Physics I Sem Fullnotes June 2010

Dept. of Physics

CMR INSTITUTE OF TECHNOLOGY

BANGALORE

DEPARTMENT OF PHYSICS

COURSE MATERIAL

ENGINEERING PHYSICS PHY 12/22(NEW SCHEME)

External Exam : 100 Marks Internal Test : 25 Marks

AUTHORRAVEESHA.K.H.

[email protected] OF PHYSICS

CMR INSTITUTE OF TECHNOLOGYIT PARK ROAD,KUNDALAHALLI

BANGALORE-560037

C.M.R Institute of Technology, Bangalore 1

Dept. of Physics

CONTENTS

Chapter Page No

1. Plancks quantum theory 3

Davisson – Germer experiment 10

Debroglies theory 14

Particle in an infinite potential well 18

2. Crystal physics 21

Expression for interplanar spacing 30

Packing factor 33

3. X – rays 37

4. Electron conduction 43

Density of states 50

5. Magnetic materials 54

6. Superconductivity 61

7. Dielectrics 68

8. Optical properties of solids 78

Ruby laser 83

Helium laser 84

9. Optical fibres 88

10. Holograms 95

CHAPTER 1: Planck’s quantum theory


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Basic definitions:

Opaque objects: They do not transmit any radiation.

Black body : It is an object which absorbs all radiations incident on it and emit those

radiations on heating. Its absorption coefficient is 100%.

White body : It reflects all the incident energy.

Black body :

A good approximation of a black body is a small hole leading to the inside of a hollow object as

shown in the above fig. Any radiation that falls inside through the hole gets reflected in every

direction and finally all the energy is absorbed. The nature of the radiation emitted from the hole

depends only on the temperature of the cavity walls. The distribution of the radiated energy varies

with wavelength and temperature as shown in the following graph.

Features of Black body spectrum:

C.M.R Institute of Technology, Bangalore

Intensity

4000K

3000K

2000K

Wavelength

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Interpretation of the graph:

1. A black body emits over wide range of wavelengths at different temperatures.

2. At each temperature, there exists a wavelength at which maximum energy is radiated.

3. As the temperature increases, the amount of energy radiated (the area under the curve)

increases and the peak shifts towards shorter wavelengths.

4. As temperature increases, energy emitted also increases.

Note:

Stefans law: Total energy radiated per unit area is proportional to the fourth power of the

temperature.

where J/m2/s

Weins displacement law: The product of the wavelength at which maximum energy is radiated

and the temperature is a constant.

Rayleigh –Jeans Law: Intensity of radiation from a hot body is inversely proportional to the

fourth power of the wavelength .

As a consequence, the energy radiated by a hot body must become very high at lower

wavelengths (ultraviolet region) leading to ‘ultraviolet catastrophe’. However, experimentally

the intensity of radiation decreases with decrease in temperature.

Planck’s radiation law:

In 1900, Max Planck developed a structural model for black body radiation that leads to a

theoretical equation for the wavelength distribution that is in complete agreement with the

experimental results at all wavelengths.

According to his theory

1. a black body is imagined to be consisting of large number of electrical oscillators.

2. an oscillator emits or absorbs energy in discrete units. It can emit or absorb energy by making

a transition from one quantum state to another in the form of discrete energy packets known

as photons whose energy is an integral multiple of hν where h is the planks constant and ν is


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the frequency.

3. the key point in Planck’s theory is the radical assumption of quantized energy states. This

development marked the birth of quantum theory.

Based on these ideas, Planck was able to derive an expression that agreed remarkably well

with the experimental curves. It is given by

Deduction of Weins law:

It is applicable at smaller wavelengths.

For smaller wavelengths

So Planck’s radiation law becomes

Deduction of Rayleigh Jeans Law:

It is applicable at longer wavelengths.

For longer wavelengths

Photoelectric effect:


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It is a process in which when an electromagnetic radiation of suitable wavelength is incident on

certain metals, they emit electrons.

Work function: It is the minimum energy required to remove an electron from an atom.

Threshold wavelength: It is a particular wavelength above which photoelectric emission is not

possible.

Stopping potential: It is the negative potential which has to be applied to stop the electrons which

are moving towards the positive electrode.

Laws of photoelectric emission:

1. Photoelectric current is proportional to the intensity of light.

2. For a given metal, there exists certain minimum frequency below which photoelectric

emission is not possible.

3. Kinetic energy of electrons is proportional to the frequency of radiation.


Quartz bulb

Radiation of suitable wavelength

electrons

Galvanometer

Stopping potential

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4. Rate of emission of electrons is independent of temperarture.

Einstein’s photoelectric equation:

Incident energy = workfunction + kinetic energy

hν = Ф +

Types of photoelectric cells:

1. Photoemissive cell:

2. Photovoltaic cell :


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3.Photo conductive cell :

Note :

If V is the stopping potential , then

eV = kinetic energy =

Einsteins equation becomes

hν = Ф +

Physical significance of photoelectric effect:

1. It supports the quantum theory.

2. It confirms the fact that the light radiation consists of particles called as photons.

Compton Effect:


Copper sheet

Cu2O film radiation

Metallic bar

Metallic transparent film radiation

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It is an effect in which when x rays are incident on electron, the scattered radiation will have the

wavelength equal to or greater than the incident wavelength.

Change in wavelength of x ray after scattering =

Here h is plancks constant

m0 is rest mass of electron

c is velocity of light

θ is angle of scattering

Physical significance:

It supports the particle nature of light.


λ1

Incident X-ray of wavelength λ

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DUAL NATURE OF MATTER

Davisson and Germer experiment:

(Davisson and Germer were working at Bell Labs studying structure of crystals)

This experiment confirms the wave nature of electrons.

The apparatus used by Davisson and Germer is shown below. A beam of electrons from a

heated filament ‘F’ is accelerated through a potential difference V .It passes through a narrow

aperture and strikes nickel crystal .Electrons are scattered in all directions by the atoms of the

crystal. These scattered electrons are detected by a device which can be moved to any angle

relative to the incident beam and it can also measure the intensity of the scattered beam.

The results of the experiment are shown in the following graphs.


F

+, V

Crystal

Detector

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From these graphs, it is clear that initially at 38V, 46V, electrons are scattered uniformly in all

directions. When the accelerating voltage is set at 54V, there is an intense reflection of the beam

at =50˚. This maximum intensity caused at this angle can only be accounted for constructive

interference of electron waves. If we assume that each atom of the crystal acts as a scatterer ,

then all the electron waves which get scattered from different layers of atoms which are in phase

undergo constructive interference producing maximum intensity. By knowing the angle of

maximum intensity and the inter atomic spacing, the wavelength of the electron waves can be

calculated using the relation

where D is the interatomic distance, is the angle between the incident beam and scattered

beam.

For Nickel D=0.215 nm, = 50˚.

λ= 0.215 x sin 50˚ = 0.165 nm

Theoretically, using Debroglie’s law we can calculate the wavelength of the waves associated

with electrons of energy 54ev.

λ= 0.165 nm


38ev 46ev 54ev 64ev

Spherical polar graphsHere the length of the position vector to a point on the graph is a measure of number of electrons scattered along that vector direction

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The theoretical and experimental values are in excellent agreement. This confirms the existence

of matter waves.

Phase velocity(Vp): It is the speed of a single pulse in a medium. Generally waves observed

in nature like light waves, sound waves, electromagnetic signals etc travel as groups. So phase

velocity is rarely used.

A single pulse ishown in this diagram .It is represented as

Y = A sin [wt – kx]

where Y is the displacement of a particle at a distance ‘x’ from the origin at a time ‘t’, A is the

amplitude , w is the angular velocity and k is the wave number.

Vp =

Group Velocity(Vg): It is the velocity with which the resultant amplitude of group of waves

propagates .

Consider two waves of same amplitude but of slightly different wavelengths travelling in the

same direction.


Y

X

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Let the waves be represented as

Y1 = A sin (wt – kx) … (1)

Y2 = A sin [(w+Δw) t – (k + Δk) x]

= 2A cos[( )t – ( )x] sin ( )t – ( ) …. (2)

But Δw and Δk are small

2w + Δw 2w, 2k+ Δk 2k

Y = 2A cos [( ) t – ( ) x] sin (wt-kx) .....(2)

Comparing equations (1) and (2), the coefficient of sin (wt-kx) in equation (2) can be considered

as the amplitude of the wave.

