Body-centered cubic packing calculations

BODY-CENTERED CUBIC PACKING

Early in the planning of the body-centered cubic packing animations, a literature search was done and the unit cell found for that packing was the one shown on Figure BCC1.

Figure BCC1 - Body-centered cubic packing unit cell found in the literature.

Because the unit cell of Figure BCC1 was found in several sources, there would be no reason to suspect that it was not correct. In addition, this unit cell forms a crystal lattice, as shown in the animation on Link BCC1.

Link BCC1 - Crystal lattice formed from the body-centered cubic packing unit cell found in the literature.

What caught our attention was the tetragonal geometry of the unit cell, which should be cubic because the name of the packing is body-centered cubic. If the name assigned to the unit cell is correct, the unit cell of Figure BCC1 is incorrect. On the other hand, if the unit cell is correct, the name of the packing is incorrect.

To solve this problem, we assumed that all ions in a crystal lattice must have the same coordination number, i.e., all ions are at an equal and minimal distance from any coordinator ion of the crystal lattice.

Inspecting the unit cell of the body-centered cubic packing found in the literature, we see that the coordination number of the ion in the center of the unit cell is equal to 8, as shown in the animation on Link BCC2.

Link BCC2 - Coordination number of the ion located in the center of the body-center cubic packing unit cell found in the literature.

Let's now inspect the coordination number of an ion located in the vertices of the unit cell. Let's suppose that the coordinator ion is located at vertex "A" in Figure BCC2.

Figure BCC2 - Parameters of the unit cell of the body-centered cubic packing found in the literature.

The nearest neighbors of a coordinator ion located at vertex "A" are the ions located at vertices "B" and "D". In order to an ion located at "P" be also coordinated to an ion located at "A", it is necessary that AP=AB=AD. The following calculations show that this condition can be mathematically satisfied.

Considering the result in equation (BCC10) and the results in equations (BCC02) and (BCC06), we see that the unit cell is tetragonal and that the coordination number of the ion located at "A" is equal to 12, as shown in the animation on Link BCC3.

Link BCC3 - Coordination number of an ion located at one of the vertices of the body-centered packing unit cell found in the literature.

As we have seen, the body-centered cubic packing found in the literature has two different coordination numbers, what means that the unit cell found in the literature is not correct.

Let's now consider the cubic unit cell shown on Figure BCC3. If the coordination number of any ion in the crystal lattice is unique, condition that must be satisfied for metals, then this is the correct unit cell.

Figure BCC3 - Parameters of a cubic unit cell.

Inspecting Figure BCC3, we see that the coordination number of the ion located in the center of the unit cell is equal to 8, as shown in the animation on Link BCC4.

Link BCC4 - Coordination number of an ion located in the center of a cubic unit cell.

We need yet verify the coordination number of the ions located in the vertices of the unit cell. Let's choose as a coordinator ion, the one located at vertex "A" in Figure BCC3. We have to calculate the distance between that ion and all its neighboring ions to discover which ones are located at an equal and minimum distance. The following calculations show the ions that are coordinated to the ion located at vertex "A".

Calculations show that the nearest neighboring ions of the ion located at vertex "A" are those located in the center of eight adjacent unit cells, what means that the coordination number of the ion located at vertex "A" is also equal to 8, as shown in the animation on Link BCC5.

Link BCC5 - Coordination number of an ion located at one vertex of the body-centered cubic packing unit cell.

We can then conclude that the unit cell of the body-centered cubic packing is cubic and not tetragonal, as usually found in the literature.

Let's see how 3D Chemistry animations may show us that metals are made of cations and not made of neutral atoms. If we model an iron unit cell with neutral atoms, we will see that spheres intersect one each other, as shown in the animation on Link BCC6. We must note that spheres should be pushed away to not intersect one each other, what would lead the unit cell of Figure BCC1.

Link BCC6 - 3D animations show that metals could not be made of neutral atoms.

Modeling unit cells with neutral atoms was, probably, the source of the mistake of those who created a tetragonal unit cell for the body-centered cubic packing. The question that we must do is: do we have to worry about the intersection of spheres in a cubic unit cell or not? The answer for this question is: spheres, which represent neutral atoms, have no physical meaning because metals are made of cations and so, an unit cell of metals should always be represented by cations.