Lec-3
Atomic Radius
It is defined as half the distance between nearest neighbours in a crystal of a pure element.
Usually, it is expressed in terms of the cube edge a. It must be remembered that any two nearest
neighbouring atoms touch each other.
Atomic Radiusfor different structures
Simple Cubic Structure:In fig.1.(a) and (b), AB represents the atomic radius. It is obvious from
𝑎
fig.1. (a) and (b) that 𝑟 = .
2
Fig.1.
Body Centered Cubic Structure:A s seen from fig.2 (a), atoms C and N are the nearest
𝐶𝑁
neighbours. By definition, 𝑟 = .
2
(a)
Fig. 2.
2
Now, 𝐴𝐶 = √𝐴𝐵 2 + 𝐵𝐶 2 = √(√2𝑎) + 𝑎2 = √3𝑎
𝐶𝑁 𝐴𝐶 √3𝑎
𝐵𝑢𝑡 𝑟 = = =
2 4 4
4𝑟
∴ 𝑎=
√3
A more detailed view is shown in fig. 2 (b), where solid diagonal AC equals four times the
radius of a single atom.
1
Susmita Bhattacharjee
Lecturer, Dept.of Physics, DUET
Lec-3
Face Centered Cubic Structure: Here, as seen from fig. 3 (a), atoms A and C are the nearest
neighbours.
𝐴𝐶 𝐴𝐵
Hence, 𝑟 = 𝐴𝐷 = =
2 4
√2𝑎
Now, 𝐴𝐵 = √2𝑎 ∴𝑟=
4
4𝑟
∴ 𝑎=
√2
(a)
Fig. 3.
Fig. 3 (b) shows more detailed view of the same. Here surface diagonal AC equals four times
the atomic radii.
Packing factor
The fraction of the space occupied by the atoms in the unit cell is known as atomic packing
factor or simply packing factor. It is the ratio of the volume of the atoms occupying the unit
cell to the volume of the unit cell relating to that structure.
Volume of all the atoms
∴ Packing factor =
Volume of the unit cell
Packing factor for different structures
Simple Cubic Structure: In simple cubic structure, eight atoms are at the eight corners of the
cell and each atom is common for eight cubes.
0 8
Thus the total number of atoms in a unit cell of simple cubic lattice is = 0 + + = 1
2 8
4
Volume of one atom = 3 πr 3
4 4
Volume of all the atoms = 1 × πr 3 = πr 3
3 3
𝑎
From the figure, atomic radius,𝑟 =
2
Volume of the unit cell = 𝑎3 = (2r)3 = 8r 3
4 3
Volume of all the atoms 3 πr 𝜋
∴ Packing factor = = 3
= = 0.52
Volume of the unit cell 8r 6
2
Susmita Bhattacharjee
Lecturer, Dept.of Physics, DUET
Lec-3
Thus, the packing factor of simple cubic structure is 0.52 means that, 52% of the total volume
of the simple cubic unit cell is filled with the atoms.
Body Centered Cubic Structure: In body centered cubic structure, eight atoms are at the eight
corners and one atom is inside the cell.
0 8
Thus the total number of atoms in a unit cell of body centered cubic lattice is = 1 + + = 2
2 8
4
Volume of one atom = πr 3
3
4 8
∴ Volume of all the atoms = 2 × πr 3 = πr 3
3 3
2
From the figure, 𝐴𝐶 = √𝐴𝐵 2 + 𝐵𝐶 2 = √(√2𝑎) + 𝑎2 = √3𝑎
4𝑟 = √3𝑎
4𝑟
∴ 𝑎=
√3
4𝑟 3 64r 3
Volume of the unit cell = 𝑎3 = ( ) =
√3 3√3
8 3
Volume of all the atoms 3 πr 8𝜋√3
∴ Packing factor = = 3 = = 0.68
Volume of the unit cell 64r 64
3√3
Thus, the packing factor of body centered cubic structure is 0.68 means that, 68% of the total
volume of the body centered cubic unit cell is filled with the atoms.