Amplitude of the resultant wave =2A cos [(Δw/2) t – (Δk/2) x]

This amplitude varies as a wave .The velocity with which the variation in amplitude is propagated

is referred as group velocity

Vg = (Δw/2) / (Δk/2)

Vg = Δw / Δk =

Note: The energy carried by a wave packet travels at the group velocity, not the phase velocity.


Y1= A Sin(ωt – kx) Y2 = A sin [(ω+∆ω)t+(K+∆K)x]

Superposition

Resultant Envelope with varying amplitude

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Relation between phase velocity and group velocity:

We have phase velocity Vp =

Vg = = = Vp + K. ………. (1)

But = X and k = ;

Substituting these in equation (1) we get

Vg = Vp -

Relation between group velocity and particle velocity:

By definition Group velocity is Vg = …………………………… (1)

w = 2Πf = 2Π

dw =

k = = 2 ; dk =

Substitute for dw and dk in equation (1)

Vg =

Using the expression E =

Also p = m

=

This shows that Group velocity and Particle velocity are equal.


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Note: The velocity of group of waves representing a particle cannot be different from particle

velocity as both are related to one physical entity.

Relation between Vg, Vp and velocity of light :

Vp =

Vp.

Debroglie’s theory:

Statement: By the law of symmetry of nature, a particle must exhibit wave like properties in

addition to its particle properties.

The wavelength of the group of waves associated with particle of mass m moving with a velocity

v is given by the expression

=

where h is the Planck’s constant

Derivation: According to debroglie theory, a moving material particle is associated with a group

of plane waves .The group velocity of the waves associated with an object of mass m moving

with a velocity v is given by

Vg = ……… (1)

But w = 2 k = ;

dw = 2 dk = 2 . d

Substituting this in equation (1) we get

d = …………………………… (2)

The total energy of the particle is given by

E = m v2 + V where V is the potential energy

The energy present in the waves associated with the particle is given by

E = hf


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Since energy of a particle remains the same either in particle form or the wave form

hf = m v2 + V

If the particle is moving in a region free of fields V = 0

hf = m v2

Differentiating h df = m v dv

df = dv

Substituting this in equation (2)

d =

Integrating

= + Constant

Neglecting the constant.

Heisenberg’s uncertainty principle:

The position and momentum of a particle cannot be determined accurately and simultaneously.The product of uncertainty in position and momentum is always greater than or equal

to .

.

This uncertainty is not due to discrepancy with the apparatus or with the method of measurement, but because of the very wave nature of the object. This uncertainty persists as long as matter possesses wave nature.

Different forms of Heisenberg’s Principle:

.

Here ΔL is the uncertainty in angular momentum


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Δθ is the uncertainty in angular displacement

ΔE is the uncertainty in the energy

Δt is the uncertainty in the time interval during which the particle exists in the state E

Time independent Schrödinger equation

A matter wave can be represented in complex form as

Differentiating wrt x

…………………….. (1)

From debroglie’s relation

=

k = =

………………………. (2)

Total energy of a particle E = Kinetic energy + Potential Energy

E = m v2 + V

Substituting in (2)

From (1)

Significance of :

According to Max Born, is a measure of probability of finding a particle in specified region.

Normalisation: Total probability of finding a particle in a closed volume is unity.


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A function is said to be a normalisable wave function if it is

1. Single valued

2. Continuous

3. Finite

Application of Schrödinger’s equation:

Particle in an infinite potential well problem:

Consider a particle of mass m moving along X-axis in the region from X=0 to X=a in a one

dimensional potential well as shown in the diagram. The potential energy is zero inside the region

and infinite outside the region.

Applying, Schrodingers equation for region (1) as particle is supposed to be present in region (1)

But

The general solution to this expression is given by

At x=0,


X = 0 X = a

Region (1)

V =0

V

Region (2)

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At x=a, D sin ka = 0 ka = n where n = 1, 2 3…

E =

To evaluate the constant D: Normalisation : For one dimension

But

0 =1

D =

Eigen function: It is the physically acceptable solution to Schrodinger’s equation. It represents

the matter wave corresponding to a quantum particle in a specific state.

Ex: For a particle in an infinite potential well, the eigen function is

Eigen Value:It represents the energy of a particle in corresponding to its Eigen function.


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Eigen value for a particle in an infinite potential well is E =

Assignments:

1. Study the applications of De broglies theory with regard to electron microscopes.

2. Heisenberg’s theory and Bohr’s theory are contradictory. Comment.

3. How do you relate the Newtonian Mechanics and Schrödinger’s Wave mechanics?

4. Wave nature and Particle nature are the properties of our interaction with light.

Explain.

5. Shape of S orbital is said to be spherical. Understand this through probability concept.


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Contents

CHAPTER 2: CRYSTAL PHYSICS

Basic definitions:

Crystal : It is that form of an object having regular arrangement of atoms.

Lattice point : These are the points in a crystal at which atoms are located.

Bravais lattice: It is a lattice with identical atoms at all the lattice points.

Space lattice : It is the regular three dimensional arrangement of atoms in a crystal.

Unit cell : It is the smallest portion of a space lattice which can generate the

complete crystal when repeated along the three perpendicular directions.

Crystal Lattice:

In a crystal, there is a regular arrangement of atoms. It is convenient to imagine three dimensional

array of points in space about which these atoms are located. Such points are known as lattice

points and the totality of such points forms the pace lattice. If all the lattice atoms are identical,

the lattice is called a Bravias lattice. Bravais showed that total number of different space lattices

with each atom having identical environment is only fourteen.

For the above mentioned two dimensional Lattice


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where are the fundamental translation vectors ,n1 and n2 are integers.

Basis: Every lattice point could be associated with on or more atoms known as basis which when

repeated in all directions, gives the crystal structure.

Unit cell: It is convenient to describe the crystal structure by assuming the crystal to be a

combination of small repeating entities known as unit cells. A unit cell is chosen such that all the

atoms in the structure are generated by translations of the unit cell through integral distances.

A crystal may have many unit cells. Generally, a unit cell with highest symmetry will be selected.

A unit cell with all the atoms at the corners is known as primitive unit cell.

A unit cell can be completely described by three vectors (representing the length) and α, β,

γ (representing the angles between the vectors). These parameters constitute lattice parameters.

This is a cubic unit cell .The intercepts a,b,c formed along the axes X,Y,Z by the intersection of

the faces are called as lattice parameters .

are the interfacial angles.

Crystal systems:

To represent lattice atoms in a given material, the following seven coordinate systems are used.

1. Cubic 5. Triclinic

2. Tetragonal 6. Rhombhohedral

3. Orthorhombic 7. Hexagonal


B

B

c

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4. Monoclinic

CUBIC :

TETRAGONAL:

ORTHORHOMBIC:


a = b = c α = β = γ = 900

a = b ≠ c α = β = γ = 900

a ≠ b ≠ c α = β = γ = 900

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MONOCLINIC:

TRICLINIC :

RHOMBOHEDRAL:


a ≠ b ≠ c α = γ = 900 ≠ β

a ≠ b ≠ c α ≠ β ≠ γ ≠900

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a ≠ b ≠ c α = γ = 900 ≠ β

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HEXAGONAL:

Bravais lattice systems:

Cubic:


a = b = c α = β = γ ≠ 900

a = b ≠ c α = β = 900 , γ = 1200

Simple Body centered Face centered

a = b = c α = β = γ = 900

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Tetragonal:

Orthorhombic:

Monoclinic:


Simple Body centered

a = b ≠ c α = β = γ = 900

Simple Base centered Body centered Face centered

a ≠ b ≠ c α = β = γ = 900

Simple Base centered

a ≠ b ≠ c α = γ = 900 ≠ β

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Triclinic:

Rhombhedral:

Hexagonal:

Coordination number:

It is the number of nearest equidistant neighboring atoms for any atom in the lattice.

Simple cubic structure:

It has got lattice points at all the corners. Coordination number is 6.

Number of atoms per unit cell is 1

Ex: Polonium


a ≠ b ≠ c α ≠ β ≠ γ ≠900

a = b = c α = β = γ ≠ 900

a = b ≠ c α = β = 900 , γ = 1200

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Body centered cubic lattice:

It has got lattice points at all the corners and one at the geometrical centre of the cube.

Coordination number is 8. Number of atoms per unit cell is 2.

Ex: Li, Na, K, Cr

Face centered cubic lattice:

It has lattice points at all the corners and also at the centre of all the faces. The Coordination

number is 12.

