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# The Model for Polonium Crystal Structure

### Skills to develop

• 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

# Body Centred Cubic Structure

At room temperatures, elements Li, Na, K, Rb, Ba, V, Cr and Fe have structures that can be described as body centre cubic (bcc) packing of spheres. The other two common ones are face centred cubic (fcc) and hexagonal closest (hcp) packing. This type of structure is shown by the diagram below.

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:

fd2 = a2 + a2 = 2 a2
bd2 = fd2 + a2
= a2 + a2 + a2
= 3 a2

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.

bd = 4 R The relationship between a and R can be worked out by the Pythagorean theorem:

(4 R)2 = 3 a2 Thus, 4 R = sqrt(3) a
or
a = 4R/sqrt(3)
Recognizing these relationships enable you to calculate parameters for this type of crystal. For example, one of the parameter is the packing fraction, the fraction of volume occupied by the spheres in the structure.

Example 1

What is the packing fraction for a body centred packed structue?

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

Packing fraction = Vsphere / Vunit cell
= 2*4/3 p R3 / (4/sqrt(3) R3)
= Ö3 p / 8 = 0.6802

Discussion
The packing fractions are
fcc and hcpbccsimple cubic
74.05 % 68.01 %52%

### Confidence Building Questions

• 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 -
fd2 = a2 + a2 = 2 a2

• 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 -
bd2 = fd2 + a2
= a2 + a2 + a2
= 3 a2

• 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 = a2

• 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