Multiple choice

A small sphere of radius $10\ \mathring{A}$ was found to fit perfectly in the largest void of simple cubic arrangement. Find the volume (in $\mathring A ^3$) of the unit cell.

  1. $1000(\sqrt{3}-1)^3$
  2. $1000(\sqrt{3}+1)^3$
  3. $1000\begin{pmatrix}\sqrt{\displaystyle\frac{3}{2}}+1\end{pmatrix}^3$
  4. $1000\begin{pmatrix}\sqrt{\displaystyle\frac{3}{2}}-1\end{pmatrix}^3$
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B Correct answer
AI explanation

In a simple cubic arrangement, the largest void fits into the body diagonal, so the body diagonal length is a + 2r where a is the cube side and r is the sphere radius. Since r is 10, we have a(3^(1/2)) = a + 20, which gives a = 20 / (3^(1/2) - 1) = 10(3^(1/2) + 1). The volume of the unit cell is a^3, which evaluates to 1000(3^(1/2) + 1)^3.

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