# Crystal structures

Entry by Emily Redston, AP 225, Fall 2011

Crystalline materials are characterized by repeating (or periodic) arrays of atoms over large atomic distances. Typically we think about these materials in the context of solids, though there are also liquid crystals. Crystallization is the process by which crystal structures form; atoms will position themselves in a repetitive three-dimensional structure upon solidification. The system is said to display long-range order. If a solid does not have long-range order, it is considered amorphous. The crystal structure of a material is the unique way in which atoms, ions, or molecules are spatially arranged. There is an extremely large number of different crystal structures that all have long-range order. Some of these are relatively simple structures (e.g. metals) while others can be exceedingly complex (e.g. ceramic and polymeric materials). When describing crystal structures, it is useful to think about the atoms as solid spheres having well-defined radii. We call this the atomic hard sphere model, where spheres representing nearest neighbor atoms touch one another. The term lattice is used to describe a three-dimensional array of points that coincide with atom positions.

In describing crystal structures, it is often convenient to sub-divide the structure into small repeat entities called unit cells. We typically choose a unit cell such that it represents the symmetry of the crystal structure. The unit cell is a small structure that repeats itself by translation through the crystal; if the unit cells are stacked together in three-dimensions, they describe the bulk arrangement of atoms of the crystal. Thus the unit cell is the basis structural unit of the crystal structure, and it defines the crystal structure by its geometry and the atom positions within. The unit cell is described by its lattice parameters, which are the length(s) of the cell edges and the angles between them.

Figure 1 – The hcp lattice (left) and the fcc lattice (right). The outline of each respective Bravais lattice is shown in red. The letters indicate which layers are the same. There are two "A" layers in the hcp matrix, where all the spheres are in the same position. All three layers in the fcc stack are different. Note the fcc stacking may be converted to the hcp stacking by translation of the upper-most sphere, as shown by the dashed outline. (http://en.wikipedia.org/wiki/Close-packing_of_equal_spheres)

Many crystal structures are based on a close-packing of atoms, which we describe using hard spheres. Figure 1 shows two of the most common crystal structures: face-centered cubic (fcc) and hexagonal close-packed (hcp). These structures are especially relevant to colloidal packing, and are important since they have the highest possible packing density. Both structures begin from the hexagonal sheet of atoms we see in the figure, but they differ in how we stack these sheets on top of one another. Let's call these positions of this first layer A. We can see that there are two possible positions to fit another sheet on top of this one called B and C.

In both the fcc and hcp arrangements each sphere has twelve neighbors. For every sphere there is one gap surrounded by six spheres (octahedral) and two smaller gaps surrounded by four spheres (tetrahedral).

Relative to a reference layer with positioning A, two more positionings B and C are possible. Every sequence of A, B, and C without immediate repetition of the same one is possible and gives an equally dense packing for spheres of a given radius.

The most regular ones are:

   fcc = ABCABCA (every third layer is the same)
hcp = ABABABA (every other layer is the same)


In close-packing, the center-to-center spacing of spheres in the x–y plane is a simple honeycomb-like tessellation with a pitch (distance between sphere centers) of one sphere diameter. The distance between sphere centers, projected on the z (vertical) axis, is:

   \text{pitch}_Z = \sqrt{6} \cdot {d\over 3}\approx0.81649658 d,


where d is the diameter of a sphere; this follows from the tetrahedral arrangement of close-packed spheres.

The coordination number of hcp and fcc is 12 and its atomic packing factor (APF) is the number mentioned above, 0.74.

## References

[1] Callister, William D. Materials Science and Engineering: an Introduction. New York: John Wiley & Sons, 2007.