Phy Notes Syl 10

ENGINEERING PHYSICS PH14/24

Dept. of Physics, Jain University 1

Engineering Physics Engineering Physics Engineering Physics Engineering Physics

STUDY MATERIAL

For B.E. I/II SemesterFor B.E. I/II SemesterFor B.E. I/II SemesterFor B.E. I/II Semester

As per JainJainJainJain University University University University Syllabus (sep 2010)

Department of Physics

Sri Bhagawan Mahaveer Jain College of Engineering

Jakkasandra Post, Kanakapura Taluk, Ramanagaram Dist.

562 112



MODULE I

QUANTUM PHYSICS

Black Body Radiation

A hot body emits thermal radiations which depend on composition and the

temperature of the body. The ability of the body to radiate is closely related to its ability to

absorb radiation. A Body which is capable of absorbing almost all the radiations incident on

it is called a black body. A perfectly black-body can absorb the entire radiations incident on

it. Platinum black and Lamp black can absorb almost all the radiations incident on them.

Emissive power of a black body:

It is defined as the total energy radiated per second from the unit surface area of a

black body maintained at certain temperature.

Absorptive power of a black body:

It is defined as the ratio of the total energy absorbed by the black body to the

amount of radiant energy incident on it in a given time interval. The absorptive power of a

perfectly black body is 1.

Spectral Distribution of energy in thermal radiation

(Black Body radiation spectrum)

A good absorber of radiation is also a good emitter. Hence when a black body is

heated it emits radiations. In practice a black body can be realized with the emission of

Ultraviolet, Visible and infrared wavelengths on heating a body. German physicists Lummer

and Pringsheim studied the energy density as a function of wavelength for different

temperatures of a black body using a spectrograph and a plot is made. This is called Black

Body radiation spectrum.



The Salient features of black body radiation spectrum are as below

1) The energy density increases with wavelength then takes a maximum value Em for

a particular wavelength λm and then decrease to a value zero for longer wavelengths. Hence

the Energy distribution in the spectrum is not uniform

2) As the Temperature increases the Wavelength (λm) corresponding to the

maximum emission energy (Em) shifts towards shorter wavelength side. Thus the λm is

inversely proportional to temperature (T) and is called Wein’s Displacement Law.

Mathematically . Here b is Wein’s Constant of value 2.898x10-3 mK.

3) The total energy emitted by the black body at a given temperature is given by the

area under the curve and is proportional to the fourth power of temperature. This is called

Stefan’s law of radiation. Mathematically E = σ T4 here ‘σ’ is the Stefan’s constant of value

5.67 x 10-8 Wm2K-4.

Explanation of Black Body Radiation Spectrum

Classical Theories

Wein’s Distribution Law:

In the year 1893 Wein using thermodynamics showed that the energy emitted per

unit volume in the wavelength range λ and λ+dλ

Here C1 and C2 are empirical constants. A suitable selection for these constants helps

to explain the experimental curve in the shorter wavelength region. The drawback of this

law is it fails to explain the curve in the longer wavelength region. Also according to this

equation the energy density at high temperatures tends to zero which contradicts

experimental observations.

Rayleigh-Jeans Law:

British Physicists Lord Rayleigh and James Jeans made an attempt to explain the

Black Body radiation spectrum Based on the concepts formation of standing electromagnetic

waves and the law of equipartition of energy. According to this law the energy density of

radiation is given by

Here ‘k’ is Boltzmann constant with value 1.38 x 10-23 JK-1. This law successfully

explains the energy distribution of the black body radiation in the longer wavelength region.

According to this law black body is expected to radiate large amount of energy in the

shorter wavelength region thus leading to no energy available for emission in the longer

wavelength region. Experimental observations show that the most of the emissions of the

black body radiation occur in the visible and infrared regions. This discrepancy is called

Ultraviolet Catastrophe.



Quantum theory of radiation

Planck’s law of radiation:

German physicist Max Plank successfully explained the energy distribution in the

black body radiation based on the following assumptions

1) The surface of the black body contains oscillators

2) These oscillators absorb or emit energy in terms of integral multiples of discrete packets

called quanta or photons. The energy ‘E’ of photons is proportional to the frequency ‘ν’ of

the radiation. Mathematically E=nhν here ‘h’ is a constant called Planck’s constant and its

value is 6.625 x 10-34 Js, and ‘n’ can take integer values

3) At thermal equilibrium the rate of absorption and emission of radiation are equal.

According to Planck’s law of radiation the expression for energy density of radiation

is given by

Where ‘c’ is the velocity of light, ‘k’ is Boltzmann constant and ‘h’ is Planck’s

constant. This law explains the distribution of energy in the black body radiation spectrum

completely for all wavelengths and at all temperatures. Also this law can be reduced to

Wein’s distribution law in the shorter wavelength region and to Rayleigh-Jeans law in the

longer wavelength region.

Deduction of Wein’s law, Rayleigh-Jean’s law from Planck’s law

(i) For shorter wavelengths, is very large

If ν is very large, i.e., is very large

⇒ , ≈ ≈ ……….(1)

Using Planck’s equation,

, by substituting equation (1)

Where, and

(ii) For longer wavelengths, is small,

If ν is very small, i.e., is very small

Now expand as power series

We have, ……….....