There are 4 atoms per unit cell = 4

Ex: Al, Pb, Ag, Ni

Hexagonal close packed:


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It has lattice points at each corner, one atom each at the centre of the hexagonal face and three

atoms within the body of the unit cell.

Coordination number =12

Number of atoms per unit cell = 6atoms.

Miller indices:

Miller introduced a system to represent a plane in a crystal .He defined a set of three numbers to

specify a plane in a crystal.

Procedure:

1. Determine the intercepts of the plane along the X,Y and Z axes.

2. Find the reciprocal of these numbers.

3. Find the least common denominator (l.c.d) and multiply each by this l.c.d.The resulting

integers are called Miller indices and denoted as (h,k,l).

Example: If the intercepts of a plane are given by 4, 1, 2 then take the reciprocal.

l,c,d = 4

Miller indices 1, 4, 2


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Definition: Reciprocals of the intercepts made by the plane on crystallographic axis when

reduced to small numbers.

Expression for interplanar spacing in terms of Miller indices:

Let ABC be one of the parallel planes represented by the Miller indices [h,k,l].Let its intercepts

be x,y,z. Imagine another plane passing through the origin O .OD is the perpendicular from O to

the plane ABC and OP is the interplanar distance .Let the angle made by OP with X,Y and Z axis

be and respectively.

Now [h, k, l] = where a, b, c are constants.

[x,y,z] = …………..(1)

Also from figure d=x cos

cos , cos , cos

Squaring and adding after Substituting for x, y, z from (1)


Normal to the plane ABC

Normal to the plane ABC

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d2

If a = b =c, then

Expression for space lattice constant ‘a’ for a cubic lattice:

Density of material = Total mass of molecules belonging to unit cell / Volume of the unit cell

Mass of each molecule = Molecular weight / NA

Let the number of molecules in a unit cell be n.

Total mass of each molecule =

Density =

a =

Structure of Nacl:


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It is a face centered cubic structure. Na+ ( ) and Cl- ( ) ions occupy alternate corners and face

centered positions. Each ion has 6 ions of other type as its nearest neighbors. So its coordination

number is 6. It consists of two interpenetrating FCC sublattices; one made up of sodium atoms

and the two sub lattices are displaced by The positions of four sodium atoms are (0 0 0),

, , , while those of the chlorine atoms are )2

1

2

1

2

1( ,(0 0 ),( 0 0 ),

(0 0) . Each ion has 6 ions of other type as its nearest neighbors .So its coordination

number is 6 .Its lattice constant is a = 5.63 Å

Structure of Cesium chloride:

Structure of Diamond:


Cs+

Cl-

It is a cubic structure.Coordination number is 8 .

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Dept. of Physics

Diamond structure consists of two inter penetrating face centered cubic lattices. Each carbon

atom is surrounded by 4 other carbon atoms situated at the corners of a regular tetrahedron.

The unit cell for this structure is an FCC with a basis made up of two carbon atoms associated

with each lattice site .The positions of two basis atoms are( 0 0 0 )and .Each atom is

surrounded by four nearest atoms which form a regular tetrahedron.

Structure of Zinc Sulphide:

Packing factor:

It is the ratio of total volume occupied by the atoms in the unit cell to the total volume of the unit

cell.

For simple cubic structure:

Number of atoms per unit cell = 1


Zn

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Volume of one atom =

Volume of the unit cell = a3

Here a = 2R,

Volume of the unit cell = 8R3

Packing factor =

For BCC structure:

BCC:


Volume of two atoms = 2.

Volume of the unit cell = a3

For BCC, a =

Volume of the unit cell =

Packing factor = 0.68

For FCC structure:


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Volume occupied by four atoms = 4 X

For FCC, a =

Volume of the unit cell = a3 = 16 R3

Packing factor = (16/3) R3 / 16 R3 = 0.74


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Assignments:

1. How do you relate the properties of crystals which you have studied in this chapter with

their applications ?

2. Study the different techniques used to determine the structure of a crystal.

3. How important is the study of crystal physics in understanding the DNA structure?

4. Discuss the techniques which were used by Watson and Crick to propose the double

Helix structure of DNA


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Contents

Chapter 3: X-rays (Discovered by Roentgen in 1895,Wavelength range 10 Ǻ – 1 Å )

Production :

X-rays are produced when fast moving electrons are suddenly stopped by a solid target. Coolidge

tube is as shown in the figure. The pressure inside is 10 -5mm of Hg. The cathode is Tungsten

filament heated by a high tension battery. The electrons emitted by filament through thermoionic

emission are accelerated towards the target. The target must be cooled to remove the heat

generated.

Properties of X-rays:

1. They are not deflected by electric and magnetic fields.

2. They have no charge.


Electron beam

X-rays

Target

Cooling plant

Coolidge tube apparatus

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Dept. of Physics

3. They cause fluorescence.

4. They have high penetrating power.

5. They affect photographic plates

X- Ray spectrum:

The graph of Intensity of X- rays against wavelength is as shown. The smoothly varying curves

represent the continuous spectrum. But as the applied voltage is increased sharp peaks are seen.

This feature is known as characteristic spectrum representing the characteristics of the target

atoms.

Features:

1. For each potential, there is a minimum wavelength.

2. As voltage is increased, λmin is shifted towards smaller values.

3. At lower voltages the graph is continuous. Continuous spectrum is formed due to the fact that

electromagnetic radiations are emitted in all frequencies when a charged particle is

accelerated.

4. Characterstic spectrum is due to the transition of electrons within the target atoms that have

been hit by electrons. Suppose an atom in the target is bombarded by a high speed

electron and K-shell electron is removed, a vacancy is created in K-shell. This vacancy is

filled up by an electron from L,M,N shells .These possible transitions result in the

lines.

Laue spots:


Wavelength

30Kv

λmin

Intensity 70Kv

40Kv

Line spectrum or characteristic spectrum

Continuous spectrum

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X- Rays were passed through a crystal (like ZnS).The transmitted light was received on a

photographic plate. After exposure the photographic plate looked as shown in the figure. The

central dark spot arises due to direct beam. The central spot is surrounded by many fainter spots

arranged in a definite pattern .This indicates that the incident X-ray beam has been diffracted

from the various crystal planes .These spots are called as Laue’s spots.

Bragg’s law:

Let us consider a set of parallel lattice planes of a crystal separated by a distance ‘d’.Suppose a

narrow beam of X-rays of wavelength λ is incident upon these planes at an angle θ as shown in

the figure.


Coolidge tube

slitsZnS crystal

X-ray beam

Photographic plate

Atomic planes

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Developed photographic plate

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Consider the ray PQ incident on the first plane. It is reflected along QR .The rays P1Q1and Q1R1

are the incident and reflected rays. QT and QS are the normals.

Total path difference between the two rays = T Q1 + Q1S = 2d sinθ

From triangles Q T Q1 ,

Q Q1 S Q1 S = d sinθ

If the path difference is nλ, then constructive interference takes place and maximum

intensity is produced.

2d sinθ =nλ

X-ray diffraction spectrometer : Apparatus: A source of X-ray, slits, crystal mounted on a circular turn table spectrometer with

vernier scale.

Construction: X –ray beam after reflection from the crystal enters the ionization chamber

mounted on a mechanical arm which can turn co axially with the turn table .This ionization

chamber is coupled with the turn table so that if the turn table rotates through an angle ‘θ’, the

ionization chamber rotates through ‘2θ’.The ionization current produced by X-rays is recorded by

the electrometer.

Working: The ionization current is measured for different values of glancing angle ‘θ’. A plot is

then obtained between ‘θ’ and ionization current .For certain values of ‘θ’, the intensity of

Ionization current increases abruptly.

Whenever the crystal receives X-rays at an angle of incidence satisfying Bragg’s law

2d sinθ = nλ ,constructive interference takes place and maximum intensity occurs .The rise in

current occurs more than once as ‘θ’ is varied because the law is satisfied for various values of ‘n’

i.e., 2d sin θ = 1λ ,2λ,3λ etc.


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Application of X-rays :

1. To detect cracks and cavities in different structures.

2. Used to study the structure of alloys.

3. To detect homogeneity of welded points.

4. To study the structure of solids and organic molecules.

5. To study bone structures.

6. To treat cancer.

7. To study the structure of genes.

Moseley’s law:

According to this law, the frequency of a spectral line in x-ray spectra varies as the square of

atomic number of the element emitting it.

