CRYSTAL STRUCTURE | Classification of solids| Lattice and lattice points, Crystal structure| Bravais space lattice| Unit cell, Primitive cell| Miller indices.

Classification of solids

The solids are categorised into categories –

A. Crystalline solids.         B. Amorphous solids.

Distinction between crystalline and amorphous solids –

i) Crystalline solids have a regular arrangement of particles whereas the amorphous solids have a completely random particles arrangement.

ii) Crystalline solids have different physical properties ( eg. – Thermal conductivity, electrical conductivity refractive index ) in different directions, whereas the amorphous solids have same in all directions.

iii) Crystalline solids have very sharp melting point whereas amorphous solids do not have sharp melting point.

What is crystal ?

Crystal : “A crystal is a homogeneous; an-isotropic body having natural shape of a polyhedron.” or

A crystal is a stable composed of a periodic array of atoms i.e, a representative unit is repeated at regular duration along any and all suggestions inside the crystal.

What is crystallography?

Crystallography : The study of the geometric form and other physical properties of crystalline solids by using x-rays, electron beams, neutron beams, etc. constitute the crystallography.

Space lattice :

An array of points in space such that the environment about each point is the same, is called space lattice.

Lattice and lattice points :

(Since a point, being an infinitesimal spot in space, is imaginary, a lattice of points is an imaginary concept.)

Each atom in the crystal structure is replaced by a point, then we get an infinite array of points in space; which is known as lattice and each such points are called as lattice points.

Basis :

The crystal structure is fashioned by means of associating with every lattice point as a unit assembly of atoms same in composition, association and orientation. This unit assembly is known as the basis.

Crystal structure :

The periodic arrangement of basis in all direction gives the actual crystal structure.

i.e.                       crystal structure = Lattice + Basis.

 

Bravais space lattice : — (In two dimension)

There is no natural restriction on lengths of translation vector (a and b) or angle between them (φ); then an unlimited number of lattice is possible. ●

But the requirement that a lattice should be invariant under a rotation operation 2π/n ; where n=1,2,3,4,6.

These five types of special lattices are called Bravais Lattice. They are

i) Oblique lattice –

Invariant under rotation of 2π about any lattice points.

ii) Square Lattice –

iii) Hexagonal lattice

Invariant under a rotation of 2π/6.

{v) Rectangular lattice –

After mirror reflection about x axis

vi) Centered rectangular lattice –

The choice of a, b will be such that which is invariant under reflection that

 

Unit cell :

A unit cell is the smallest geometric figure, the repetition of which in three dimensions will give crystal structure.

 

Primitive cell :

The primitive cell is defined as unit cell which contains lattice points at the corner.

1. All primitive cells are unit cell but all unit cells are not primitive cell.

2. It contains one atom.

3. Choice of primitive is unit.

Number of atoms  or lattice points per unit cell –

i) Simple cubic cell – (s.c.)

In a simple cubic cell there are 8 atoms at the 8 corners of the cell. Thus contribution of each atom to  a unit       cell = 1/8

Thus no. of atoms per unit cell = 1/8×8 =1

ii) Body centre cubic ( bcc. structure) –

Here 8 corner atoms and one atom at the centre of the cell. No. of atoms per unit cell = 1/8 × 8 + 1 = 2

iii) Face centered cubic (fcc structure) –

Here 8 corner atoms and 6 face atoms each at the centre of each face.

No. of atoms per unit cell. =1/8×8+1/2×6 = 1+3 = 4

Miller indices –

Definition – In order to specify the orientation of a crystal plane in space, we attribute indices on variables, which are the reciprocals of the intercepts made by the plane on the three rectangular axes. These are called Miller Indices.

How will you attribute Miller Indices to a plane ?

Step-I : first choose any atom of the crystal as the origin and draw three crystallographic axis.

Step-II : Choose the particular crystal plane and note the intercepts on the axes.

Step-III : Take the reciprocals of that intercepts and convert them to smallest set of integers in ratio.

 

Estimate :

 

Let OX, OY and OZ of are the three axes parallel to crystal axes.

Let

lattice constant along three axes respectively.

We want to find the Miller indices of the plane ABC. Let

Ratio of the intercepts,   

Ratio of the reciprocal of the intercepts,

where (h,k,l) are called Miller Indices.

Space lattice: Three dimensional lattice type :

Bravis lattice:

Bravis showed that in three dimensions, point symmetry groups are associated with fourteen different types of lattices; one general and thirteen special types, collectively called Bravis lattices.

General type is – triclinic lattice

These fourteen lattice types are conventionally grouped into seven crystal systems, according to seven types of conventional unit cells.

N.B. – In two dimensions, point groups are associated with five different types of lattices; one general (oblique) and four special –  discussed before.

Name the seven lattice systems :

In three dimension, seven lattice systems are –

1. Regular and cubic system.              [a= b = c ,    α = β = γ = 90°]

2. Trigonal and Rhombohedral system        [ a = b = c ,      α= β = γ = 90°]

3. Hexagonal system                       [ a = b ≠ c ,      α= β =90° , γ = 120°]

4. Tetragonal system                        [ α= β = γ = 90° ; two lateral ares equal ]

5. Orthorhombic system                 [ a ≠ b ≠ c , α= β = γ = 90° ]

6. Monoclinic system                  [a ≠ b ≠ c , α = γ = 90° , β ≠ 90°]

7. Triclinic system           [ a ≠ b ≠ c , α= β = γ = 90° ] 4. 5. 7.

 

Regular or cubic systems have 3 types of lattice depending upon simple cubic upon the shape of unit cell –

i) simple cubic

ii) body centred cubic

iii) face centred cubic

Example – Diamond, Zinc sulphide, alkali metals .

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