Question

Atoms A, B, C and D are present at corners, face centres, body centre and edge centres respectively. If atoms touching one of the 2-fold axis, 3-fold axis and 4-fold axis are removed, then total number of atoms remaining per unit cell is $X+\frac{1}{4}$ .The value of X is

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Answer 1

2-Fold axis is along diagonally opposite edge centres.So,
C removed= 1 and D removed = $\frac{2}{12}\times \frac{12}{4}=\frac{1}{2}$
3-fold axis is along the body diagonal.
So, A removed = $\frac{2}{8}\times \frac{8}{8}=\frac{1}{4}$
4-fold axis passes through opposite face centres.So,
B removed = $\frac{2}{6}\times \frac{6}{2}=1$
Originally the molecule is $A{B}_{3}C{D}_3$, after removal it will become $A_{\frac{3}{4}}{B}_2{D}_{\frac{5}{2}}$
Remaining atoms = $\frac{3}{4}+2+\frac{5}{2}$
= 5+$\frac{1}{4}$
So, the answer is 5.25.