When systems crystallize, they adopt the structure
whose configuration has the lowest energy crystal structure. In
ionic solids, this is mainly determined by the coulombic interaction
between the ions: a maximization of the attractive forces between
oppositely charged ions and a minimization of the repulsion between
like charged ions.
The arrangement of ions in a crystal is known as
the crystal structure.
The crystal structure is made up of a periodically
repeating structural motif known as the unit
cell. This is the basic component of the structure,
which, when repeated in three dimensions by translational moves
only, makes up the complete structure. There may be a choice of
unit cells for a given system, such that when repeated give the
same overall crystal structure. The one usually adopted as the
standard is the one which has the same symmetry as the crystal
as a whole.
The pattern of ions or atoms within the structure
is given by the lattice. A lattice
point need not be a particular ion, but may be a set of
ions or atoms, which can be repeated to give the structure. The
group of ions associated with a given lattice point are known
as the basis for the cell.
Thus if the lattice and the unit cell is known,
the entire structure may be elucidated.
In discussing the structures of metals and ionic
solids, we generally consider the ions to be spherical objects,
or a size given by the ionic radius of the ion, and some of the
simplest structures can be understood in terms of the best possible
packing arrangements of these spheres. These structures are the
ones in which the occupied volume is minimized, and lead to the
idea of close-packing of spheres.
Close Packed Structures
Close Packed structures
have the highest occupied volume and each sphere has the maximum
number of neighbours, or the highest coordination
number. In many metals, for example, the close packing
of spheres leads to there being twelve nearest
neighbours, and hence a coordination number of 12.
When a layer of close packed spheres is formed,
another layer may be placed on top, with each of the spheres in
the second layer falling into the basin made by three of the spheres
in the second row. However, when a third layer is placed on top
of the second layer, there are two different orientations of this
third layer with respect to the first layer. If the third layer
spheres are in position 1, then they are directly above
the spheres in the first layer, and this gives a layer structure
known as ABA (ie. the third
layer is the same as the first, but the second is different to
both). If the third layer is in position 2, then the spheres
of the third layer are over the holes in the first layer, and
this gives a layer structure known as ABC
(ie. all three layers are in different relative orientations).
|Close packing of spheres
|Close packing of two
layers of spheres. The two possible positions for the
spheres of the third layer are labeled 1 and 2
||Third layer spheres occupy
positions 1. This gives ABA packing.
||Third layer spheres occupy
positions 2. This gives ABC packing.
The ABA packing is also known as hexagonal
close packing, and the ABC packing is also known as cubic
packing of spheres: 3D structures
|hexagonal close packing
||cubic close packing
|The ABA arrangement of spheres
is shown in three dimensions.
||The ABC arrangement of atoms
is shown, rotated relative to the hcp view. The different
colours show the A, B, and C layers.
|The hcp structure shown in
lattice form. The atoms in the second layer are at the
center of three of the triangular prisms.
||The ccp structure shown in
lattice form. There are atoms at the 8 corners of the
cube, and also in the centers of the 6 faces of the cube.
If one considers the lattice form of the ccp structure
shown, we can see that the unit cell contains 8 atoms
at the corners, and 6 atoms in the faces, specifically one in
the center of each face. This structure is also known as the
face centered cubic structure (fcc).
It should be noted that the fcc unit cell, as shown, contains
This is based on the following calculation, where
Na is the number of ions in position a, and
na share is the number of unit cells where
share a given position a.
It can be seen, however, that when generating
structures by packing spheres, although the spheres may be close
packed, there is still empty space within the structures. This
empty space may be categorized in terms of holes
within the structure.
Holes in close packed structures
When we place one layer of spheres on top of another,
there are two possible orientations. The hole between layers in
position 2 on the diagram above is in the first orientation shown
below, and the hole in position 1 on the diagram above is in the
second orientation shown below.
|The hole is surrounded by
six spheres, and is known as an octahedral
||This hole is surrounded by
four spheres, and is known as a tetrahedral
In the three dimensional lattice, these octahedral
and tetrahedral holes occupy characteristic positions in the
unit cell. These are shown for the fcc lattice.
of holes in the fcc lattice
holes are shown as black, and the fcc array of spheres
There is one octahedral hole for each atom in
the fcc lattice.
The holes are shown as black, and the fcc array of
spheres as red.
There are two tetrahedral holes for each atom
within the fcc lattice.