SELVAMM ARTS SCEINCE COLLEGE (AUTONOMOUS)

Construction of brillouin zones

DEGREE OF MASTER OF SCEIENCE IN PHYSICS Submitted by S.SOUNDARRAJAN

• Solid state

• 1.crystalline substance

• 2.amorphous substance

1.crystalline substance

A substance is said to be crystalline when the • arrangement of units of matter is regular and • periodic.

It possesses a regular shape and if it is broken, all • broken pieces have the same regular shape.

A crystalline material can either be a single • (mono) crystal or a polycrystal.

A single crystal consists of only one crystal, • whereas the polycrystalline material consists of • many crystals separated by well-defined • boundaries.

• Examples• Metallic crystals – Cu, Ag, Al, Mg etc,

• Non-metallic crystals – Carbon,Silicon,Germanium,

2.amorphous substance

In amorphous solids, the constituent particles are not arranged in an orderly manner. They are randomly distributed

(a)mono (or) single crystals

(b) polycrystalline solids (c) amorphous solids

• What is crystal ?

•A crystal is a solid composed of a periodic array of atoms.

•Lattice + basis = crystal structure

= +

Crystal structure = Lattice + Basis

• Brillouin zonesA Brillouin Zone is defined as a Wigner-Seitz primitive cell in the reciprocal lattice.• What is primitive cell ?• The primitive cell is defined as unit cell which contains lattice points at corner only

•

reciprocal lattice

Construction of brillouin zones • The brillouin zones are constructed from the planes which are the perpendicular or bisectors of all reciprocal lattice vectors

• The first zones is the smallest volume about the origin enclosed by these planes

• The second zone is the volume between the first zone and next set of planes

THEORY• The primitive translation vectors of this lattice are• a=a• b=a• The corresponding translation vectors of the reciprocal

lattice are, =(2)• =(2)• Therefore the reciprocal lattice vector is written as• G=2)• Where h and k are integer• K=+

• Braggs equavation• 2K . G + = 0• 2( +) . 2) + (

• [( +) . ] + (

• ( h + k ( = 0

• ( h + k ( • The k values which are Bragg reflacted are obtained by

considering all possible combinations of h and k

• For h =±1 and k = 0

• = ± and is arbitrary

• h = 0 and k = ±1

• = ± and is arbitrary• That is first brillouin zones

• For second brillouin zone ,the integers h and k are given the next higher values , h and k is ±1

• When h = 1 and k = 1, then • +2• When h = - 1 and k = 1, then • +2• When h = 1 and k = - 1, then • 2• When h = - 1 and k = - 1, then • 2• That is second brillouin zones

The numbersindicate the Brillouin zone to which each region belongs.

The numbersindicate the Brillouin zone to which each region belongs.

The numbersindicate the Brillouin zone to which each region belongs.