Crystalline solid consists of a large number of small units, called crystals, each of which possesses a definite geometric shape bounded by plane faces. The crystals of a given substance produced under a definite set of conditions are always of the same shape.
In this topic, we would be studying certain properties of a solid which depend only on the constituents of the solid and the pattern of arrangement of these constituents.
To study these properties, it would be convenient to take up a small amount of the solid as possible because this would ensure that we deal with only the minimum number of atoms or ions.
This smallest amount of the solid whose properties resemble the properties of the entire solid irrespective of the amount taken is called a unit cell.
It is the smallest repeating unit of the solid. Any amount of the solid can be constructed by simply putting as many unit cells as required.
The following characteristics define a unit cell:
Bravais (1848) showed from geometrical considerations that there are only seven shapes in which unit cells can exist.
Moreover, he also showed that there are basically four types of unit cells depending on the manner in which they are arranged in a given shape. These are:
He also went on to postulate that out of the possible twenty-eight unit cells (i.e. seven shapes, four types in each shape = 28 possible unit cells), only fourteen actually would exist. These he postulated based only on symmetry considerations. These fourteen unit cells that actually exist are called Bravais Lattices.
In a primitive cubic unit cell, the same type of atoms are present at all the corners of the unit cell and are not present anywhere else. It can be seen that each atom at the corner of the unit cell is shared by eight unit cells (four on one layer, as shown, and four on top of these).
Therefore, the volume occupied by a sphere in a unit cell is just one-eighth of its total volume. Since there are eight such spheres, the total volume occupied by the spheres is one full volume of a sphere.
A primitive cubic unit cell is created in the manner as shown in the figure. Four atoms are present in such a way that the adjacent atoms touch each other.
Therefore, if the length of the unit cell is 'a', then it is equal to 2r, where r is the radius of the sphere.
Four more spheres are placed on top of these such that they eclipse these spheres. The packing efficiency of a unit cell can be understood by calculating the packing fraction.
It is defined as the ratio of the volume occupied by the spheres in a unit cell to the volume of the unit cell.
Let ‘a’ be the edge length of the unit cell and r be the radius of the sphere.
As spheres are touching each other. Therefore a = 2r
No. of spheres per unit cell = 1/8 × 8 = 1
The volume of the sphere = 4/3 πr3
Volume of the cube = a3= (2r)3 = 8r3
∴ The fraction of the space occupied
∴ % occupied = 52.4 % .
(This implies that 52 % of the volume of a unit cell is occupied by spheres).
Void fraction = 1 - Packing fraction
Therefore, V.F = 0.48
Primitive cubic unit cell
A Body-Centered unit cell is a unit cell in which the same atoms are present at all the corners and also at the center of the unit cell and are not present anywhere else. This unit cell is created by placing four atoms that are not touching each other.
Then we place an atom on top of these four. Again, four spheres eclipsing the first layer are placed on top of this. The effective number of atoms in a Body-Centered Cubic Unit Cell is 2 (One from all the corners and one at the center of the unit cell).
Moreover, since in BCC the body-centered atom touches the top four and the bottom four atoms, the length of the body diagonal (√3a) is equal to 4r.
The packing fraction in this case is =
VF ≈ 0.32
In an fcc unit cell, the same atoms are present at all the corners of the cube and are also present at the center of each square face and are not present anywhere else. The effective number of atoms in fcc is 4 (one from all the corners, 3 from all the face centers since each face-centered atom is shared by two cubes).
Since, here each face-centered atom touches the four corner atoms, the face diagonal of the cube is equal to 4r.
VF ≈ 0.26
Each corner atom would be common to 6 other unit cells, therefore their contribution to one unit cell would be 1/6. Therefore, the total number of atoms present per unit cell effectively is 6.
Figure 6(b) ABCD is the base of the hexagonal unit cell AD=AB=a. The sphere in the next layer has its center F vertically above E and it touches the three spheres whose centers are A, B, and D.
The height of the unit cell (h) =
The area of the base is equal to the area of six equilateral triangles, = .
The volume of the unit cell = .
VF ≈ 0.26
The density of crystal lattice is the same as the density of the unit cell which is calculated as
Z = no. of atoms present in one unit cell
m = mass of a single atom
Mass of an atom present in the unit cell = m/NA
The seven crystal systems are given below.
The table given below can be used to summarize types of lattice formation.
Illustration 1: Lithium borohydride crystallizes in an orthorhombic system with 4 molecules per unit cell. The unit cell dimensions are a = 6.8 Å, b = 4.4 Å and C = 7.2 Å. If the molar mass is 21.76 g. Calculate the density of the crystal.
Since, Density, Here z = 4, Av. No = 6.023 x 1023 &
Volume = V = a x b x c
= 6.8 x 108 x 4.4 x 108 x 7.2 x 108 cm3
= 2.154 x 1022 cm3
= 0.6708 gm/cm3