Chapter 12, Problem 12.110QP

A quantitative measure of how efficiently spheres pack into unit cells is called packing efficiency, which is the percentage of the cell space occupied by the spheres. Calculate the packing efficiencies of a simple cubic cell, a body-centered cubic cell, and a face-centered cubic cell. (Hint: Refer to Figure 12 .21 and use the relationship that the volume of a sphere is πr³, where r is the radius of the sphere.)

Interpretation Introduction

Interpretation:

The packing efficiency of simple cubic unit cell, body-centered cubic unit cell and face-centered cubic unit cell has to be determined.

Concept Introduction:

The simplest and basic unit of a crystalline solid is known as unit cell.  It is cubic in shape.  It is the building block of crystalline solids.  The unit cells repeat themselves to build a lattice.  Crystalline solids consist of many of such lattices.  There are three types of unit cell – simple cubic unit cell, body – centered cubic unit cell and face – centered cubic unit cell.

In packing of the components in a solid, the components are imagined as spheres.  Close packing of atoms refers to the packing of atoms with most possible minimal space between them.

A simple cubic unit cell is the simplest form of a cubic unit cell.  A cube has eight vertices, twelve edges and six faces.  Similarly a cubic unit cell has eight vertices, twelve edges and six faces.  If in a cubic unit cell, the components occupy only the eight vertices, then the unit cell is known as simple cubic unit cell.  So, each simple cubic unit cell has $\frac{\text{1}}{\text{8}}\text{th}$ of an atom at each vertex.  Thus the number of atoms per simple cubic unit cell is –

$\text{8}\text{vertices}\text{×}\frac{\text{1}}{\text{8}}\text{th}\text{of}\text{an}\text{atom}\text{=}\text{1}\text{atom}$

The edge length of simple cubic unit cell is represented by the formula “ $\text{a}\text{=}\text{2r}$”.  The length of body diagonal is $\sqrt{3}l$.

In a body – centered cubic unit cell is another type of unit cell in which atoms are arranged in all the eight vertices of the unit cell with one atom per vertex. Further one atom occupies the center of the cube.  Thus the number of atoms per unit cell in BCC unit cell is,

$\text{8}\text{×}\frac{\text{1}}{\text{8}}\text{atoms}\text{in}\text{corners}\text{+}\text{1}\text{atom}\text{at}\text{the}\text{center}\text{=}\text{1}\text{+}\text{1}\text{=}\text{2}\text{atoms}$

$\begin{array}{l}\text{The edge length of BCC unit cell is given by}\\ \text{a}\text{=}\frac{\text{4r}}{\sqrt{\text{3}}}\\ \text{where a}\text{=}\text{edge length of unit cell}\\ \text{r}\text{=}\text{radius}\text{of}\text{atom}\end{array}$

In a face – centered cubic unit cell the atoms are arranged in all the eight vertices of the unit cell with one atom per vertex. Further all the six faces of a cubic unit cell are occupied with one atom per face.  Thus the number of atoms per unit cell in FCC unit cell is,

$\text{8}\text{×}\frac{\text{1}}{\text{8}}\text{atoms}\text{in}\text{corners}\text{+}\text{6}\text{×}\frac{\text{1}}{\text{2}}\text{atoms}\text{in}\text{faces}\text{=}\text{1}\text{+}\text{3}\text{=}\text{4}\text{atoms}$

$\begin{array}{l}\text{The edge length of one FCC unit cell is given by}\\ \text{a}\text{=}\text{2r}\sqrt{\text{2}}\\ \text{where a}\text{=}\text{edge length of unit cell}\\ \text{r}\text{=}\text{radius}\text{of}\text{atom}\end{array}$

The measure of efficiency of packing of atoms in solid is termed as packing efficiency. It is represented as follows –

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

Atoms are considered as spheres and so its volume is equivalent to $\frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}$. Volume of cubic unit cell is equivalent to ${\text{a}}^3$.

