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(C) $(i)$ For $SC$,the body diagonal length is $\sqrt{3}a$. The atoms touch along the edge,so $2r = a$. The fraction of the body diagonal occupied by

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Packing fraction along the face diagonal in $fcc$ is $\frac{1}{\sqrt{2}}$

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(C) $(i)$ For $SC$,the body diagonal length is $\sqrt{3}a$. The atoms touch along the edge,so $2r = a$. The fraction of the body diagonal occupied by atoms is $\frac{2r}{\sqrt{3}a} = \frac{a}{\sqrt{3}a} = \frac{1}{\sqrt{3}}$. This is correct.$(ii)$ For $bcc$,the body diagonal is $\sqrt{3}a = 4r$,so $r = \frac{\sqrt{3}a}{4}$. The edge length is $a$. The fraction of the edge occupied is $\frac{2r}{a} = \frac{2(\sqrt{3}a/4)}{a} = \frac{\sqrt{3}}{2}$. This is correct.$(iii)$ For $fcc$,the face diagonal is $\sqrt{2}a = 4r$,so $r = \frac{\sqrt{2}a}{4}$. The face diagonal length is $\sqrt{2}a$. The fraction of the face diagonal occupied is $\frac{4r}{\sqrt{2}a} = \frac{4(\sqrt{2}a/4)}{\sqrt{2}a} = 1$. The given option states $\frac{1}{\sqrt{2}}$,which is incorrect.$(iv)$ The $3-D$ packing efficiency order is $fcc$ $(74\%)$ > $bcc$ $(68\%)$ > $pc$ $(52\%)$. This is correct.

Select the $INCORRECT$ option regarding the cubic crystal system-

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