Let us consider the
packing of fruits for display in fruit stalls. They are in a closest packed
arrangement as shown in the following fig. we can extend this analogy to
visualize the packing of constituents (atoms / ions / molecules) in crystals,
by treating them as hard spheres. To maximize the attractive forces between the
constituents, they generally tend to pack together as close as possible to each
other. In this portion we discuss how to pack identical spheres to create cubic
and hexagonal unit cell. Before moving on to these three dimensional
arrangements, let us first consider the two dimensional arrangement of spheres
for better understanding.
In a specific direction,
there is only one possibility to arrange the spheres in one direction as shown
in the fig. in this arrangement each sphere is in contact with two neighbouring
spheres on either side.
Two dimensional planar
packing can be done in the following two different ways.
Linear arrangement of spheres
in one direction is repeated in two dimension i.e., more number of rows can be
generated identical to the one dimensional arrangement such that all spheres of
different rows align vertically as well as horizontally as shown in the fig. If
we denote the first row as A type arrangement, then the above mentioned packing
is called AAA type, because all rows are identical as the first one. In this
arrangement each sphere is in contact with four of its neighbours.
In this type, the second
row spheres are arranged in such a way that they fit in the depression of the
first row as shown in the figure. The second row is denoted as B type.
The third row is
arranged similar to the first row A, and the fourth one is arranged similar to
second one. i.e., the pattern is repeated as ABAB….In this arrangement each
sphere is in contact with 6 of its neighbouring spheres.
On comparing these two
arrangements (AAAA...type and ABAB….type) we found that the closest arrangement
This type of three
dimensional packing arrangements can be obtained by repeating the type two
dimensional arrangements in three dimensions. i.e., spheres in one layer sitting
directly on the top of those in the previous layer so that all layers are
identical. All spheres of different layers of crystal are perfectly aligned
horizontally and also vertically, so that any unit cell of such arrangement as
simple cubic structure as shown in fig.
In simple cubic packing,
each sphere is in contact with 6 neighbouring spheres - Four in its own layer,
one above and one below and hence the coordination number of the sphere in
simple cubic arrangement is 6.
There is some free space
between the spheres of a single layer and the spheres of successive layers. The
percentage of total volume occupied by these constituent spheres gives the
packing efficiency of an arrangement. Let us calculate the packing efficiency
in simple cubic arrangement,
Let us consider a cube
with an edge length ‘a’ as shown in fig. Volume of the cube with edge length a
is = a × a × a = a3
Let ‘r’ is the radius of
the sphere. From the figure, a=2r ⇒ r = a/2
∴ Volume of the sphere
with radius ‘r’
In a simple cubic
arrangement, number of spheres belongs to a unit cell is equal to one
i.e., only 52.31% of the
available volume is occupied by the spheres in simple cubic packing, making
inefficient use of available space and hence minimizing the attractive forces.
In this arrangement, the
spheres in the first layer ( A type ) are slightly separated and the second
layer is formed by arranging the spheres in the depressions between the spheres
in layer A as shown in figure. The third layer is a repeat of the first. This
pattern ABABAB is repeated throughout the crystal. In this arrangement, each
sphere has a coordination number of 8, four neighbors in the layer above and
four in the layer below.
Here, the spheres are touching
along the leading diagonal of the cube as shown in the fig.
Number of spheres belong
to a unit cell in bcc arrangement is equal to two and hence the total volume of
i.e., 68 % of the
available volume is occupied. The available space is used more efficiently than
in simple cubic packing.
In this arrangement, the
first layer is formed by arranging the spheres as in the case of two
dimensional ABAB arrangements i.e. the spheres of second row fit into the
depression of first row. Now designate this first layer as ‘a’. The next layer
is formed by placing the spheres in the depressions of the first layer. Let the
second layer be ‘b’.
In the first layer (a)
there are two types of voids (or holes) and they are designated as x and y. The
second layer can be formed by placing the spheres either on the depression
(voids/holes) x (or) on y. let us consider the formation of second layer by
placing the spheres on the depression (x).
Wherever a sphere of
second layer is above the void (x) of the first layer (a), a tetrahedral void
is formed. This constitutes four spheres – three in the lower (a) and one in
the upper layer (b). When the centers of these four spheres are joined, a
tetrahedron is formed.
At the same time, the
voids (y) in the first layer (a) are partially covered by the spheres of layer
(b), now such a void in (a) is called a octahedral void. This constitutes six
spheres – three in the lower layer (a) and three in the upper layer (b). When
the centers of these six spheres are joined, an octahedron is formed.
Simultaneously new tetrahedral voids (or holes) are also created by three
spheres in second layer (b) and one sphere of first layer (a)
The third layer of
spheres can be formed in two ways to achieve closest packing
(i) aba arrangement -
(ii) abc arrangement –
The spheres can be
arranged so as to fit into the depression in such a way that the third layer is
directly over a first layer as shown in the figure. This “aba’’ arrangement is
known as the hexagonal close packed (hcp) arrangement. In this arrangement, the
tetrahedral voids of the second layer are covered by the spheres of the third
Alternatively, the third
layer may be placed over the second layer in such a way that all the spheres of
the third layer fit in octahedral voids. This arrangement of the third layer is
different from other two layers (a) and (b), and hence, the third layer is
designated (c). If the stacking of layers is continued in abcabcabc… pattern,
then the arrangement is called cubic close packed (ccp) structure.
In both hcp and ccp
arrangements, the coordination number of each sphere is 12 – six neighbouring
spheres in its own layer, three spheres in the layer above and three sphere in
the layer below. This is the most efficient packing.
The cubic close packing
is based on the face centered cubic unit cell. Let us calculate the packing
efficiency in fcc unit cell.
Total number of spheres
belongs to a single fcc unit cell is 4
The structure of an
ionic compound depends upon the stoichiometry and the size of the
ions.generally in ionic crystals the bigger anions are present in the close
packed arrangements and the cations occupy the voids. The ratio of radius of
cation and anion (rC/rA) plays an important role in
determining the structure. The following table shows the relation between the
radius ratio and the structural arrangement in ionic solids.