# 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. for metals) while others can be exceedingly complex (e.g. for 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 the three-dimensional array of points that coincide with the 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 conventional unit cell). 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 the atoms in the crystal. Thus the unit cell is the basic structural unit of the crystal structure. The unit cell is described by its lattice parameters, which are the lengths of the cell edges and the angles between them. Using these parameters, we can define 14 Bravais lattices in three-dimensional space that describe the geometric arrangement of the lattice points. The 32 crystallographic point groups describe the rotation and mirror symmetries of the unit cell. To describe the full symmetry of a crystal (including translational symmetry operations), we have to consider the 230 space groups. 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 the 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 because they have the highest possible packing density. Both structures begin from the hexagonal sheet of atoms we see in Figure 1, but they differ in how we stack these sheets on top of one another. Let's call the 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 (place atoms on downward-facing triangular spaces) and C (place atoms on upward-facing triangular spaces).

fcc = ABCABCA... stacking
hcp = ABABABA... stacking

Figure 2 shows a nice animation of how hcp is generated. In both fcc and hcp, each atom has twelve nearest neighbors, so we say that the coordination number is 12. The atomic packing factor (APF) of both structures is 0.74.