CAcT HomePage
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 = 4/sqrt(3) R
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 hcp | bcc | simple 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
©cchieh@uwaterloo.ca