Close Packed Structures
(a)
Hexagonal and Cubic Close Packing
The structures of many metals can be described as close packed
arrays of spherical atoms. There are two alternative ways to
maximize the packing efficiency of a collection of equally sized
spheres.
Cubic Close Packing (CCP) ABCABC
Hexagonal Close Packing (HCP) - ABABAB
Fm3m
P63/mmc
The figure below illustrates the two packing sequences. The
spheres of the second layer can be centered over either the sites
marked b or c. The symmetry is exactly the same regardless of
the choice, however, when a third layer is added it now has a
choice of packing over the spheres in the original layer or over
the sites marked as c in the diagram of the first layer. The
former option leads to the ABAB... stacking characteristic of
hexagonal close packing, while the latter option leads to the
ABCABC... packing which defines a cubic close packed structure.
The packing efficiency (volume of space occupied by the
spheres/total volume) is 74.05 % for both cubic and hexagonal
close packing schemes. The number of (12) and distance between
nearest neighbors is also identical for both HCP and CCP, but the
second nearest neighbor coordination is different.
In addition to these two schemes a fair number of metals adopt a
body centered cubic arrangement. In this structure type the
number of nearest neighbors has been reduced from 12 to 8.
This does not result in a close packed array of spheres, and the
packing efficiency drops to 68.02%.
(b)
Crystallographic Unit Cells
Now that we are familiar with cubic and hexagonal close packing,
let us turn to their crystallographic descriptions.
Cubic close packing leads to a structure with a face centered
cubic unit cell as shown below. For this reason cubic close packing
is sometimes called face centered cubic (fcc) packing. This
structure represents the simplest of all structures based on a
face centered cubic Bravais lattice, one with a single atom as the
basis (asymmetric unit).
Space Group = Fm3m
Composition = M
Atom
M
site x
4a 0
y
0
z
0
It is not immediately obvious how the fcc unit cell is related to
ccp stacking described in the previous section. To see the
relationship begin by finding the 3-fold rotation axes
perpendicular to the layers in the ABC… stacking scheme. They
should be fairly easy to spot. Next consider that the 3-fold axes
run along the body diagonal in a cubic unit cell. Combining these
two we see that the layers from the ccp stacking are
perpendicular to the body diagonal of the fcc unit cell. This
relationship is highlighted in the figure below where the atoms in
layer A are shaded green, those in layer B are blue, and those in
layer C are red.
The unit cell and crystallographic description corresponding to
hexagonal close packing are shown below.
Space Group = P63/mmc
Composition = M
Atom
M
site x
2a 0
y
0
z
0
Here the relationship between the crystallographic unit cell and
the layer stacking is easily seen (atoms in layer A are shaded blue
and those in layer B are shaded red in the figures above). The
atoms in layer A are located at the vertices of the hexagonal unit
cell, while one atom from layer B is contained in each unit cell at
1/3, 2/3, 1/2.
(c)
Body Centered Cubic Packing
Another highly symmetric packing arrangement, sometimes,
adopted by metals, is the body centered cubic structure shown
below.
Space Group = Im3m
Composition = M
Atom
M
Site x
2a 0
y
0
z
0
Unlike ccp and hcp arrangements each atom in the bcc
arrangement has 8 nearest neighbors (rather than 12).
Consequently, the packing efficiency is lower (as we shall see in
the next section) and body centered cubic packing cannot be
considered a close packed arrangement.
(d)
Packing Efficiency and Theoretical Density
The packing efficiency of a structure can be calculated if we
assume the atoms are hard spheres. In that case the packing
efficiency is given by
 = (volume occupied by spheres) / (total volume)
This calculation is carried out on the contents of the unit cell.
Therefore, our first step is to count the number of atoms
contained within the unit cell. To do this you should keep in mind
that atoms located at the corners, edges and faces of the unit
cell are not fully contained in a single unit cell, so that they only
partially occupy the unit cell.