- Frequency

Z - Atomic number


Coolidge tube (Source of X-ray)

slit

Electrometer

Turn table on which powdered crystal is taken

Vernier scale

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Dept. of Physics

Applications:

1. From this law it was understood that it is the atomic number which determines the physical

and chemical properties of elements.

2. The discrepancy in the position of certain elements like Argon and Potassium:

i.e., should come before

3. It helps in predicting the existence of undiscovered elements.

Assignments:

1. Study the application of X – ray diffraction technique in the field of Bio technology.

2. Collect information about CHANDRA X- RAY SPACE TELESCOPE launched few

years back by NASA.


Dept. of Physics

Contents

CHAPTER 4: Electron conduction in solids:

Classical free electron theory:(Drude – Lorentz theory)

Postulates:

1. A metal is assumed to possess a three dimensional array of ions in between which there

are freely moving valence electrons confined to the metallic boundary.

2. These free electrons are treated as equivalent to gas molecules and they are assumed to

obey the laws of kinetic energy of gases. In the absence of any electric field the energy

associated with electrons is equal to

Kinetic energy =

3. The electric current in a metal is due to the drift of electrons in a direction opposite to

Electric field.

4. The electric field due to all the ions is assumed to be constant.

Drift velocity:


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The net displacement in the position of electrons per unit time caused by the application of

electric field is known as drift velocity.

Explanation:

Consider a cylindrical metal connected to a voltage source. The free electrons experience a force F = qE

From Newton’s second law F = ma = m

If u = 0 , v = vd , t = then

qE = m

vd =

Where e – charge on the electron, E – Electric intensity, - Mean collision time.

Mean Collision Time: It is the average time taken between two consecutive collisions of

electrons.

Relaxation time: It is the time taken for the drift velocity to decay to (1/e) times after the

removal of electric field.

Mean free path: The average distance traveled by the electrons between two successive

collisions.

Expression for the electric current through a metal :

Consider a cylindrical conductor with area of cross section A , electron concentration ‘n’

connected to a voltage source. Let the drift velocity be Vd.


Dept. of Physics

The number of electrons crossing unit area of a imaginary cylinder of length vd per second

is given by n A vd

Total charge crossing unit area per second = neAVd

By definition Electric current =

Electric current = neAVd

Expression for electrical conductivity :

Step 1: Derive the expression for Vd.

Step 2: Derive the expression for Electric Current.

Step 3 : From the definition of current density J =

From Ohms law J = E

=

Variation of resistivity with temperature:


V

T

For metals

45

Dept. of Physics

Matthisen’s rule: It states that the total resistivity of a metal is due to the sum of resistivity due

to scattering of electrons with lattice vibrations which is temperature dependent and resistivity

due to the scattering of electrons by impurity atoms/ defects which is temperature independent.

=

Note:

1. Thermal conductivity: K =

2. Electrical conductivity:

3. Widemann Franz law: The ratio of the thermal conductivity to the electrical conductivity of

metals is proportional to the absolute temperature.

Mobility of electrons:

It is defined as the ratio of the drift velocity to the electric field applied.

It can be shown that .

Failures of Classical free electron theory:

1. Prediction of low specific heats for metals:

Classical free electron theory assumes that conduction electrons are classical particles similar to

gas molecules. Hence,they are free to absorb energy in a continuous manner. Hence metals

possessing more electrons must have higher heat content. This resulted in high specific heat given

by the expression CV = .

This was contradicted by experimental results which showed low specific heat for metals.

2. Temperature dependence of electrical conductivity:

From the assumption of kinetic theory of gases


Dept. of Physics

Also mean collision time τ is inversely proportional to velocity,

However experimental studies show that

3. Dependence of electrical conductivity on electron concentration:

As per free electron theory,

The electrical conductivity of Zinc and Cadmium are 1.09 x 107 /ohm m and .15 x 107 /ohm m

respectively which are very much less than that for Copper and Silver for which the values are

5.88x107 /ohm m and 6.2 x 107 /ohm m. On the contrary, the electron concentration for zinc and

cadmium are 13.1x1028 /m3 and 9.28 x1028 /m3 which are much higher than that for Copper and

Silver which are 8.45x1028 /m3 and 5.85 x1028 /m3.

These examples indicate that does not hold good.

4. Mean free path, mean collision time found from classical theory are incorrect.

Quantum free electron theory:

Assumptions:

1. The energy of conduction electrons in a metal is quantized.

2. The distribution of electrons amongst various energy levels is according to Pauli’s exclusion

principle and Fermi – Dirac statistical theory.

3. The average kinetic energy of an electron is equal to

4. The attraction between the electrons and ions, the repulsion between electrons are ignored.

Fermi – Dirac theory:


Dept. of Physics

According to this theory, the free electrons are quantum particles .The distribution of electrons

amongst various energy levels is according to Pauli’s exclusion principle. the number of

conduction electrons per unit energy range per unit volume is given by

n(E)de = g(E)dE . f (E)

where g (E) is density of states

f(E) is the Fermi probability factor .

Fermi energy ( ):

It is the highest energy possessed by an electron at zero Kelvin.

Fermi probability factor: It represents the probability of occupation of an energy level.

f (E) =

Density of energy of states:

It represents the number of energy levels per unit energy range per unit volume.

g (E)=

To show that energy levels below Fermi energy are completely occupied:

For E < EF, at T = 0,

f (E) = =

To show that energy levels above Fermi energy are empty:

For E > , at T=0

f (E) = =

At ordinary temperatures, for E = EF,

f(E) =

Fermi energy for T > 0k, Ef = Ef0


Dept. of Physics

Success of quantum theory:

1. Specific heat:

Classical theory predicted high values of specific heat for metals on the basis of the assumption

that all the conduction electrons are capable of absorbing the heat energy as per Maxwell -

Boltzmann distribution i.e., CV=

But according to the quantum theory, only those electrons occupying energy levels close to

Fermi energy (EF ) are capable of absorbing heat energy to get excited to higher energy levels.

Thus only a small percentage of electrons are capable of receiving the thermal energy and specific

heat value becomes small.

It can be shown that CV = .

This is in conformity with the experimental values.

2. Temperature dependence of electrical conductivity.

According to classical free electron theory,

Electrical conductivity

Where as from quantum theory

Electrical conductivity

This is in agreement with experimental values.

3. Dependence of electrical conductivity on electron concentration:

According to classical theory,


T= 0 K

T> 0 K

EF

49

Dept. of Physics

But it has been experimentally found that Zinc which is having higher electron concentration

than copper has lower Electrical conductivity.

According to quantum free electron theory,

Electrical conductivity where VF is the Fermi velocity.

Zinc possesses lesser conductivity because it has higher Fermi velocity.

Application of Quantum free electron theory :

Thermionic emission:

It is the process of emission of electrons from the surface of metallic conductors on heating to

high temperature.


EF

Φ

Fermi level

Highest energy level beyond which electrons are free

50

Dept. of Physics

At T = 0K all the levels up to the Fermi level are filled. As the temperature is increased, the

electrons are excited thermally and move to higher energy states. For the electrons to be ionized,

they must acquire a minimum energy known as work function.

The current density J = AT2 where

A – Emission constant ,T– Temperature, Φ – work function, K – Boltzmann constant.

Density of states:

It is defined as the number of energy levels per unit energy range per unit volume.

To determine the density of states, we first count the number of states that have energy below this

energy level. This is done using the diagram given below in quantum number space.

The lattice consists of points with positive integer coordinates (n1,n2,n3) and occupies one octant

of three dimensional space .Each point corresponds to one energy level for a particular value of

n1,n2,n3 such that it is the square of the distance of the point from the origin.

So for a 3 dimensional cube of a length L

E = ……………………(1)


n1

n2

n3

51

Dept. of Physics

Therefore all lattice points within the octant of radius R correspond to energy, less than or equal

to E .If each cubic cell in the figure is of unit volume , and one point (one energy level)

corresponds to each cubic cell , then the total number of points in the octant of radius R is equal

to the volume of the octant.

Total number of energy levels less than energy E = Total number of points or total number of

cells

= volume of the octant

=

Since each level can accomodate two electrons of opposite spin

Total number of energy levels less than energy E is N =

Substituting for R

N =

Density of states g(E) = Number of energy levels of energy E in the range

E = dN/dE =

g(E)(dE) = dE


Dept. of Physics

Assignments:

1. Study the applications of quantum theory to understand the properties of metals.

2. The density of states is a universal constant. Explain.


Dept. of Physics

ContentsCHAPTER 5: Magnetic materials :

Diamagnetism:

Langevein’s theory:

The atoms of the diamagnetic material do not possess magnetic moment .They have completely

filled shells. Diamagnetism is the result of Lenz’s law operating in atomic scale.

