Geos 306 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
packing. The two models are cubic closest packing (ccp) and hexagonal
closest packing (hcp). The models were formalized by John Barlow in the 1800's. Barlow was an
Englishman who was funded by the French government to determine the most efficient way to pack
cannonballs on the deck of a warship.
ccp, with packing sequence abcabcabc eg. Cu, Au, Ag, Al, Ni, Pt
hcp, with packing sequence abababab eg. Mg, Be, Ti, Zr
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
Image from https://www.seas.upenn.edu/~chem101/sschem/metallicsolids.html
Next, we can add a 2nd 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 3rd layer is put directly over
the 1st layer, then we have the sequence aba.
The 3rd 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 3rd layer is placed over the
last remaining set of holes then we call the sequence abc. The 3rd 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
CO3 group found in
CaCO3) coordinated sites
can be found within a monolayer, the
sites (e.g. SiO4 or MgO6 groups that are found in
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 Fe, V, Cr, Mo, W, and the diamond structure, C, Si, Ge.
After a lecture a couple years ago, one of the students sent me this photo. He realized that closest-packing principles
can be extrapolated from cannon balls to beer cans. The photo shows how he got the maximum packing density of beer cans into a
crisper drawer in his fridge.
The Sizes of Atoms
Figure from Gibbs G V, Spackman M A, Boisen M B (1992) Bonded and promolecule radii for molecules and crystals. American Mineralogist 77, 741-750
- There is no "true" radius of the atom. There are only models,
and we usually choose the model that best illustrates the point that we want to make.
However, you should never mix the radii from different models.
- Table of Radii
- For a given element the ionic radius decreases with increasing positive charge and increases with increasing negative charge.
e.g. r(Pb4-) = 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(O2-) = 1.40 Å, r(F1-) = 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(O2-) = 1.40 Å, r(Cl1-) = 1.81 Å
Wenk and Bulakh, Chapter 2.
Klein, p 64-75
Nesse, Chapter 4.