Face Centered Cubic Structure: In face centered cubic structure, eight atoms are at the eight
corners and all (six) the faces centered.
6 8
Thus the total number of atoms in a unit cell of face centered cubic lattice is = 0 + + = 4
2 8
4
Volume of one atom = πr 3
3
4 16
∴ Volume of all the atoms = 4 × πr 3 = πr 3
3 3
From the figure, 𝐴𝐶 = √𝐴𝐵 2 + 𝐵𝐶 2 = √𝑎2 + 𝑎2 = √2𝑎
4𝑟 = √2𝑎
4𝑟
∴ 𝑎= = 2√2𝑟
√2
3
Volume of the unit cell = 𝑎3 = (2√2𝑟) = 16√2r 3
3
Susmita Bhattacharjee
Lecturer, Dept.of Physics, DUET
Lec-3
16 3
Volume of all the atoms πr 𝜋
∴ Packing factor = = 3 = = 0.74
Volume of the unit cell 16√2r 3 3√2
Thus, the packing factor of face centered cubic structure is 0.74 means that, 74% of the total
volume of the face centered cubic unit cell is filled with the atoms.
Glass
Glass is a non-crystalline or amorphous solid material that exhibits a glass transition when
heated towards the liquid state. Glasses can be made of quite different classes of materials:
inorganic networks, metallic alloys, ionic melts, aqueous solutions, molecular liquids, and
polymers.
Liquid crystal
Liquid crystals (LCs) are state of matter that has properties between those of conventional
liquid and those of solid crystal. For instance, a liquid crystal may flow like a liquid, but its
molecules may be oriented in a crystal-like way.
Liquid crystals can be divided into thermotropic, lyotropic and metallotropic phases.
Thermotropic and lyotropic liquid crystals consist of organic molecules. Thermotropic liquid
crystals exhibit a phase transition into the liquid-crystal phase as temperature is changed.
Lyotropic liquid crystals exhibit phase transitions as a function of both temperature and
concentration of the liquid-crystal molecules in a solvent (typically water). Metallotropic liquid
crystals are composed of both organic and inorganic molecules; their liquid-crystal transition
depends not only on temperature and concentration, but also on the inorganic-organic
composition ratio.
Examples of liquid crystals can be found both in the natural world and in technological
applications. Lyotropic liquid-crystalline phases are abundant in living systems. For example,
many proteins and cell membranes are liquid crystals. Other well-known examples of liquid
crystals are solutions of soap and various related detergents.
Ceramics
A ceramic is an inorganic, non-metallic solid prepared by the action of heat and subsequent
cooling. Ceramic materials may have crystalline, partly crystalline structure or may be the
amorphous. Ceramic materials are compounds of carbon, nitrogen, boron, silicon etc.
Example: Barium titanate (BaTiO3), Zinc ferrite (ZnFe2O4) etc.
Properties of ceramics: Ceramics possess chemical, mechanical, thermal, and magnetic
properties that distinguish them from other materials, such as metals and plastics.
Manufacturers customize the properties of ceramics by controlling the type and amount of the
materials used to make them.
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Susmita Bhattacharjee
Lecturer, Dept.of Physics, DUET
Lec-3
Polymer
A polymer a large molecule, or macromolecule, composed of many repeated subunits, known
as monomers. Both synthetic and natural polymers play an essential role in everyday life.
Polymers range from familiar synthetic plastics such as polystyrene to natural biopolymers
such as DNA. Polymers, both natural and synthetic, are created via polymerization of many
monomers. Their consequently large molecular mass relative to small molecule compounds
produces unique physical properties, including toughness, viscoelasticity, and a tendency to
form glasses and semicrystalline structures rather than crystals.
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Susmita Bhattacharjee
Lecturer, Dept.of Physics, DUET