When a diamagnetic material is kept in a magnetic field, the magnetic flux through the surfaces

bounded by the molecular circulatory currents is increased and a current is induced in each

elementary circuit as to oppose the external field. This reduces the density of flux lines. Hence

diamagnetic materials get repelled by the magnetic field.

They are characterized by negative susceptibility, magnetic permeability less than one.

Ex: Cu, Au, Ge, diamond, Nacl, Al2O3

The magnetic moment induced by the external field B is given by

The negative sign shows that the induced moment is opposite to the applied field.

According to Langevein theory, diamagnetism is attributed to the influence of the magnetic field

on the orbital momentum of the electrons.


Dept. of Physics

Diamagnetic susceptibility = χ =

Paramagnetic materials:

Paramagnetism occurs in materials where the atoms or molecules possess a feeble magnetic

moment.

The orbital and spin motion of electrons contributes to that.

Langevin’s Theory of Paramagnetic materials:

When a paramagnetic material is kept in a magnetic field, atoms with individual magnetic

moments tend to align in the direction of magnetic field and the specimen acquires magnetization

in the direction of magnetic field.

Characteristics:

1. Low magnetization, low susceptibility.

2. Susceptibility is small, positive and varies inversely with temperature.

3. Permeability is greater than 1.

Paramagnetic susceptibility χ =

Where C is curie constant .

This result namely the magnetic susceptibility of atoms varies as is known as Curies law.

Ex: CuSO4 , O2

Paramagnetic susceptibility = χ =

Weiss theory of Paramagnetism:


Absence of magnetic field Presence of magnetic field

55

Dept. of Physics

Langevin theory failed to explain the complex relation between susceptibility and temperature

exhibited by paramagnetic substances. Also this theory does not explain the relationship between

para and ferromagnetism.

Weiss introduced the concept of internal molecular field in order to explain the dependence of

susceptibility. In a gas, the molecules are influenced by their magnetic moments and

consequently, there should exist a field with which they interact.

Weiss derived the expression for susceptibility as

Paramagnetic susceptibility = χ =

Here θc is known as Curie point.

This result is in agreement with the experimental results.

Ferromagnetism:

These are the permanent magnets which exhibit hysteresis. It arises due to the self alignment of

groups of atoms carrying permanent magnetic moment in the same direction. The magnetic

moment is an account of spin of the electrons. Ferromagnetic materials are characterized by curie

temperature above which they become paramagnetic materials.

Ex: Fe, Ni,gd,Dy

According to Weiss law, the resultant field in a ferromagnetic material is the sum pf applied field

and field due to all the magnetic dipoles .

where γ is Weiss molecular constant.

Ferromagnetic susceptibility χ =

B – H graph in Ferromagnetic materials


Dept. of Physics

1. A ferromagnetic solid can be assumed to be comprised of small number of small regions

called domains each of which is spontaneously magnetized. The magnetic moments of all

the exams are all aligned in a particular direction. However the different domains are so

oriented as to make the net magnetization zero.

2. The process of magnetization consists in rotating the different domains in the direction of

applied field so that the specimen exhibits net magnetization.

3. When a magnetic field is applied on a ferromagnetic material, the domains nearly parallel

to H can grow in size at the expense of antiparallel domains and gradually all the domains

align along the applied field at which the material is said to be saturated.

The hysteresis can be explained as follows.

1. As H is further increased, the rate of increase of B falls and ultimately becomes zero and

the flux density B reaches a saturation value indicated as Bsat in the figure.

2. As the applied field H is reduced from the saturation value to zero, the reduction of flux

density does not follow the same path.

3. When H becomes zero, their remains certain amount of flux in the material called the

remnant flux density Br . The material remains magnetized even in the absence of an

external field.


BsatBr

-Hc

57

Dept. of Physics

4. To reduce the remnant flux Br to zero, it is necessary to apply H in the reverse direction

and the amount Hc required to make Br zero is called the coercive force.

5. As the field is increased beyond Hc , the flux density reaches saturation.

6. When a ferromagnetic material is taken over one cycle of magnetic field, it exhibits

hystresis loop.

7. The area of the hystresis loop signifies the amount of energy required for magnetization.

Ceramic magnets:

These are the magnetic materials with metallic and non metallic elements. They possess

ferromagnetic properties like domain structure, hysteresis curve etc.

They are expressed by the general formula MO Fe2 O3 .They possess strong ferromagnetic

properties such as domain structure, hystresis.

Properties: 1.High magnetic permeability

2. High electrical resistance

3. High remnant magnetism, Coercivity

Ex: Ferrites – Manganese ferrite, Nickel Ferrite

Applications:

1. Hard ferrites are used in the production of permanent magnets.

2. Soft ferrites are used to make cores of transformers.

3. They are used in magnetic films, magnetic discs, magnetic tapes.

Soft magnetic materials:


Dept. of Physics

These are temporary magnets which can retain magnetism for a short interval of time. They

possess small hysteresis loss.

Properties:

- Low remnant magnetism

- High permeability

- High susceptibility

Ex: Soft iron, Si, Steel, Alloys

Uses: They are used in the construction of cores of transformers.

Hard magnetic materials :

These are the magnetic materials which can retain magnetism permanently.

They possess large hysteresis loop, large remnant magnetization, and high coercivity.

Ex: Carbon, tungsten.

Uses: They are used in making permanent magnets, microphones, loud speakers.


Dept. of Physics

Assignment:

1. List out the importance of magnetic materials in electronic circuits.

2. An atom is a magnet. Comment.


Dept. of Physics

Contents

CHAPTER 6: SUPER CONDUCTIVITY :[KAMERLINGH IN 1914]

It is a phenomenon in which some materials loose their resistance completely below certain

temperature.

Temperature dependence of resistivity:

Critical temperature: (TC)

It is the temperature at which a normal material transforms to superconducting state.

Material TC (K)


T

For metals

T

For superconductors

61

Dept. of Physics

HgBa2 Ca2Cu3O8 134

Bi-Sr-Ca-Cu-O 105

YBa2Cu3O7 92

Nb3Ge 23.2

Nb 9.46

Pb 7.18

Hg 4.19

Zn 0.88

BCS Theory :[Bardeen , Cooper, Schrieffer]

According to this theory superconductivity occurs when an attractive interaction known as

electron-lattice-electron interaction is established resulting in the formation of cooper pairs.

In a lattice, an electron passing close to a lattice atom is attracted towards it and displaces it. This

lattice atom will interact with another electron and in turn forms an electron – lattice –

interaction. This system of two electrons of equal and opposite momentum attached to a lattice

atom is known as a cooper pair. The electrons are bound to the lattice atom through the exchange

of phonons (Lattice vibrations).When electrons flow in the form of cooper pairs they do not get

scattered as the energy required to break it up is large enough. This reduces the resistance.

Properties of superconductors:

1. When a superconductor is placed in a magnetic field, it expels the magnetic flux out of its body

and behaves like a diamagnet.This effect is known as Meisseners effect.


Lattice atom

ElectronElectron

Phonons

62

Dept. of Physics

2. They have abnormal specific heats.

3. Isotope effect:

Isotopes of an element possess different critical temperatures.

The critical temperature is inversely proportional to the isotopic mass.

where

Ex: for is 4.185K

For is 4.16K

Types of superconductors

Type 1 Superconductor:

These are pure superconductors.


SuperConductor

Normal state

Superconducting state

63

Dept. of Physics

When kept in magnetic field, initially they continue to exhibit superconductivity and the negative

magnetic moment increases. At critical magnetic field there is a sharp transition to normal

state .These possess low critical magnetic fields.

Ex: Al,Pb

Type 2 superconductor:

These are generally alloys.

When kept in magnetic field, initially they continue to exhibit superconductivity and the negative

magnetic moment increases. At HC1, the flux lines start penetrating .As the magnetic field is

increased, the super conductivity coexists with magnetic field and this phase is known as mixed

state(vortex state). At higher critical magnetic field HC2, the penetration is complete and the

material transforms to normal state.

Ex: Nb3Ge, YBaCu2O3

Applications of superconductivity:

1. Super conducting magnets:

Superconducting magnets are used to develop strong magnetic fields required in various

techniques such as MRI scanning, detection of ore deposits etc.

