# Solid State Notes

## Amorphous Solids

Solids in which the constituent particles of the matter are arranged in the random manner and it is a pseudo – solids or super-cooled liquids. They don't have a sharp melting point

Examples: plastics, glass, rubber, metallic glass, polymers, gel, fused silica, pitch tar, thin film lubricants, wax etc.

## Crystalline Solids

Solids in which the constituent particles of the matter are arranged in the specific manner and it is a true solid. They have a sharp melting point

Examples: quartz, calcite, sugar, mica, diamonds, snowflakes, rock, calcium fluoride, silicon dioxide, alum etc.

## Crystal Lattice

Crystal Lattice is a three-dimensional representation of constituent particles (atoms, molecules or ions) arranged in a specific order. Or the geometric arrangement of constituent particles of crystalline solids as point in space is called crystal lattice. There are total 14 possible three-dimensional lattices. Crystal lattices are also known by Bravais Lattices.

## Characteristics of crystal lattice:

☛ Each constituent particle is represented by one point in a crystal lattice.

☛ These points are known as lattice point or lattice site.

☛ Lattice points in a crystal lattice are joined together by straight lines.

☛ By joining the lattice points with straight lines the geometry of the crystal lattice is formed.

## Unit Cell

The smallest portion of a crystal lattice is called Unit Cell. By repeating in different directions unit cell generates the entire lattice.

## Characteristics of Unit Cell

☛ A unit cell has three edges a, b and c and three angles α, β and γ between the respective edges.

☛ The a, b and c may or may not be mutually perpendicular.

☛ The angle between edge b and c is α, a and c is β and that of between a and b is γ.

## Number of Particles Per Unit Cell (Z)

**☛ For Simple Cubic Unit Cell**

In Simple cubic unit cell, particles are present at corners only. In a crystal lattice every corner is shared by eight adjacent unit cells. Therefore, only 1/8 of constituent particles, belong to a particular unit cell. Therefore, Z=8 X 1/8=1

So, the number of particles per unit cell for simple cubic cell is 1.**☛ For Body Centered Cubic (BCC) Unit Cell:**

In Body Centered Cubic unit cell, particles are present at corners as well as at the center of the body. Therefore,

Z=(8 X 1/8) + 1=2

So, the number of particles per unit cell for BCC Unit cell is 2.**☛ For Face Centered Cubic (FCC) Unit Cell:**

In Face Centered Cubic unit cell, particles are present at corners as well as at the center of the face which is shared between adjacent two particles.

Therefore,

Z = (8 X 1/8) + (6 X 1/2) = 4

So, the number of particles per unit cell for FCC Unit cell is 4.**☛ For End Centered Cubic (ECC) Unit Cell:**

In Eace Centered Cubic unit cell, particles are present at corners as well as at the center of two face which is opposite to each other. Therefore,

Z = (8 X 1/8) + (2 X 1/2) = 2

So, the number of particles per unit cell for ECC Unit cell is 2.

## Limiting Radius Ratio:

The limiting radius ratio is the minimum allowable value for the ratio of cationic radii to anionic radii (ρ=r^{+}/r^{-}) for the structure to be stable. Here, r^{+} is the radius of the cation and r^{-} is the radius of the surrounding anions.

## Bragg's Equation:

When a beam of light falls on a crystal plane composed of regularly arranged particles, the X- rays are diffracted. If the waves are in phase after reflection,the difference in distance travelled by the two rays must be eqaul to an integral number of wavelength, nλ for constructive interference.

Thus, Path difference = WY + YZ

= XY sinθ + XY sinθ

= 2XY sinθ

= 2d sinθ

So, nλ = 2d sinθ

This equation is Known as Bragg's equation.

where, n = 1,2,3...(diffrection order)

λ = wavelength of X-rays

d = distance between planes

θ = angle at which interference occurs.

## Law of rational indices:

The law of rational indices was deduced by Hauy.*The law of rational indices states that the intercepts of any face of a crystal along the crystallographic axes are either equal to the unit intercepts or some simple whole number multiples of them.*

The faces of crystals and also planes within crystals can be characterized by means of a suitable set of coordinates.

Let consider the three axes OX, OY, OZ which are cut by a crystal face ABC at distances OA, OB, and OC from the origin. Let OX, OY, and OZ represent the three crystallographic axes and let ABC be a unit plane. The unit intercepts will then be a, b and c.

According to the above law, the intercepts of any face such as KLM, on the same three Axes will be simple whole number multiples of a,b and c respectively. These distances at called intercepts. It is found that if the axes are suitably chosen, the intercepts of different faces upon them bear a simple ratio to each other or a given face may cut an axis at infinity. This is called the law of rational intercepts or indices.

## Structure of Diamond

All the carbon atoms attached with four other carbon atoms, thus making a perfect tetrahedron structure.

All the carbon atoms are sp^{3} hybridized.

Bond angle between two carbon atoms is 109.5^{o}.

Bond lengths of carbon-carbon atom in diamond are equal and is equal to 154pm or 1.54A^{o}.

Diamond forms a three-dimensional network of strong covalent bonds.

Diamond has a very high melting point of about 3843K.

Diamond has a high density of about 3.51 g/cm^{3}.

Diamond is a poor conductor of electricity since its valence electrons get involved in C-C sigma covalent bonds, and hence they are localized and are not free to conduct the electricity.

## Structure of Graphite

All the carbon atoms attached with three other carbon atoms with covalent bonds.

Each carbon atom is sp^{2} hybridized.

All carbon atoms form a layer like structure with a hexagonal arrangement of carbon atoms. These layers have weak forces between them. Due to these weak forces, the layers can slip over each very easily.

The distance between two carbon atoms in the same sheet or layer is 0.14nm or 141.5pm and between two sheets is 0.34nm or 340pm.

Each carbon atom has a one non bonded electron.

Graphite is a good conductor of heat and electricity due to its free delocalized electron which is free to move throughout the sheets.

Graphite has a density of 2.09 – 2.23 g/cm^{3}.

Graphite is insoluble in organic solvents and water because the attraction between solvent molecules and carbon atoms is not strong enough to overcome the covalent bonds between the carbon atoms in the graphite.

Graphite is having a high melting point of 3650^{o}C.

Due to its sheet or layer-like structure, it is soft and slippery in nature.

Due to the slippery nature of Graphite, it is used as a lubricant in the machine parts.

Graphite has the ability to absorb high-speed neutrons.