Chapter 11, Problem 136AP

Interpretation Introduction

Interpretation:

The packing efficiencies of a simple cubic, body-centered, and face-centered cell are to be calculated

Concept introduction:

A crystal lattice is made of small repeating unit cells.

A unit cell is of primitive or centered type. The spheres in a unit cell can represent an atom, a molecule, or an ion.

Each sphere is shared by adjacent cells. A sphere at corners is shared by eight unit cells. A sphere in the center of each face is shared by two unit cells.

The volume of the unit cell is calculated as:

$V=a^{\text{3}}$;

Here, $a$

is the edge length.

The formula to calculate the volume of a sphere is as:

$V=\frac{\text{4}}{\text{3}}\text{π}{r}^{\text{3}}$

Here, $r$

is the radius of the sphere.

Packing efficiency is the ratio of the volume of atoms in a unit cell to the volume of the unit cell. Its expression is:

$\text{Packing efficiency}=\frac{\text{volume of atoms in unit cell}}{\text{volume of unit cell}}\times 100\%$

Answer to Problem 136AP

Solution: $52.4\%$; $68.0\%$;$74\%$

Explanation of Solution

Given information: For SCC, $a=2r$; for BCC, $a=\frac{4r}{\sqrt{3}}$; for FCC, $a=\sqrt{8r}$.

In a simple cubic cell, spheres are present only at the eight corners of the unit cell.

The contribution of 8 spheres at the corners to one unit cell is:

$\frac{1}{8}\times 8=1$

Thus, one sphere is present in the unit cell.

The formula to calculate the volume of a unit cell is as follows:

$V=a^{\text{3}}$ ……(1)

Substitute $2r$

for $a$

in the above expression as:

$\begin{array}{c}V={\left(2r\right)}^{\text{3}}\\ =8r^3\end{array}$

The volume of a sphere is $\frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}$

The formula to calculate packing efficiency is as:

$\text{Packing efficiency}=\frac{\text{volume of atoms in unit cell}}{\text{volume of unit cell}}\times 100\%$ …… (2)

Substitute $\frac{\text{4}}{\text{3}}\text{π}{r}^{\text{3}}$

for the volume of atoms in the unit cell and $8r^3$

for the volume of the unit cell in the above expression as:

$\begin{array}{c}\text{Packing efficiency}=\frac{\frac{\text{4}}{\text{3}}\text{π}\cancel{r^{\text{3}}}}{\text{8}\cancel{r^{\text{3}}}}\times 100\%\\ =52.4\%\end{array}$

In BCC, there are spheres at the eight corners of the unit cell and one sphere is present in the center of the unit cell.

The contribution of 8 spheres at the corners to one unit cell is:

$\frac{1}{8}\times 8=1$

One sphere is in the center of the unit cell.

Thus, two spheres are present in a body centered unit cell.

The volume of a sphere is $\frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}$.

Thus, the volume of two spheres will be as:

$\begin{array}{c}V=2\times \frac{\text{4}}{\text{3}}\text{π}{r}^{\text{3}}\\ =\frac{8}{3}\text{π}{r}^{\text{3}}\end{array}$

Substitute $\frac{4r}{\sqrt{3}}$ for $a$

in equation (1) as:

$\begin{array}{c}V={\left(\frac{4r}{\sqrt{3}}\right)}^3\\ =\frac{\text{64}{r}^{\text{3}}}{3\sqrt{3}}\end{array}$

Substitute $\frac{8}{\text{3}}\text{π}{r}^{\text{3}}$

for the volume of atoms in the unit cell and $\frac{64r^3}{3\sqrt{3}}$

for the volume of the unit cell in equation (2) as follows:

$\begin{array}{c}\text{Packing efficiency}=\frac{\frac{\text{8}}{\text{3}}\text{π}\cancel{r^{\text{3}}}}{\frac{\text{64}}{\text{3}\sqrt{\text{3}}}\cancel{r^{\text{3}}}}\times 100\%\\ =\frac{\text{8}}{\text{3}}\text{π}\times \frac{\text{3}\sqrt{\text{3}}}{\text{64}}\times 100\%\\ =67.9\%\\ \approx 68.0\%\end{array}$

In FCC cubic cell, there are spheres at the eight corners of the unit cell and one sphere in the center of six faces.

The contribution of $8$ spheres at the corners to one-unit cell is:

$\frac{1}{8}\times 8=1$

The contribution of $6$

spheres present in the center of each face to one-unit cell is:

$\frac{1}{2}\times 6=3$

Thus, four spheres are present in a face-centered unit cell.

The volume of a sphere is $\frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}$.

Thus, the volume of four spheres is:

$4\times \frac{\text{4}}{\text{3}}\text{π}{r}^{\text{3}}=\frac{16}{3}\text{π}{r}^{\text{3}}$

Substitute $\sqrt{8r}$

for $a$

in equation (1):

$V\text{=}{\left(\sqrt{8r}\right)}^{\text{3}}$

Substitute $\frac{16}{3}\text{π}{r}^{\text{3}}$

for the volume of atoms in the unit cell and ${\left(\sqrt{8r}\right)}^{\text{3}}$

for the volume of the unit cell in equation (2) as follows:

$\begin{array}{c}\text{Packing efficiency}=\frac{\frac{\text{16}}{\text{3}}\text{π}{r}^{\text{3}}}{{\left(\sqrt{8r}\right)}^{\text{3}}}\times 100\%\\ =74\%\end{array}$

Conclusion

The packing efficiencies of simple cubic, body-centered, and face-centered cell are $52.4\%$, $68.0\%,$

and $74\%$, respectively.