A superconducting solenoid consists of superconducting filaments embedded in copper matrix.

The matrix prevents mechanical fractures that may happen in the superconducting material due to

the flow of large currents.

High field magnetic applications

Nuclear magnetic resonance (medical diagnostics)

Magnetic levitation

Ore refining


Superconducting state

Vortex state

64

Dept. of Physics

Magnetic shielding

2. Maglev vehicles:

These are the vehicles which are set afloat above the track reducing the friction. With such an

arrangement, great speeds could be achieved with low energy consumption.

The vehicle consists of superconducting magnets built at the base. Large currents are passed

through aluminum guide way. Due to the interaction between the magnetic fields produced by the

superconducting magnet and the aluminum guide way, the vehicle is set afloat. These magnetic

fields also propel the vehicle.

3. Lossless power transmission:

Since there is no resistance to the flow of current in a superconductor, they can transfer electrical

power without any loss. In ordinary conductors, 30-40% of the power is lost on account of Joules

heating effect.

DC Josphsons effect:

When the distance between the two superconducting bars is reduced below 1nm, the voltmeter

suddenly due to the passage of electric current in the energy gap.

AC Josephsons effect:

When the superconducting bars are insulators, on application of a voltage between the

superconducting bars, a high frequency electromagnetic radiation emanates from the gap.

SQUIDS :( Superconducting quantum interference devices)

SQUID is a bi junction quantum interferometer formed from two Josephson’s junctions.


Dept. of Physics

When magnetic field is applied, current is induced in the SQUID which opposes the external

magnetic field. Magnetic Flux through the SQUID is quantized .The current through the SQUID

varies periodically with the external magnetic field.

Uses: Squids are used to detect very low magnetic fields .They are used to study magnetic signals

in the brain. They are used by miners to locate ore deposits.


J

J

J1 J2

Josephons junction

I

B

66

Dept. of Physics

Assignment:

1. Elaborate on the applications of SQUIDS in MRI - NMR scanning.

2. All the cooper pairs have the same wave function. Comments.


Atom in a dielectric in presence of electric field

E

Dept. of Physics

ContentsCHAPTER 7: Dielectrics :

These are the materials which do not conduct electricity.

Dipole: Two equal and opposite charges separated by a small distance constitutes a dipole.

Polarisation : The separation of effective centre of positive and negative charges in a substance

by the application of electric field is known as polarization.

Dipole moment is the product of charge and the separation distance.

Atomic polarisability p = where is the polarization constant.


Atoms in a dielectric in the absence of electric field

68

Dept. of Physics

Polar dielectrics: These possess permanent dipole moment .They are permanently polarized in

nature.

Ex: Water, Kcl, NH3

Non Polar dielectrics:

These are the materials which do not possess permanent dipole moment.

They get polarized only in the presence of external electric field.

Ex: O2, N2, He, Ne

Expression for static dielectric constant:

Consider a parallel plate capacitor of area A, Charge density and total charge Q.

From Gauss’s law, Electric intensity between the plates E =

Potential difference V = E . d = . d

(a)In the absence of a dielectric

Capacitance C without dielectic =

(a) In the presence of a dielectric Cwith dielectric =

Static dielectric constant =

It is defined as the ratio of capacitance of a capacitor with a dielectric to its capacitance in

the absence of a dielectric.

Internal fields in a dielectric:

It is the resultant of the applied field and the field produced due to all the dipoles.

Consider a dielectric material kept in an electric field Ea.


Dept. of Physics

The dipole is assumed to be in a one dimensional array and are oriented in the same direction.

The electric field at O due to dipole A1 is given by

EA1= = as θ = 0

The electric field at O due to A2

EA2 = as all dipoles are oriented in the same direction.

Field at O due to A1 and A2 is E A1 ,A2 = = E1

Similarly field at O due to B1 and B2 is = = E2

The resultant field due n dipoles is given by

ER = E1 + E2 + E 3+ E 4+ E5 + E6+ …………….

= + +……………..

=

The internal electric field is Ei = Ea +

Ei = Ea +

In three dimensional case , (1/d3) could be replaced by N, the number of atoms per unit volume

and (1.2/Π) by a constant γ which depends on the crystal structure.


B1

Ea

A1 O A2 B2

Er

+Q -Q

Eθ

70

Dept. of Physics

Hence Ei = Ea + =

Clausius – Mosotti relation:

This expression relates dielectric constant of an insulator (ε) to the polarization of individual

atoms(α) comprising it.

where N is the number of atoms per unit volume

α is the polrisability of the atom

εr is the relative permittivity of the medium

εo is the permittivity of free space.

Proof:

If there are N atoms per unit volume ,the electric dipole moment per unit volume –known as

polarization is given by

P = NαE

By the definition of polarization P, it can be shown that

P =

…………………..(1)

The internal field at an atom in a cubic structure(γ =1/3) is of the form

Substituting for in equation (1)


Dept. of Physics

Different polarization mechanisms:There are 4 mechanisms.

1. Electronic polarization

2. Ionic polarization

3. Orientation polarization

4. Space charge polarization

Electronic polarization: These are generally seen in the case of covalent compounds.

When a covalent compound is placed in electric field, displacement of electron cloud takes place

relative to the nucleus. This displacement creates a dipole which develops dipole moment.

Electronic polarisability αe =

Ionic polarization:

This is exhibited by ionic compounds.

When ionic compounds are kept in an electric field, displacement of positive and negative ions

occurs developing a dipole moment.

Ionic polarisability αi=

Orientation polarization:

Polar molecules exhibit this mechanism.


Dept. of Physics

When polar molecules are kept in an electric field, already existing dipoles tend to align in the

direction of applied electric field .This increases the dipole moment.

Orientation polarization αo =

Space charge polarization:

This polarization exists in materials possessing different phases due to difference in temperatures.

In such materials charge carriers drift and accommodate in certain regions of higher conductivity

causing dipole moment.

Dielectric loss:

It represents the loss of electrical energy due to collisions between the dipoles which occur during

the reorientation of dipole in resonance with changing electric field.

Expression for dielectric loss:

Consider a material kept in alternating field. As a result the dipoles engage in switching action.

This movement is opposed by the collisions .This opposition is equivalent to the presence of a

resistance accompanying the resistor.

The energy lost in a capacitor is equivalent to loss of power across a resistor in an equivalent

circuit shown below.


Dielectric

73

Dept. of Physics

The current through resistor and the capacitor are shown in the phasor diagram.

The angle θ between Ic and I is known as dielectric loss angle.

Dielectric loss = V IR = VI cos (90-θ) = VI sinθ

Also Ic = I cosθ

Hence Dielectric loss = V Ic = VIctanθ

Frequency dependence of polarization:


Ic

IR

74

Dept. of Physics

From the graph, it can be observed that

orientation polarization is active till micro frequencies. Beyond this frequency, the atomic

dipoles cannot keep in step with the changing field and the orientation polarization is

destroyed

at infrared frequencies, the ionic polarization disappears as the ions will no longer remain

separated.

Beyond ultraviolet frequencies, electronic polarization diminishes.

Ferroelectrics:

These are the dielectric materials which easily get polarized by the application of electric field.

They exhibit hysteresis with respect to P and E similar to ferromagnets.

Ex: NaK(C4H4).4H2O, BaTiO3 , KH2 PO4

In these materials, there exist large regions which are oriented in specific directions. When kept in

electric field, such regions align in the direction of field increasing the flux density. The

polarization rises rapidly and attains saturation at specific point. On subsequent reduction in the

field to zero, polarisation does not diminish and remanent polarization remains. The negative field

to reduce the polarization to zero is known as coercive field. In one cycle of polarisation, these

materials show hysteresis curve.


Microwave frequencies

Infrared Frequencies

Ultraviolet frequencies

αe+ αi + αo

αe+ αi

αe

Frequency

Dielectric Loss

75

Dept. of Physics

The curie –Weiss law for ferroelectrics is given by

Ferroelectric susceptibility χ =

Piezoelectrics: These are the dielectrics which when subjected to mechanical stress; develop

voltage across the surface on account of polarization. This is observed with crystals without

centre of symmetry. When a mechanical stress is applied on an array, the crystal develops net

dipole moment due to each ion .

Few Applications:

1.Specifically cut quartz discs are used as oscillators.

2. Acoustic pulses which are used in underwater search operations are produced by piezoelectric

transducers.


