The face-centered cubic unit cell is a cube (all sides of the same length and all face perpendicular to each other) with an atom at each corner of the unit cell and an atom situated in the middle of each face of the 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. In the case of the face-centered cubic unit cell, the atoms lying along the diagonal of each face are in contact with each other. Thus the diagonal of each face has a length of 4 r, where r is the radius of an atom.
Atoms, of course, do not have well-defined bounds, and the radius of an atom is somewhat ambiguous. In the context of crystal structures, the diameter (2 r) of an atom can be defined as the center-to-center distance between two atoms packed as tightly together as possible. This provides an 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 unit cell. None of these atoms lies completely within the unit cell. Each atom exists partially inside the unit cell and partially 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.
Silver crystallizes in a cubic closest-packed structure. The unit cell for the cubic closest-packed structure is the face-centered cubic cell, which is illustrated in the virtual reality display shown below. 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. Consult the Description of Controls or simply experiment with the features of the display.
The atomic mass of silver is 107.8682 and the length of a side of the unit cell is 4.07 Å.
Use the face-centered cubic unit cell to answer the following questions.
- What is the volume of the unit cell?
- What is the atomic radius of a silver atom?
- What is the volume of a silver atom (based upon the atomic radius)?
- How many silver atoms are contained in the unit cell?
- What fraction of the volume of the unit cell is "occupied" by silver atoms? (This fraction is the packing efficiency.)
- What is the density (g/cm3) of metallic silver?
This virtual reality display requires Java3D. If the display is not visible, consult the Java3D FAQ. Dragging an object with the left mouse button rotates the object. Dragging with the center mouse buttons expands the display, and dragging with the right mouse button moves the display.
Unit Cells: Simple Cubic Body-Centered Cubic Face-Centered Cubic Hexagonal Closest-Packed