Crystal Lattices And Unit Cells

Solid State

Solids

Solids are the chemical substances which are characterized by definite shape and volume, rigidity, high density, low compressibility. The constituent particles (atoms, molecules or ions) are closely packed and held together by strong inter-particle forces.

Types of Solids

The solids are of two types: Crystalline solids and amorphous solids

Structure Determination by X-ray Diffraction (Bragg's Equation)

When a beam of X-rays falls on a crystal plane composed of regularly arranged atoms or ions, the X-rays are diffracted. If the waves are in phase after reflection, the difference in distance travelled by the two rays (i.e. path difference) must be equal to an integral number of wavelength, n$\lambda$ for constructive interference.

Thus, path difference = WY + YZ

= XY sin $\theta$ + XY sin $\theta$

= 2 XY sin $\theta$ = 2d sin $\theta$

$\therefore$ n $\lambda$ = 2d sin $\theta$

This equation is called Bragg's equation.

Where, n = 1, 2, 3… (diffraction order),

$\lambda$ = wavelength of X-rays incident on crystal and

d = distance between atomic planes

$\theta$ = angle at which interference occurs.

Crystal Lattices

A regular three dimensional arrangement of points in space.

Unit Cell

The smallest geometrical portion of the crystal lattice which can be used as repetitive unit to build up the whole crystal is called unit cell.

Types of Unit Cell

(a) Simple or primitive unit cell: In which the particles are present at the corners only.

(b) Face centred unit cell: In which the particles are present at the corners as well as at the centre of each of six faces.

(c) Body centred unit cell: In which the particles are present at the coners as well as at the centre of the unit cell.

(d) End centred unit cell: In which the particles are present at the corners and at the centre of two opposite faces.

Seven Crystal Systems and Possible Variations

There are about 230 crystal forms, which have been grouped into 14 types of space lattices, called Bravais Lattices, on the basis of interfacial angles and axes.

Coordination Number (CN)

It is defined as the number of particles immediately adjacent to each particle in the crystal lattice.

[In simple cubic lattice, CN is 6, in body centred lattice, Cn is 8 and in face centred cubic lattice, CN is 12.]

High pressure increases CN and high temperature decreases the CN.