Atom Location
Corner
Edge
Face
Anywhere else
Fraction Inside Unit Cell*
1/8
1/4
1/2
1
*For unit cells with non-orthogonal axes it is not strictly true
that an atom on a corner (edge) is exactly 1/8 (1/4) inside the
unit cell, but it is still true that 8 (4) such atoms add up to one
atom in the unit cell.
Now we can determine the number of atoms inside the unit cell
for each of the packing arrangements we have discussed.
Face Centered Cubic (ccp)  (8 ×1/8) + (6 ×1/2) = 4
Body Centered Cubic (bcc)  (8 ×1/8) + 1 = 2
The volume of occupied space contained within the unit cell can
now be easily calculated by multiplying the number of atoms per
unit cell by the volume of a sphere.
V = 4/3  r3
The next step is to determine the size of the unit cell as a
function of the atomic radius, r:
Face Centered Cubic  4r = (2)1/2/a
Body Centered Cubic  4r = (3)1/2/a
Face Centered Cubic  volume = a3 = [4r/(2)1/2]3 = 16(2)1/2 r3
Body Centered Cubic  volume = a3 = [4r/(3)1/2]3 = 64/[3(3)1/2]r3
Now we can easily calculate the packing efficiency:
Face Centered Cubic  4[4/3 r3] / [16(2)1/2 r3]
=  / (3(2)1/2) = 0.7405
= 74.05%
Body Centered Cubic  2[4/3 r3] / [64/(3(3)1/2)r3]
=  (3)1/2/8 = 0.6802
= 68.02%
As promised the packing efficiency of the bcc structure is lower
than the fcc structure. With a little bit more work it can be
shown that the hcp structure has the exact same packing density
as the fcc arrangement.
From this point with a relatively easy extension of our methods,
it is possible to calculate the theoretical density of a compound
from its structure. Since density is a mass per unit volume we
need to calculate the total atomic weight of all atoms in the unit
cell then divide by the volume of the unit cell.
Example
Silver crystallizes with a fcc structure and a lattice parameter
a=4.086 Å. What is its theoretical density?
Density = mass/volume
= 4 (107.87 g/mol)(1 mol/6.0221023 atoms)/
(4.08610-8 cm)3
= 10.50 g/cm3
(e)
Body Centered Cubic Packing
Since the number and arrangement of nearest neighbors are
identical in ccp and hcp, it is reasonable to expect that the free
energies of these two structures would be very similar. Thus it
should not come as a surprise that both structure types (along
with bcc) are commonly observed. Furthermore, the addition of
small levels of impurities can lead to a change in the structure
type. This phenomena has important technological implications in
several areas, including making steel from iron.
Cubic close packed,
a
Cu 3.6147
Ag 4.0857
Au 4.0783
Al 4.0495
Ni 3.5240
Pd 3.8907
Pt 3.9239
Pb 4.9502
(f)
Hexagonal close
packed, a, c
Be 2.2856, 3.5832
Mg 3.2094, 5.2105
Zn 2.6649, 4.9468
Cd 2.9788, 5.6167
Ti 2.506, 4.6788
Zr 3.312, 5.1477
Ru 2.7058, 4.2816
Os 2.7353, 4.3191
Re 2.760, 4.458
Body centered
cubic, a
Fe 2.8664
Cr 2.8846
Mo 3.1469
W 3.1650
Ta 3.3026
Ba 5.019
Ordered Structures
Several structures can be described as ordered derivatives of
the fcc, bcc and hcp structures. Examples of this include the
CsCl structure, which is an ordered derivative of the bcc
structure, as well as the Cu3Au and CuAu structures, which are
ordered derivatives of the fcc structure.
Note that in each of these structures the symmetry is lower than
in the parent structure. This is a consequence of the fact that
two atoms which would be equivalent in an elemental compound,
are no longer equivalent in an ordered compound (i.e., the atoms
at 0,0,0 and ½, ½, ½ in CsCl). Thus the symmetry operation which
related these two atoms in the parent structure (i.e., body
centering translation) is destroyed, and a reduction in symmetry
results.