PsatPr

-Ec

76

Dept. of Physics

Assignment :

1. Study the variation of dielectric constant with (a) temperature:

2. Dielectrics are good conductors of electro magnetic energy. Discuss.

3. Air as a dielectric is facilitating electromagnetic wave propagation. Comments.


Dept. of Physics

CHAPTER 8: Optical properties of solids:

Luminescence: It is a process in which a substance absorbs energy and emits a part of this energy

in the form of light.

Fluorescence: It is a kind of luminescence in which the material emits light simultaneously with

the incidence of light and the emission time does not extend beyond 10-8s after the incident light

is cut off.

Phosphorescence:

It is a process in which when a light radiation falls on a material, it continues to emit light for a

time greater than 10-8s after the incident radiation is cut off.


Dept. of Physics

LASERS: [LIGHT AMPLIFICATION BY STIMULATED EMISSION OF RADIATION]

Spontaneous emission: It is the emission of photons by an atom making a transition

from a higher level to a lower level without any external aid.

Stimulated emission: It is the emission of photons by an atom making a transition from a higher

level to a lower level when stimulated by another photon.

Population inversion: It is a state in which number of atoms in an excited energy state is more

than the number of atoms in the ground state.

Properties of Laser:

1. High degree of monochromaticity.

2. High power.

3. High intensity.

4. Coherence.

5. Directionality

Basic principles:

Optical pumping:

The population inversion is achieved by the method of optical pumping. In this process the active

medium is excited by the irradiation with light or through electrical discharge .The atoms of the

active medium absorb energy and rise to higher energy state. As a result the number of atoms in

the higher energy states increases and the population inversion is said to be achieved.

Population inversion is achieved in certain systems which possess metastable

states. The life time of the excited atoms in these energy levels is higher(10 -3 s).Hence atoms stay

for a longer time .

Induced absorption:

It is a process in which an atom at a lower level absorbs a photon to get excited to the higher

level.

Let E1 and E2 be the energy levels in an atom and N1 and N2 be the number density in these

levels respectively. Let Uγ be the energy density of the radiation incident.


Dept. of Physics

Rate of absorption is proportional to the number of atoms in lower state and also on the energy

density Uγ.

Rate of absorption = B12 N1 Uγ

Here B12 is a constant known as Einsteins coefficient of spontaneous absorption.

Spontaneous emission:

It is a process in which ,atoms at the higher level voluntarily get excited emitting a photon. The

rate of spontaneous emission representing the number of such deexcitations is proportional to

number of atoms in the excited state.

Rate of spontaneous absorption = A21 N2

Here B12 is a constant known as Einsteins coefficient of spontaneous emission.

Stimulated emission:

In this process, an atom at the excited state gets deexcited in the presence of a photon of same

energy as that of difference between the two states.


E1

E2

h

EE 12

E1

E2

h

EE 12

E1

E2

h

EE 12

80

Dept. of Physics

The number of stimulated emissions is proportional to the number of atoms in higher state and

also on the energy density Uγ.

Rate of stimulated emission = B21 N2 Uγ

Here B21 is the constant known as Einsteins coefficient of stimulated emission.

Expression for energy density:

Induced absorption:

It is a process in which an atom at a lower level absorbs a photon to get excited to the higher

level.

Let E1 and E2 be the energy levels in an atom and N1 and N2 be the number density in these

levels respectively. Let Uγ be the energy density of the radiation incident.

Rate of absorption is proportional to the number of atoms in lower state and also on the energy

density Uγ.

Rate of absorption = B12 N1 Uγ

Here B12 is a constant known as Einsteins coefficient of spontaneous absorption.

Spontaneous emission:

It is a process in which ,atoms at the higher level voluntarily get excited emitting a photon. The

rate of spontaneous emission representing the number of such deexcitations is proportional to

number of atoms in the excited state.

Rate of spontaneous absorption = A21 N2

Here B12 is a constant known as Einsteins coefficient of spontaneous emission.


E1

E2

h

EE 12

81

Dept. of Physics

Stimulated emission:

In this process, an atom at the excited state gets deexcited in the presence of a photon of same

energy as that of difference between the two states.

The number of stimulated emissions is proportional to the number of atoms in higher state and

also on the energy density Uγ.

Rate of stimulated emission = B21 N2 Uγ

Here B21 is the constant known as Einsteins coefficient of stimulated emission.

At thermal equilibrium,

Rate of absorption = Rate of spontaneous emission + Rate of stimulated emission

B12 N1 Uγ = A21 N2 + B21 N2 Uγ

Rearranging this, we get


E1

E2

h

EE 12

E1

E2

h

EE 12

82

Dept. of Physics

By Boltzmans law ,

Hence

From Planck’s radiation law,

Comparing these expressions, we get

and

Ruby Laser :[Mariman in 1960]Ruby is a Al2O3 crystal with 0.05%chromium atoms.

- Solid laser -

Construction:


Dept. of Physics

Construction: It consists of a ruby rod of diameter of 0.5cm and length 5 to 20cm.The end faces

are grounded and polished and are parallel to each other. One of the end face is partially silvered

and the the other is fully silvered. The ruby rod is placed along the axis of a high intensity Xenon

flash lamp.

Working: When the light from the Xenon lamp falls on the ruby rod, it absorbs photons and

raises the chromium ions to the energy bands and .These chromium ions in these bands

have a life time of 10-8s and make non radiative transitions to metastable states. As the life time of

chromium ions in the metastable states is high, number of such chromium atoms increases and

population inversion is achieved between meatstable states and ground states which are called as

lasing energy levels.

The laser transition occurs between these energy levels and a pulsal out put is given.

Generally Ruby laser generates laser beam of wavelengths 6943Ǻ and 6928Ǻ.

Chromium atoms are fixed in the lattice of Aluminum and aluminum provides metastable states.


Laser beam

Highly silvered face

Power supply

Partially silvered face Ruby rod

Helium flash lamp

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Dept. of Physics

Draw Back: The output is pulsal.

Helium – Neon Laser (Gas laser – 4 level system)

Construction: It consists of fused quartz tube of length nearly 50cm and diameter 1.5cm.The tube

is filled with Helium and Neon under pressure1mm and 0.1mm of Hg respectively. The ends of

the tube are sealed with transparent material made of quartz and are called Brewster windows.

Two plane mirrors are fixed on either side of the tube. Among them one is partially silvered .

Optical pumping is done by electrical discharge.


Metastable states

Non radiative transitions

6943Ǻ 6928Ǻ

Energy bands

He – Ne Mixture

Fully silvered mirror

Brewsters window

Partially silvered mirror

Polarised laser beam

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Dept. of Physics

Working: When a voltage of about 1000V is applied across the electrodes, due to discharge of

gases electrons are released .These free electrons accelerate towards the positive electrode and

they collide with Helium atoms exciting them to metastable state.

The Neon atoms possessing metastable energy levels almost same as the Helium atoms, receive

the energy through collisions with helium atoms and rise to their metastable states and population

inversion is achieved. The laser transition takes place between 3S and 2P levels yielding

predominant laser beam of wavelength 6325Ǻ.

Semiconductor laser :

It is the only device which can be used for amplification in the infrared and optical ranges.

Amplification is possible if the population of the valence and conduction bands could be inverted

as shown in the diagram.


2 S

2 S

2S

Laser Transitions

6923 Ǻ

11152 Ǻ

Non radiative transitions

Meta stable states of Neon

3S

Meatstable states of Neon

E

k

p regionn region

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Dept. of Physics

The first laser action was observed in a GaAs junction(8400Å) which is a direct gap

semiconductor.

When a heavily doped junction is forward biased, electrons from n side combine with holes on

the p side releasing photons. The junction region is the active region .The optical cavity is formed

by the faces of the crystal itself which are taken on the cleavage plane and are then polished. The

wavelength of the radiation depends on temperature. The wavelength of laser increases as the

temperature increases as the energy gap decreases. The frequency can be increased to the optical

region by alloying with phosphor according to the relation .

Requirements: The population inversion must be strong enough such that the downward

transitions are larger than absorption effects by free carriers. There is a threshold carrier

concentration (n) which can be related to junction current.

where τ is the carrier recombination time.

Ex: ZnS, CdS,InSb,PbS

Advantages: It is small size, high efficiency, can be easily connected to electronic circuits.

It is possible only in the case of direct gap semiconductors because of the requirement of

conservation of momentum.

