## Face-Centered Cubic Unit Cell

### Face-Centered Cubic Unit Cell

Silver crystallizes in a cubic closest-packed structure. The unit cell for this structure is cubic (all sides of the same length and all face perpendicular to each other) with a silver atom at each corner of the unit cell and a silver atom in the center of each face of the unit cell. This unit cell is also called a face-centered cubic (fcc) unit cell.

The unit cell completely describes the structure of the solid, which can be regarded as an almost endless repetition of the unit cell.

The volume of the unit cell is readily calculated from its shape and dimensions. This calculation is particularly easy for a unit cell that is cubic. The dimensions of the unit cell are dictated by the size of the atoms in the cell. In the case of the fcc unit cell the atoms lying along the diagonal of each face are in contact with each other. Thus the diagonal of the face of the unit cell has a length of 4r, where r is the radius of a silver atom.

Atoms, of course, do not have well-defined bounds, thus the radius of an atom is somewhat ambiguous. In the context of crystal structures, the diameter (2r) of an atom can be defined as the center-to-center distance between two atoms packed as tightly together as possible. This provides a type of effective radius for the atom and is sometime called the atomic radius.

A more challenging task is to determine the number of atoms that lie in the unit cell. As described above, an atom is centered on each corner and in the middle of each face of the fcc unit cell. In this case, however, none of these atoms lies completely within the cell. Part of each atom lies within the unit cell and the remainder lies outside the unit cell. In determining the number of atoms inside the unit cell, one must count only that portion of an atom that actually lies within the unit cell.

The density of a solid is the mass of all the atoms in the unit cell divided by the volume of the unit cell.

### Properties of Silver

The virtual reality image below illustrates the face-centered cubic unit cell, which is the unit cell that describes the structure of silver metal. The positions of the individual silver nuclei are shown by small dots. The silver atoms or sections of silver atoms are shown by the spheres or sphere sections.

The atomic mass of silver is 107.8682 and the length of a side of the unit cell is 4.07 angstroms.

Use this structure to answer the following questions.

1. What is the volume of the unit cell?

2. What is the atomic radius of a silver atom?

3. What is the volume of a silver atom (based upon the atomic radius)?

4. How many silver atoms are contained in the unit cell?

5. What fraction of the volume of the unit cell is "occupied" by silver atoms?

6. What is the density (g/cm3) of silver metal?

The animation controls at the left of the display perform the following operations.

 Reset Resets the animation to the unit cell containing the various sections of silver atoms. Show/Hide Sides Shows or hides the solid sides of the unit cell. Clear Unit Cell Shows only the outline of the unit cell. Show Sections Shows the sections of the silver atoms that lie within the unit cell. Show Atoms Shows the silver atoms that lie in the unit cell (no atom lies completely within the unit cell in this case). View Sections The sections of silver atoms that lie within the unit cell are spread out for display and examination. The user may step through this process. Form Atoms The sections of silver atoms that lie within the unit cell are pieced together to form whole atoms.

To run through the animation in sequence,

1. Select Show Sections to display the portions of the atoms in the unit cell.
2. Select View Sections or step through this sequence to spread the atom sections across the display for viewing.
3. Select Form Atoms or step through this sequence to assemble the atom sections into full atoms.

 Face-Centered CubicUnit Cell

Unit Cells:     Simple Cubic     Body-Centered Cubic     Face-Centered Cubic     Hexagonal Closest-Packed

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