Question

Predict the general formula for the given unit cell.

• A. $X_2{Y}_2{W}_{2}Z$
• B. $X_3{Y}_3{W}_{2}Z$
• C. $XYWZ$
• D. $X_3{Y}_2{W}_{2}Z$

Answer 1

The correct option is D $X_3{Y}_2{W}_{2}Z$

$\text{Contribution of a corner particle is}=\frac{1}{8}$
$\text{Contribution of a face centre particle is}=\frac{1}{2}$
$\text{Contribution of a edge centre particle is}=\frac{1}{4}$
$\text{Contribution of a body centre particle is}=1$

In the given unit cell,
Atom Z is present only at the body centre
Thus,
$\text{No. of Z atom per unit cell}=1$

Atom X occupies 8 corners of the cube and 4 face centres of the cube.
$\therefore$
$\text{No. of X atoms per unit cell}=\frac{1}{8}\times 8+\frac{1}{2}\times 4=3$
Atom Y occupies 8 edge centres of the unit cell.
$\therefore$
$\text{No. of Y atoms per unit cell}=\frac{1}{4}\times 8=2$

Atom W occupies 2 face centre and 4 edge centres.
$\therefore$
$\text{No. of W atoms per unit cell}=\frac{1}{4}\times 4+\frac{1}{2}\times 2=2$

Hence, the formula of unit cell is
$X_3{Y}_2{W}_{2}Z$