Note: Recently, a new semiconductor laser ‘spin flip Raman’ is discovered. Here the transition of

electrons between two spin states in the presence of magnetic field results in laser.

Applications:

1. He-Ne laser beam is used as a carrier of sound waves in magneto optic modulators to

transfer messages.

2. Laser beam is used as heat source in welding to drill holes, tailoring industry for

stitching.

3. To detect unwanted tissues and foreign particle in human body.

4. Used in eye surgery (lasik), dental treatment, skin diseases.

5. Laser guided missiles play important role in warfare.

6. used in holography.

7. LIDAR systems are used to study atmosphere.


Dept. of Physics

Measurement of pollutants in the atmosphere: LIDAR (Light detection and Ranging):

A LIDAR can be used to evaluate the distance, altitude and angular coordinates of the object in

space.

In the LIDAR system, the pulsed laser beam is radiated in space, the laser beam undergoes

scattering at regions having particulate matter in high density. The scattered light is analyzed

through a receiving system consisting of a concave mirror, a photo detector, a band filter and a

processing circuit. Through the data analysis, the concentration of the pollutants is studied.

The composition of the pollutants is determined by studying the absorption spectra of the sample

collected from the atmosphere. Raman scattering method could also be used to identify the

pollutants in the sample.

Contents


Dept. of Physics

Chapter 9:- Optical fiber

It is a device used to transmit information in the form of waves .Its principle is total internal

reflection.

Construction:

It consists of

1. Innermost region which is the light guiding region.

2. The core region is surrounded by middle region called cladding. The refractive index of

cladding is less than that of core.

3. The outermost region is sheath. It is made of plastic which protects the core and cladding from

moisture.

Working : The light signal which enters the core falls at an angle greater than the critical angle .It

undergoes repeatedly total internal reflection as shown in the figure.

Types:

1. Single mode fiber:

It consists of a core which is made of glass having refractive index n1.The core is surrounded by a

cladding made of glass which is of refractive index n2 where n1 > n2.

The core is narrow and hence it can guide just a single mode.


Sheath

Core

Cladding

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Dept. of Physics

Step index multimode fibre :

Here the diameter of core is larger so that large number of rays can propagate.

Graded index multimode fiber:

In this type, the refractive index decreases in the radially outward direction from the axis and

becomes equal to that of the cladding at the interface. The refractive index of the cladding

remains uniform.

Applications:

1. They carry large amount of information.

2. Cost is low.

3. Light weight and compact.

4. Not affected by lightning and sparking

5. Endoscope uses optical fibers to study internal parts of human body.


Cladding

Core

Cladding

Variation of refractive index

Cladding

Core

Cladding

Cladding


Cladding

Core

Cladding

Cladding

Axis

90


Dept. of Physics

6. There is no interference between the communication channels and others and also the purity of

signals are unaffected.

7. There is no energy radiation from the fiber .Hence the loss of information by radiation is ruled

out.

Disadvantages:

1. Fibres are expensive.

2. Whenever there is line break, operations required to establish connection are highly skillful and

time consuming.

3. They undergo expansion and contraction with temperature that upset the alignments which will

lead to line break.

4. Maintainance costs are high.

Numerical aperture:

Acceptance angle (θ0):

It is that angle of incidence above which if the rays fall in to the fibre, they do not undergo total

internal reflection and it will be lost.

sinθ0 is known as numerical aperture of the fibre.

Expression for condition for propagation :

Consider a light ray falling in to the optical fibre at an angle of incidence θ0 equal to acceptance

angle. Let n0 be the refractive index of the surrounding medium .

Let n1 be the refractive index of the core.

Let n2 be the refractive index of the cladding.


A light ray

Cladding n2

Core n1

Cone of acceptance

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Dept. of Physics

From Snell’s Law:

For the ray OA n0 sinθ0 = n1 sinθ1

= n1(1-cos2 θ1 ………………….(1)

For the ray AB

n1 sin(90 - θ1) = n2 sin 90

[ here the angle of incidence is (90 - θ1) for which angle of refraction is 900].

n1 cosθ1 = n2

Substituing for cosθ1 in equation (1)

n0 sinθ0 = n1

sinθ0 =

If the medium surrounding the fibre is air then n0 = 1,

Numerical aperture = sinθ0 =

The total internal reflection will take place only if the angle of incidence θi < θ0

sin θi < sin θ0

sin θi <

This is the condition for propagation.


A light ray

Cladding n2

Core n1

Cone of acceptance

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Dept. of Physics

Fractional index (∆):

It is the ratio of difference in the refractive index of core and cladding to the refractive index of

the core.

∆ =

Relation between numerical aperture and refractive index:

Numerical aperture = =

If n1 n2

Numerical aperture: =

Point to point communication system using optical fibers

This system is represented through a block diagram as follows.

The analog information in the form of voice gives rise to electrical signals through the

transmitter. The analog signal is converted in to binary data with the help of electronic device

coder. The binary data in the form of electrical pulses are converted in to pulses of optical power

using the optical transformer. This optical power is fed to the optical fiber. Out of this incident


Binary electrical signals

voice

Binary electrical signals

Optical fiber

Voice information in analog form coder

Optical transmitter

Photo detector

Detector Information in analog form

Optical signals in pulse form

voice

Light source

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Dept. of Physics

light, only those modes within the angle of acceptance cone will be sustained for propagation by

means of total internal reflection. At the receiving end of the fiber, the optical signal is fed in to a

photo detector where the signal is converted to pulses of current which is then fed to decoder

which converts the sequence of binary data stream in to an analog signal which will be the same

information such as voice.

Attenuation

Attenuation is the loss of power suffered by the optical signal as it propagates through the fiber.

Different mechanisms:

1. Absorption losses:

In this case, the loss of signal power occurs due to absorption of photons associated with the

signal .The photons are absorbed by impurities such as transitional metal ions, hydoxy ions.

2. Scattering losses:

This occurs due to the scattering of the signal because of the changes in refractive index of the

glass .The scattered photon moves in random direction and escapes from the fiber reducing the

intensity. The changes in the refractive index is induced by the presence of inhomogeneity in the

Glass constitution.

3. Radiation losses:

Radiative losses occur due to bending of fiber.

Macroscopic Bends:

This refers to the bends having radii that are large compared to the fibre diameter.

Microscopic bends:

These are repetitive small scale fluctuations in the linearity of the fibre axis.


Dept. of Physics

Assignments:

1. Study the application of Laser in LIDARS.

2. Explain LASIK eye surgery technique.

3. Compare the applications of optical fibres and copper cable as carriers of electric

signals.

Contents


Dept. of Physics

Chapter 9:- Holography

It is a technique in which the wave front from the object interferes with another wavefront from a

reference source producing a interference pattern on a photographic plate which gives a three

dimensional picture of the object.

Recording of the image of an object :

The given object , mirror and photographic plates are arranged as shown in the figure .An

expanded laser beam is directed on this arrangement in which a part of the beam is incident on the

mirror and the rest falls on the object.

The photographic plate is placed such that it receives the reflected beam (reference beam) and

light scattered from the object (object beam).

Due to the interference between plane wavefronts of reference beam and spherical wavefronts of

object beam , an interference pattern is formed on the photographic plate. This will be called as a

hologram.


Mirror

Photographic plate

object

Expanded laser beam

Reference beam

Object beam

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Dept. of Physics

Hologram :

Reconstruction of the image:

The image of the object is reconstructed by passing the reference beam from the same laser

through the hologram ,which is oriented with respect to the reference beam .The reference beam

is diffracted and two images of the object , real image and virtual image are seen.

Applications :

1. Holographic interferometry

2. Holographic diffraction gratings

3. Acoustical holography

4. Information coding


Mirror

Virtual image

laser beam

Reference beam

Object beam

Hologram

eyes

Real image

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Dept. of Physics

Assignments:

1. Study the applications of BOSE – EINSTEIN CONDENSATION THEORY in

holography.

2. List out the applications of HOLOGRAPHY techniques.


Dept. of Physics

Symmetry in crystals:

Three dimensional crystal will have three primitive vectors. There are 230 possible symmetries.

These can be grouped into seven classes. The lattice with least symmetry is triclinic. Its unit cell

is parallelepiped. The primitive vectors are of different lengths and no two of the angles between

them are equal.


Physics I Sem Fullnotes June 2010

Documents

radiated energy

kt kt kt

black body radiation

incident energy

hk p kt

minimum energy

maximum energy

hk p p