The unit cell in a body centred cubic lattice is given in the figure. Each sphere has a radius, r and the cube has a side, a. What fraction of the total cube volume of empty?

Engineering

Chemistry

Voids or Interstitial Site and Radius Ratio

Question

The unit cell in a body centred cubic lattice is given in the figure. Each sphere has a radius, r and the cube has a side, a.
What fraction of the total cube volume of empty?

$1 - \frac{8}{3} π \frac{r^{3}}{a^{3}}$

$\frac{4}{3} \cdot π \frac{r^{3}}{a^{3}}$

$\frac{r}{a}$

$2 - \frac{4}{3} π \frac{r^{3}}{a^{3}}$

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Z_eff of BCC = 2

Volume occupied by atoms =

2 \times \frac{4}{3} {πr}^{3}

Volume of unit cell = a³

Packing fraction=

\frac{8}{3} π \frac{r^{3}}{a^{3}}

Fraction of empty unit cell =

1 - \frac{8}{3} π \frac{r^{3}}{a^{3}}

Answer: (A)

1 - \frac{8}{3} π \frac{r^{3}}{a^{3}}

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