Primitive (Simple) Cubic

• As its name suggests, this is the simplest type of lattice point arrangement in a crystalline solid. Picture a cube with a sphere at each corner. Only one-eighth of each sphere is contained within the cube; 7/8 of each sphere is outside the cube. This is a simple cubic structure. It is the least efficient packing structure; only 52 percent of the unit cell is filled. The element Polonium exhibits this type of packing structure.
  
  

Body-centered Cubic

• The body-centered cubic structure is the same as the primitive (simple) cubic structure but with the addition of a single atom in the center of the unit. Whereas the primitive structure contains only one atom (8 x 1/8 = 1), the body-centered packing structure contains two atoms: one complete atom in the center, and one-eighth of each of the eight atoms at the corners of the unit. Common table salt (NaCl) exhibits this type of crystal lattice structure. It is the second most efficient packing structure, as 68 percent of the unit cell is filled.
  
  

Face-centered Cubic

• This is the most efficient crystalline solid structure for packing and a little more difficult to visualize. Imagine a primitive cubic structure with a single atom in the center of each face of the cube. Half of each centered atom projects into the unit cube; half of each projects out. These six halves add up to three atoms (6 x 1/2 = 3). Add these six halves to the single atom already contained in the corners of the primitive structure (8 x 1/8 = 1) and you have four atoms in the face-centered cubic packing structure. This structure is also called cubic close packed; it has a packing efficiency of about 74 percent. Calcium fluoride exhibits this structure.
  
  

Packing Efficiency

• You can calculate the packing efficiency of a given lattice structure if you know the volume of the atoms contained in the unit structure and the volume of the unit structure itself. For example, if the volume of an atom is assumed to be equivalent to the volume of a sphere (or 4/3pi * r^3), and the area of the unit cube itself is two times the radius of one of the corner atoms of the unit cubed (or (2r)^3), you can write this equation for the packing efficiency of a primitive cubic structure: [(8 x 1/8) x (4/3 pi r^3)]/[(2r)^3] x 100. This equation simplifies to pi/6 x 100. If pi equals approximately 3.14, pi/6 x 100 = 52.3. The packing efficiency of the primitive cubic lattice structure is approximately 52 percent.