Day 10

Atomic Packing Arrangements

Anions are generally larger than cations, and the outer orbitals of electrons about most anions are s-orbitals. For these reasons, it is convenient to describe and view many atomic structures as consisting of identical spheres packed in some arrangement.

There are many examples in nature and the world around us where similar objects pack and arrange to fill a particular volume. One trip through a grocery store illustrates some simple packing schemes of atoms.

 Simple Cubic Packing, or 'Body Centered Cubic Packing', results when each layer of atoms has a regular, square lattice arrangement. Alternate layers are translated along a body diagonal (see the body-centered isometric lattice, Fig. 3.17-13, p. 126).

 Closest packing structures result when layers of atoms are arranged in a hexagonal pattern. In closest packing arrangements, each atom ( or grapefruit, in the upper right box) is in closest-contact with six others in the layer. (Imagine that each layer extends indefinitely, and count how many grapefruit are arranged about each individual fruit). Closest packing arrangements lead to higher density structures, as the spaces between atoms are smaller.

  There are two closest packing arrangements - Cubic Closest Packing (CCP) and Hexagonal Closest Packing (HCP). In hexagonal closest packing, every other layer is identical, and thus the upward 'repeat' of layers in a structure is ABABABAB.... (As in the two layers at left).

In cubic closest packing, it is every third layer that is identical, and thus the repeat is ABCABCABCABC.... Considering the stacks of grapefruit above, the third layer would come into a position different than the first two.

In both HCP and CCP, each sphere is in contact with tweleve others in the three-dimensional structure.

  The arrangements of spaces, or voids, in HCP and CCP is fundamentally different. You can see the spaces by comparing the figure above with the one at the left. A space with six corners is empty above, and filled by a Key lime, at left. This is an octahedral void in the closest-packing arrangement of grapefruit, and the lime is octahedrally coordinated with respect to the surrounding grapefruit.

Tetrahedral voids, those with four corners and which are smaller, are also are present throughout the structure. In the photos above, each grapefruit sits on top of three in the layer below, thus surrounding a space with four corners. (Herein lies another way to 'see' differences between octahedral and tetrahedral sites, viewed from above - you can see the octahedral sites, but the tetrahedral ones are 'hidden' by the top layer.)

The Key lime above would not fit into one of the tetrahedral sites. However, a smaller round fruit (maybe a muscadine grape) would fit into a tetrahedral site. Thus the sizes of voids is dependent on the number of atoms arranged around the void. Similarly, the size of speres that can be placed into the void depends on the relative size of the void. Therefore, polyhedra of higher coordination number accomodate larger atoms, as indicated in the table below.

Oxygen is, by far and by any measure, Earth's most abundant atom. Therefore, when considering values of radius ratio and the way cations fill atomic structures, it is most important to consider the radius ratio of cations to oxygen. Table 4.9 in the text tabulates the radius ratio and typical CN for various cations and oxygen.

Not all of the possible sites in an atomic structure will, necessarily, be filled. In order for a mineral to be stable, the total number of ions must be such that the whole structure is electrically neutral.

Local Charge Balance: CaF2, CN=8. Ca+2, 8 F at -1/4 each (cubic coordination). Work out examples for SiO4 (tetrahedral coordination), and CO3 (in triangular coordination).

Pauling's Rules: