# 3. Metal Packing: Three Dimensions

Note that this cube has an atom in the center of each face; hence the name, face-centered cubic.

• A cubic close-packed takes the name "cubic" from the presence of a cubic unit cell.
• This cubic unit cell is oriented diagonally to the close-packed layers.
• This cubic unit cell has an atom in the center of each face; it is also called "face-centered cubic".

## Problems

### 3.8

Take a look at your new cubic close-packed unit cell.

1. What fraction of an atom is found in each corner of the cubic close-packed cubic unit cell?
2. How many corners are there in the cell?
3. Are there atoms anywhere else in the cell?
4. What is the total number of atoms in the cubic close-packed unit cell?

### 3.1.9.

We saw previously that the number of near neighbors of a metal atom is called the coordination number. What is the coordination number in each of the following cases?

1. an atom at the corner of a simple cubic cell. Remember, it is surrounded by other unit cells, too.
2. an atom in the center of a body-centered cubic cell.
3. an atom in the corner of a body-centered cubic cell.
4. an atom in the corner of a face-centered cubic cell.
5. an atom on the face of a face-centered cubic cell.

### 3.1.10.

Coordination geometry is related to the coordination number. What is the coordination geometry in each of the following cases?

1. an atom at the corner of a simple cubic cell. Remember, it is surrounded by other unit cells, too.
2. an atom in the center of a body-centered cubic cell.
3. an atom in the corner of a body-centered cubic cell.
4. an atom in the corner of a face-centered cubic cell.
5. an atom on the face of a face-centered cubic cell.

### 3.1.11.

We learned earlier about the holes or interstitial spaces between atoms in a layer. There are also holes between atoms in a three dimensional structure. For example, how would you describe the shape of the holes in the following cases:

1. the hole in the middle of a simple cubic cell.
2. a hole between the central atom and the face of a body-centered cubic cell.
3. the hole right in the middle of the face of a body-centered cubic cell.
4. the hole between the atoms forming a valley in one hexagonal layer and the atom sitting in the valley.
5. the hole between the atoms forming an empty valley in one hexagonal layer and the atoms in the layer above.

### 3.1.12.

Packing efficiency is often determined in terms of the percentage of the volume of a unit cell that is actually occupied by atoms.

In the following cases, calculate the volume of the entire unit cell.

1. a simple cubic unit cell.
2. a body-centered cubic unit cell.
3. a face-centered cubic unit cell.
4. a hexagonal close-packed unit cell (remember, this cell is a rhombic prism, not a cube).

You can assume the atoms in the cell are titanium. Titanium has an atomic radius of 2.00 Angstroms (or 2.00 x 10-10 m).

### 3.1.13.

How many atoms are there in a unit cell in the following layers?

1. a simple cubic unit cell.
2. a body-centered cubic unit cell.
3. a face-centered cubic unit cell.
4. a hexagonal close-packed unit cell (remember, this cell is a rhombic prism, not a cube).

You may need to add up fractions of titanium atoms to arrive at the answer. The answer may or may not be a whole number.

### 3.1.14.

What is the volume occupied by a titanium atom?

### 3.1.15.

In the following cases, what percentage of the unit cell would be filled with titanium atoms?

1. a simple cubic unit cell.
2. a body-centered cubic unit cell.
3. a face-centered cubic unit cell.
4. a hexagonal close-packed unit cell.

The percentage of unit cell that is occupied by atoms is called "the packing efficiency".