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If the radius of the octahedral void is \(r\) and radius of the atoms in close-packing is \(R\), derive relation between \(r\) and R.
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Verified Answer
A sphere is fitted into the octahedral void as shown in the diagram.
\(\triangle \mathrm{ABC}\) is a right angled triangle.
\(\therefore \quad \mathrm{BC}^2=\mathrm{AB}^2+\mathrm{AC}^2\)
\((2 R)^2=(R+r)^2+(R+r)^2\)
\((2 R)^2=2(R+r)^2\)
\(\Rightarrow \quad \frac{(2 R)^2}{2}=(R+r)^2\)
\((\sqrt{2} R)^2=(R+r)^2\)
\(\Rightarrow \quad \sqrt{2} R=R+r\)
\(r=\sqrt{2} k-R\)
\(r=R(\sqrt{2}-1)\)
\(r=R(1.414-1)\)
\(r=0.414 \mathrm{R}\)
\(\triangle \mathrm{ABC}\) is a right angled triangle.
\(\therefore \quad \mathrm{BC}^2=\mathrm{AB}^2+\mathrm{AC}^2\)
\((2 R)^2=(R+r)^2+(R+r)^2\)
\((2 R)^2=2(R+r)^2\)
\(\Rightarrow \quad \frac{(2 R)^2}{2}=(R+r)^2\)
\((\sqrt{2} R)^2=(R+r)^2\)
\(\Rightarrow \quad \sqrt{2} R=R+r\)
\(r=\sqrt{2} k-R\)
\(r=R(\sqrt{2}-1)\)
\(r=R(1.414-1)\)
\(r=0.414 \mathrm{R}\)
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