Face Centered Cubic Unit Cell - Online Tutor, Practice Problems & Exam Prep

The face-centered cubic (FCC) unit cell is a fundamental structure in crystallography featuring atoms at each corner of a cube and centered on each face. This arrangement results in four atoms per unit cell: one from the corners and three from the face centers. The edge length of an FCC unit cell is expressed as:

$2\sqrt{2}r$

where r is the atomic radius. This structure is characterized by a high packing efficiency of 74% and a coordination number of 12, indicating each atom is in contact with 12 others. Understanding the FCC unit cell is crucial in material science, as it influences the properties of crystalline substances.

Face Centered Cubic Unit Cell Concept 1

The face centered cubic unit cell is composed of a cube with an atom in each corner and three atoms on the faces. Now if we took all of this into consideration, that means that a face centered cubic unit cell has a total of 4 atoms per one unit cell and again three of them are in the center.

If we take a look here at the image, remember each one of these red spheres is shared by neighboring unit cells. Each one represents 1/8 of an atom. And since there are eight corners, 18 * 8 means that this unit cell, one atom of it comes from the corners. And then here we have these six spheres in the center. This unit cell shares half of each of those spheres, right? So we'd say here, because of that, that gives us another three atoms. And that's how we come up with our total of 4 total atoms for our face centered cubic unit cell.

Now the edge length. Because there's a small little gap here, our edge length will not be just simply 2R becomes now 2 * sqrt(2) * R. And remember, as our complexity increases, both our packing efficiency as well as our coordination number will increase. Here our packing efficiency becomes 74% and our coordination number now is 12.

Now, these are the important ideas and key values associated with any type of face centered cubic unit cell, so keep them in mind anytime we're dealing with this particular type of unit cells.

Face Centered Cubic Unit Cell Example 1

Silver adopts a face center cubic unit cell structure illustrate how the number of atoms per unit cell for that silver atom are obtained. All right, so remember we have eight corners to our queue, so each one is 1/8 of an atom, and we have eight corners involved, so 18 × 8 = 1 atom.

Also remember that on each face we have half of a sphere, and there are six faces faces to a cube, so that's times 6. So that's going to give us 3 atoms when we add them together. That explains why a face center cubic unit saw is 4 atoms per one unit cell.

So this is our atoms 2 unit cell ratio. It's a four to one ratio.

Aluminum has a face-centered cubic unit structure and a density of 2.716 g/cm3. Calculate the edge length of the unit cell.

• A. 4.041×10^–8 cm
• B. 3.992×10^–8 cm
• C. 3.615×10^–8 cm
• D. 3.247×10^–8 cm
• E. 2.836×10^–8 cm

An element crystallizes in a face-centered cubic lattice and has a density of 18.44 g/cm³. The edge of its unit cell is 1.05×10^-8 cm. Calculate the atomic mass for the element.

• A. 3.21 g•mol^-1
• B. 15.5 g•mol^-1
• C. 60.8 g•mol^-1
• D. 151 g•mol^-1

Here’s what students ask on this topic:

What is the three-dimensional coordination number of the face-centered cubic arrangement?

The three-dimensional coordination number of a face-centered cubic (FCC) arrangement, also known as a cubic close-packed structure, is 12. This means that each atom in an FCC crystal structure is surrounded by 12 other atoms. To visualize this, consider that in an FCC lattice, an atom at the corner of the cube is surrounded by four atoms in the same layer (one on each face adjacent to the corner), four atoms in the layer above, and four atoms in the layer below. These neighboring atoms are all equidistant from the atom in question, forming a coordination geometry that maximizes packing efficiency and minimizes empty space in the structure. This high coordination number is one of the reasons why FCC metals typically have high density and good ductility.

What is the net number of face-centered atoms contained in a face-centered cubic unit cell?

In a face-centered cubic (FCC) unit cell, there are atoms located at each of the corners and the centers of all the faces of the cube. While it might seem like there are 6 face-centered atoms (one on each face), we have to consider that each of these face-centered atoms is shared with an adjacent unit cell.

Since each face is shared by two unit cells, only half of each face-centered atom is actually within one unit cell. Therefore, each of the 6 face-centered atoms contributes 1/2 of an atom to the unit cell, resulting in a total contribution of 3 atoms (6 x 1/2 = 3) from the face-centered positions.

So, the net number of face-centered atoms contained in a single face-centered cubic unit cell is 3.