Geos 596a, Spring 2006, Lecture 1, Closest Packing

Geos 596a, Spring 2006, Lecture 1
The Packing of Atoms

In this lecture we examine what happens when groups of atoms are brought together to form crystals.
We start with the packing of atoms in metals because they are the simplest. We assume that all atoms are spherical and have the same size since we assume that the metal is made of only one type of atom, eg Cu. Here is an image of the electron density map of Cu. Note how spherical the atoms appear.
There are 2 major packing models that collectively are called closest packing. The two models are cubic closest packing (ccp) and hexagonal closest packing (hcp).

ccp, with packing sequence abcabcabc eg. Cu
hcp, with packing sequence abababab eg. Mg

Put a layer of spheres (marbles) on a flat surface and push them together as tightly as you can, then you would find that each sphere came into contact with 6 other spheres, just like we saw in the Cu electron density map. We call this layer a monolayer and label it "layer a".

Next, we can add a 2^nd layer on top of the first. There are two places that we can add this layer. These places are equivalent, with the only difference being orientation. We call this layer b.

HCP. If a 3^rd layer is put directly over the 1^st layer, then we have the sequence aba. The 3^rd layer is called layer a because it is equivalent in position to the first layer. The sequence can be repeated over and over again, forming a pattern: abababab... and the result is a packing scheme that we call hexagonal closest packing, or hcp. It is called hcp because the arrangement has hexagonal symmetry.
Examine the structure of magnesium, an hcp metal.
CCP. If, instead, a 3^rd layer is placed over the last remaining set of holes then we call the sequence abc. The 3^rd layer is called layer c because it is in a new position, different from that of layer a or layer b. If this pattern is reapeated over and over again, then we have the sequence abcabcabc... and the result is a packing scheme that we call cubic closest packing, or ccp. This packing scheme displays cubic symmetry. The stacking direction of the closest packed layers corresponds to the body diagonal of a cube.
Examine the structure of copper, an ccp metal.
An examination of the images should lead you to believe that each closest-packed sphere is in contact with 12 other spheres.

Such packing schemes have voids or empty spaces where smaller spheres (atoms) can fit. In general, we observe that it is the anions that form the closest-packed arrays, with smaller cations found in the voids. This is a typical packing arrangement found in ionic or covalent minerals. There are spaces for 2, 3, 4 and 6 coordinated sites, as well as a 12 coordinated site if the cation substitutes for one of the closest-packed anions. The 2 (e.g. CO₂) and 3 (e.g. the CO₃group found in calcite, CaCO₃) coordinated sites can be found within a monolayer, the tetrahedral and octahedral sites (e.g. SiO₄ or MgO₆ groups that are found in forsterite, Mg₂SiO₄) require 2 layers. Note that both + and - tetrahedral sites exist, pointing up or pointing down, respectively. Each atom in the closest packed array is associated with 2 tetrahedral sites, a + and a -, so there are 2 tetrahedral sites per atom in a closest-packed array. There is one octahedral site per closest-packed atom. In the following images, the 2 and 3 coordinated sites are labeled 2 and 3 respectively, and the 4 and 6 coordinated sites are labeled T and O, respectively, (tetrahedral and octahedral). The tetrahedral site that is shown is one that points down.

Examine the structure of carbon dioxide. It represents a severely distorted closest-packed structure with cations in 2-fold coordination. Determine the stacking sequence, i.e. abab... or abcabc... or something else.

Examine the structure of calcite. It represents a distorted closest-packed structure with C in 3-fold coordination and Ca in 6-fold. Determine its stacking sequence.

Examine the structure of forsterite. It represents a distorted closest-packed structure with Si in 4-fold coordination and Mg in 6-fold. Determine its stacking sequence.

Examine the structure of jadeite. It represents a distorted closest-packed structure with Si in 4-fold coordination and Al and Na in 6-fold. Determine its stacking sequence.

Examine the structure of C2/m phlogopite.

Examine the structure of zinc sulfide. The crystal structure database has 4 different polytypes of ZnS. A polytype is defined by the IUCr as:

An element or compound is polytypic if it occurs in several different structural modifications, each of which may be regarded as built up by stacking layers of (nearly) identical structure and composition, and if the modifications differ only in their stacking sequence. Polytypism is a special case of polymorphism: the two-dimensional translations within the layers are (essentially) preserved whereas the lattice spacings normal to the layers vary between polytypes and are indicative of the stacking period. No such restrictions apply to polymorphism.

Examine the different polytypes of ZnS. Note that there are about 100 different polytypes of ZnS defined in the literature, and sphalerite is the special case of abcabc... stacking that is associated with cubic symmetry. Half the tetrahedral sites are filled.

Examine halite, ccp with all octahedral sites filled.

Examine fluorite, ccp with all tetrahedral sites filled. Note that the cations make up the closest-packed array.

Examine nickeline, hcp with all octahedral sites filled.

Other important closest-pcked minerals include chalcopyrite, corundum, spinel, perovskite, and rutile.

Homework: Examine the structure of hawthorneite, and determine its packing sequence.

Homework: Determine the number of 2-fold (6), 3-fold (3 in a monolayer), 4-fold (2), 6-fold (1), and 12-fold (1) sites in a closest-packed array per atom.

Other packing schemes

Body-centered cubic, bcc, e.g. iron, chromium.

Diamond structure, bcc, e.g. diamond, silicon.

Reading:

Wenk and Bulakh, Chapter 2.

Klein, p 64-75

Nesse, Chapter 4.

Putnis, Chapter 5.

Thompson, R.M., and Downs, R.T. (2001) Quantifying distortion from ideal closest-packing in a crystal structure with analysis and application. Acta Crystallographica B57, 119-127. ( pdf file )

Thompson, R.M., and Downs, R.T. (2001) The systematic generation of all nonequivalent closest-packed stacking sequences of length N using group theory. Acta Crystallographica, B57, 766-771. ( pdf file )