Geos 306, Fall 2007, Lecture 5, The Packing of Atoms

Geos 306, Fall 2007, Lecture 5
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 the atoms in the metal are spherical and have the same size since we assume that the metal is made of only one type of atom, eg Cu. Recall the electron density map of Cu and how spherical the atoms appeared.
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.
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.
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 assume 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. 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.

Two other important packing schemes are bcc, body-centered cubic, Eg Iron, and the diamond structure, C.

The Sizes of Atoms

Table of Radii
For a given element the ionic radius decreases with increasing positive charge and increases with increasing negative charge.
e.g. r(Pb^4-) = 2.15 Å, r(Pb⁰) = 1.74 Å, r(Pb²⁺) = 1.18 Å, r(Pb⁴⁺) = 0.78 Å
Radii of elements in the same vertical column of the Periodic Table with identical ionic charge increase in size with increasing atomic number.
e.g. r(Li¹⁺) = 0.74 Å, r(Na¹⁺) = 1.02 Å, r(K¹⁺) = 1.38 Å, r(Rb¹⁺) = 1.49 Å, r(Cs¹⁺) = 1.70 Å
Radii of ions with the same electronic configuration, but with increasing positive charge, decrease in size.
e.g. 1s² 2s² 2p⁶, r(O^2-) = 1.40 Å, r(F^1-) = 1.33 Å, r(Na¹⁺) = 1.02 Å, r(Mg²⁺) = 0.72 Å, r(Al³⁺) = 0.53 Å, r(Si⁴⁺) = 0.40 Å
Anions are usually larger than cations
e.g. r(O^2-) = 1.40 Å, r(Cl^1-) = 1.81 Å

Reading:
Wenk and Bulakh, Chapter 2.
Klein, p 64-75
Nesse, Chapter 4.