Density of Cubic Crystals :
The density based on the structure can be calculated from the mass contained in a unit cell and its volume.
If N is the number of molecules per unit cubic cell of edge length a, then the mass and volume per unit cell are Mass =
NA
M N Volume = a3
Therefore, Density = volume
mass =
3NA
a NM
The value of N for the three cubic cells can be calculated as follows :
Primitive cubic cell : In a primitive cubic cell, atoms are present at the corners of the cube. There are eight corners of a cube and thus eight atoms are present at these corners. Now, any particular corner of the cube is actually shared amongst eight such cubic unit cells placed adjacent to one another. Thus, the contribution of the atom placed at one of the corners to the single cubic unit cell is 1/8. Since there are eight corners of a cube, the number of atoms associated with a single primitive unit cell is 8/8 = 1.
Body-centred cubic cell : In a body-centred cubic unit cell, besides atoms being present at the corners, there is one atom in the centre of the cube which belongs exclusively to this cubic unit cell. Therefore, number of atoms per unit cell are two.
Face-centred cubic cell : Here, atoms, besides being at the corners, are also present at the centre of the six faces. Each of these atoms is shared between two such unit cells. Thus, their contribution to the unit cell is 6/2 = 3 atoms, making a total of 4 atoms per cubic unit cell
Packing in a simple Cubic Lattice :
In a lattice of this type, the spheres are packed in the form of a square array by laying down a base of spheres and then piling upon the base other layers in such a way that each sphere is immediately above the other sphere, as shown in fig.
Packing in a simple cubic lattice
In this structure, each sphere is in contact with six nearest neighbours (four in the same base, one above and one below). The percentage of occupied volume in this structure can be calculate as follows:
The edge length a of the cube will be twice the radius of the sphere, i.e. a = 2r. Since in the primitive cubic lattice, there is only one sphere present in the unit lattice, the volume occupied by the sphere is
V =
3
4πr3 or V =
3 4π
3
2 a
The fraction of the total volume occupied by the sphere is
φ = 3
3
a 2 a 3
4
π
= 6
π = 0.5236
or 52.36 percent
Thus, the structure is relatively open since only 52.36% (π/6) of the total volume is occupied by the spheres. The remainder, i.e. 0.4764 of the total volume is empty space or void volume.
No crystalline element has been found to have this structure.
Closest Packing :
In closest packing arrangements, each sphere is in contact with the maximum possible number of nearest neighbours. Fig. shows a closest packed layer of spheres. Each sphere is surrounded by six nearest neighbours lying in the plane, three spheres Just above it and three below it, thus making the total number of nearest neighbours equal to twelve.
If the spheres are packed in the same plane, then just above these spheres
A B
A A A
B B B
C C C C
A A A
A A A
A
A
A
A
B B
Fig. (a) Closest packed layers of spheres B
C C C C
Physical Chemistry
Fundamentals SOLID STATE
KEY CONCEPT
A B A B A B A
B A C B A C B A Fig. (b) Two types of packing
there exist two different types of voids, pointing in different directions as shown in fig. (a). Thus, we can have three different types of locations as shown by A, B and C in fig. (a). Location A is occupied by the spheres while B and C are the two different types of voids. But because of the size of the spheres, both types of voids cannot be occupied simultaneously.
The third layer of closest-packed sphere can be formed in two different ways. If, for example, we choose to place the spheres of the second layer in B sites, one of the available sets of voids for the third layer will be directly above the spheres in the original layer. These are A sites. The other set of voids will be directly above the voids designated by C in the original layer.
Types of Packing :
Thus, two types of packing (fig. b) are possible
ABABA.... or ABCABC ....
We can have many other varieties of patterns such as ABCACB..., ABAC .... etc. But for many of the common substances that form closest-packed structures, one of the above two symmetrical arrangements is observed.
Hexagonal Closest Packed Structure :
The packing ABAB.... is known as a hexagonal closest-packed structure (HCP). The unit cell of shown in figure.
A B A
Exploded view Hexagonal closest -packed Unit cell formed by ABA packing
The fraction of the volume occupied in HCP can be calculated as described in the following.
The distance C/2 (in figure) is the distance between the layers A and B. This distance will be from the centre of a sphere to the plane of the three spheres
determined by reference to a face centred cubic lattice with unit cell length a. In such a lattice, the distance between closest-packed layers (Miller indices 111)is one –third of the body diagonal, i.e.
3a./3.
Thus, 2 C =
3 a 3
Layer A
Layer B
Layer A C/2
a 2r
Hexagonal closest-packed structure
Now, in the face-centred lattice spheres touch one another along the face diagonal. Thus, we have
4r = 2a or a =
2 4 r
With this, the distance C becomes C = 2
a
3
3 = 2
2 r 4 3
3 =
6 8 r
The hexagonal base consists of six equilateral triangles, each with side 2r and with an altitude of 2r sin 60º, i.e. 3r. Therefore,
Area of the base = 612
( )
3r(2r) = 6 3r2Volume of the prism =
(
6 3r2)
r
6
8 = 24 2 r3
Number of spheres belonging to this prism 3 spheres in B layers exclusively belong to this prism.
1 from the centre of the base. There are two spheres of this type and each is shared by two prisms.
2 from the corners. There are twelve such spheres and each is shared amongst six prisms of this type.
Thus, the total number of spheres is 6.
The fraction of volume of the prism actually occupied by the spheres is
3 3
r 2 24
3 r
6 4
π
= 6
2π = 0.7405
or 70.05 percent
Cubical Closest-Packed Structure
The packing ABCABC, .... is a cubical closest-packing (CCP) or face-centred cubic closest-packing. The fraction of volume occupied in CCP can be calculated as follows :
The radius of the sphere in terms of the unit length of the face-centred cube is given by
r =
4 a 2
since the sphere will be touching each other along the diagonal of the face of the cube.
In the face-centred cubic lattice, there are four spheres per unit cell. Therefore, fraction of volume occupied by the spheres is
3 3
a 4
a 2 3 4 4
π
= 6
2π = 0.7405
or 74.05 percent A
C
B
Exploded view Cubical closest-packed stricture Face-cented cubic unit cell
formed by ABCA packing A
out of all these packings, HCP and CCP are more common for uniform spheres.
In general, the packing fraction, i.e. fraction of volume occupied, is independent of the radius of the sphere and depends only on the nature of packing.
From the values of packing fractions, it follows that the density of a substance in HCP and CCP structures will be more than in the other two packings.
Packing in a Body Centred Cubic Lattice :
Here the packing consists of a base of spheres, followed by a second layer where each sphere rests in the hollow at the junction of four spheres below it, as shown in figure.
Packing in a body-centred cubic lattice
The third layer then rests on these in arrangement which corresponds exactly to that in the first layer. In
this arrangement, spheres are touching one another along the cross diagonal of the cube, making its distance equal to 4r. This must be equal to 3a.
Thus, 4r = 3a, i.e., r =
4 3 a Volume of the cube = a3 Volume of one sphere =
34 πr3 =
34 π a 3 4
3
Since there are two spheres in each unit cell, the total volume occupied will be
2
π
3
4 a 3 3 4
The fraction of the volume occupied by the spheres
φ = 3
3
a 4 a
3 3 2 4
π
= 8 3π
= 0.6802 or 68.02 percent
In this arrangement each sphere has eight nearest neighbours