Question

Calculate the packing efficiency in a Body Centered Cubic (BCC) lattice?

Answer 1

Hint: Packing efficiency gives how efficiently the atoms in a crystal lattice are packed. To calculate it, we must know the volume of each unit cell. Volume of a unit cell is $\dfrac{8}{3}\pi {{r}^{3}}$ where r is the radius of an atom.


Complete step by step solution:
For Body Centered Cubic lattice(BCC), the relationship between the radius r and and the edge length a of a unit cell is given as
\[a=\dfrac{4r}{\sqrt{3}}\]
Now, we have to find the volume of a unit cell which is edge length raised to three times.
\[{{a}^{3}}={{(\dfrac{4r}{\sqrt{3}})}^{3}}=\dfrac{64{{r}^{3}}}{3\sqrt{3}}\]
We know that the number of atoms per unit cell in a body centered unit cell is 2.
The volume of a sphere is \[=\dfrac{4}{3}\pi {{r}^{3}}\]
So, volume of each unit cell will be \[=2\times \dfrac{4}{3}\pi {{r}^{3}}\]\[=\dfrac{8}{3}\pi {{r}^{3}}\]
$\text{Packing Efficiency=}\dfrac{\text{Volume occupied by the atoms in a unit cell}}{\text{total volume of unit cell}}\times 100$
Substituting the values in the above equation we get
Packing efficiency \[=\dfrac{\dfrac{8}{3}\pi {{r}^{3}}}{\dfrac{64{{r}^{3}}}{3\sqrt{3}}}\times 100=68.04%\]
Thus, the packing efficiency of Body Centered Cubic lattice is equal to 68.04%.


Additional Information: Constituent particles are arranged in different patterns in a unit cell. No matter how we arrange the particles, there will be some vacant space and those are called voids. With the help of packing efficiency, the quantitative aspect of solid state can be done. Mathematically it is the volume occupied by atoms in a unit cell divided by the total volume of the unit cell, the whole multiplied by 100. Packing efficiency is always calculated in percentage. It can be also defined as the percentage of space occupied by the constituent particles packed in a lattice. It depends on the volume of the cell, the number of atoms in a structure and volume of atoms.

Note: Packing efficiency depends on the number of atoms in the unit cell. For a body centered cubic system, the number of atoms in a unit cell is two. For an fcc arrangement it is 4.