- Draw a diagram of body centred cubic (bcc) packing

- Identify the bcc unit cell

- Analyze the geometry of the bcc unit cell

- Calculate geometric parameters of the bcc packing

In a crystal structure, the arrangement extends over millions and millions of atoms, and the above diagram shows the unit cell, the smallest unit that, when repeatedly stacked together, will generate the entire structure.

Actually, the unit we draw is more than a unit cell. We use the centre of the atoms (or spheres) to represent the corners of the unit cell, and each of these atoms are shared by 8 unit cells. There is a whole atom located in the centre of the unit cell.

Usually, the length of the cell edge is represented by *a*.
The direction from a corner of a cube to the farthest corner is called
body diagonal (*bd*). The face diagonal (*fd*) is a line drawn
from one vertex to the opposite corner of the same face. If the edge is
*a*, then we have:

=

= 3

Atoms along the body diagonal (*bd*) touch each other. Thus, the body
diagonal has a length that is four times the radius of the atom, *R*.

or

**
Example 1
**

*Solution*

Well, this means you should calculate the percentage of the volume occupied
by the spheres in the packing.

= 2*4/3 p

= Ö3 p / 8 = 0.6802

*Discussion*

The packing fractions are

fcc and hcp | bcc | simple cubic |
---|---|---|

74.05 % | 68.01 % | 52% |

**How many atoms are there per unit cell for the bcc structure model?****Discussion -**

Recall the number of atoms per unit cell in a primitive cubic packing.

Only 1/8 th of the atom in the corner belong to the cell, and there are 8 such atoms. The one in the centre of the cell entirely belong to the unit cell.**If the radius of the atoms is***R*, how many times*R*is equal to the body diagonal*bd*of the unit cell?**Discussion -**

The atoms touch each other along the body diagonal directions.**If the edge of the cube has a length represented by***a*, what is the face diagonal of the cube as a multiple of*a*?**Discussion -**

*fd*^{2}=*a*^{2}+*a*^{2}= 2*a*^{2}**If the edge of the cube has a length represented by***a*, what is the body diagonal of the cube as a multiple of*a*?**Discussion -**

*bd*^{2}=*fd*^{2}+*a*^{2}

=*a*^{2}+*a*^{2}+*a*^{2}

= 3*a*^{2}**If the edge of the cube has a length represented by***a*, What is the volume of the cell in terms of*a*?**Discussion -**

Area of face =*a*^{2}**In the bcc packing, atoms along the body diagonal (***bd*) touch each other. If the atomic radius is*R*, express*bd*in terms of*R*.**Discussion -**

O O | If we cut the cube along a face diagonal, the arrangement of O | atoms look like the diagram on the left. O O | Thus, bd = 4 R

**If the edge of the cube has a length represented by***a*, and if the radius of the atoms is*R*, Express*a*as a function of*R*.**Discussion -**

*a*=^{4 R}/_{Ö3}