Answer to Problem 12.110QP

• The packing efficiency of simple cubic unit cell is $\text{52}\text{.4\%}$.
• The packing efficiency of body-centered cubic unit cell is $\text{68}\text{.0\%}$.
• The packing efficiency of face-centered cubic unit cell is $\text{74}\text{.0\%}$.

Explanation of Solution

Each simple cubic unit cell has one whole atom.  As atoms are considered as spheres the volume of atom is equivalent to that of the volume of sphere. Volume of the unit cell is equivalent to cubic value of the edge length of unit cell.  Dividing these two values gives packing efficiency of the simple cubic unit cell.

Determine the packing efficiency of simple cubic unit cell.

$\begin{array}{l}\text{edge length of simple cubic unit cell, a = 2r}\\ \end{array}$

$\begin{array}{l}\text{volume of atoms in unit cell = }\frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}\\ {\text{volume of unit cell, a}}^{\text{3}}{\text{ = (2r)}}^{\text{3}}\\ \text{packing efficiency = }\frac{\text{volume of atoms in unit cell}}{\text{volume of unitcell}}\text{ × 100\%}\\ \text{ = }\frac{{\text{4πr}}^{\text{3}}}{{\text{(2r)}}^{\text{3}}}\times \text{ 100\% = 52}\text{.4\%}\\ \end{array}$

Each BCC unit cell has two atoms.

$\begin{array}{l}\text{edge length of BCC unit cell, a = }\frac{\text{4r}}{\sqrt{\text{3}}}\\ \text{ volume of atoms in unit cell = }\frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}\\ {\text{ volume of unit cell, a}}^{\text{3}}\text{ = }{\left(\frac{\text{4r}}{\sqrt{\text{3}}}\right)}^{\text{3}}\\ \text{ packing}\text{efficiency = }\frac{2\times \frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}}{{\left(\frac{\text{4r}}{\sqrt{\text{3}}}\right)}^{\text{3}}}\times \text{ 100\% = 68\%}\\ \end{array}$

$\begin{array}{l}\text{volume of atoms in unit cell = }\frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}\\ {\text{volume of unit cell, a}}^{\text{3}}{\text{ = (2r)}}^{\text{3}}\\ \text{packing efficiency = }\frac{\text{volume of atoms in unit cell}}{\text{volume of unitcell}}\text{ × 100\%}\\ \text{ = }\frac{{\text{4πr}}^{\text{3}}}{{\text{(2r)}}^{\text{3}}}\times \text{ 100\% = 52}\text{.4\%}\\ \end{array}$

Each BCC unit cell has two atoms.  As atoms are considered as spheres the volume of atom is equivalent to that of the volume of sphere.  Volume of the unit cell is equivalent to cubic value of the edge length of unit cell.  Since two atoms are present in a BCC unit cell, two times the volume of atoms in unit cell is considered.  Dividing these two values gives packing efficiency of the simple cubic unit cell.

Each FCC unit cell has four atoms.

$\begin{array}{l}\text{edge length of FCC unit cell, a = 2r}\sqrt{2}\\ \text{ volume of atoms in unit cell = }\frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}\\ {\text{ volume of unit cell, a}}^{\text{3}}\text{ = }{\left(\text{2r}\sqrt{2}\right)}^{\text{3}}\\ \text{ packing}\text{efficiency = }\frac{4\times \frac{\text{4}}{\text{3}}{\text{πr}}^{\text{3}}}{{\left(\text{2r}\sqrt{2}\right)}^{\text{3}}}\times \text{ 100\% = 74\%}\\ \end{array}$

Each FCC unit cell has four atoms.  As atoms are considered as spheres the volume of atom is equivalent to that of the volume of sphere.  Volume of the unit cell is equivalent to cubic value of the edge length of unit cell.  Since four atoms are present in a BCC unit cell, four times the volume of atoms in unit cell is considered.  Dividing these two values gives packing efficiency of the simple cubic unit cell.

Conclusion

The packing efficiency of simple cubic unit cell, body-centered cubic unit cell and face-centered cubic unit cell has been determined.