2}=3$
$\therefore$ Contribution of face center $=\frac{1}{2}$
$\therefore$ Effective number of $C=3 \times \frac{1}{2}=\frac{3}{2}$
Hence, formula of the lattice as : $A_1 B_1 C_{\frac{3}{2}}$
or $A_2 B_2 C_3$