≈



(Since is very small, higher power terms could be neglected)

⇒

Again using Planck’s equation,

is reduced to

This is known as Rayleigh-Jean’s law

Photo-Electric effect

“The emission of electrons from the surface of certain materials when radiation of

suitable frequency is incident on it is called the phenomenon of Photo-Electric effect.” The

electrons emitted are called photo electrons and the material is said to be photo sensitive.

This was discovered in the 1887 by Henrich Hertz.

The experimental observations of photoelectric effect are

1) Photo electrons are emitted instantaneously as soon as the radiation is incident

2) Photo electric emission occurs only if the frequency of the incident radiation is greater

than a certain value called Threshold frequency.

3) The kinetic energy acquired by photo electrons is directly proportional to the frequency of

the incident radiation and is independent of the intensity.

4) The number of photoelectrons emitted depends on the intensity of the incident radiation

and is independent of the frequency.

Photoelectric effect signifies the particle nature of radiation.

Einstein’s explanation of the photo electric effect

When metal is illuminated with radiation of suitable frequency, the photons of the

radiation interact with electrons in the metal. When a photon interacts with an electron, the

electron absorbs it and the photon vanishes. The energy acquired by the electron from the

photon is made use in two stages. A part of the energy is used by the electron to free itself



from the metal since it is bound within metal. Thus some minimum amount of energy is

required for the electron just to escape from the metal is called Work Function (φφφφ). The

rest of the energy is carried by the electron as Kinetic Energy (KE). Since the energy of

the photon is ‘hν’ the photoelectric effect satisfies the following equation

This is called Photoelectric Equation. Here ‘ν’ is the frequency of the incident

radiation

Here ‘ ’ is the threshold frequency and ‘v’ is the velocity of electron and ‘m’ the

mass. Thus from the photoelectric equation, if the frequency of the radiation ν < no

photoelectrons are emitted.

Compton Effect

“The phenomenon of scattering of X-rays from suitable material and hence increase

in its wavelength is called Compton Effect.”

When X-rays are incident on certain materials they are scattered and the scattered

X-rays contain two components. One component has the same wavelength as the incident

X-ray and the other with wavelength greater than the incident X-rays. This is due to the

scattering of X-ray photons from the electrons present in the material. Due to the transfer

of energy from X-ray photon to electron the wavelength of X-ray increases and the electron

recoils. This can be treated as collision between two particles. Thus Compton Effect signifies

particle nature of radiation. The change in wavelength which is also called Compton Shift is

given by

Here ‘λ’ is the wavelength of incident X-rays and ‘λ’ ‘is the wavelength of scattered X-ray ‘θ’

is the scattering angle and ‘m0’ is the rest mass of electron. The quantity is called

Compton Wavelength.



Experimental verification

A beam of monochromatic X-rays are allowed to fall on a graphite crystal as shown

in the figure. The intensity of the scattered X-rays is measure as a function of wavelength of

X-

rays, at different scattering angles. At each angle, two peaks appear corresponding to

scattered X-ray photons with two different wavelengths. The wavelength of one peak does

not change as the angle is varied. This is called primary or unmodified component. We

denote it by λλλλ. The wavelength of the other peak varies strongly with the angle and hence it

is called modified component. Ii is denoted by λλλλ’. This effect is called Compton effect.

The change in wavelength ∆∆∆∆λλλλ is called Compton shift.

Dual Nature of Radiation and de Broglie’s hypothesis

The phenomenon like Interference, Diffraction and Polarization are attributed to the

wave properties of radiation. The Quantum theory of radiation and experiments like

Photoelectric effect and Compton Effect describe the particle nature of radiation. Thus

radiation behaves like waves and like particles under different suitable circumstances.

Hence radiation exhibits dual nature.

In the year 1924 French physicist Louis de Broglie made a daring suggestion “If

radiant energy could behave like waves in some experiments and particles or

photons in others and since nature loves symmetry, then one can expect the

particles like protons and electrons to exhibit wave nature under suitable

circumstances.” This is well known as de Broglie’s hypothesis.

Therefore waves can be even associated with moving material particles called Matter

waves and the wavelength associated with matter waves is called de Broglie wavelength.

The de Broglie wavelength is given by where ‘m’ is the mass of the particle and ‘v’ is

its velocity.

Expression for de Broglie wavelength (Wavelength of matter wave)

According to the Einstein’s photon theory the energy of the photon is given by

Here ‘ν’ is the frequency of the incident radiation and ‘h’ is Planck’s constant. If ‘m’ is the

mass equivalent of the energy of the photon then



Since the frequency of the incident radiation could be expressed in terms of wavelength

‘λ’as we get,

⇒

Here ‘p’ is the momentum of the photon

Therefore,

Thus, according to de Broglie’s hypothesis, for a particle moving with velocity ‘v’ the above

equation can be modified by replacing the momentum of photon with the momentum of the

moving particle ‘mv’. Therefore the de Broglie wavelength associated with a moving particle

is given by

…… (1)

Here ‘m’ is the mass of the moving particle.

de Broglie wavelength of an electron accelerated with a potential difference of ‘V’

volt

Consider an electron accelerated by a potential difference of ‘V’ volts. The kinetic energy

acquired by the electron is given by

⇒

Here ‘m’ is the mass of the electron and is given by 9.1 x 10-31 kg

Substituting the value of ‘v’ in equation (1) we get

Since the electron acquires kinetic energy from the applied potential difference ‘V’ The

kinetic energy of the electron is also given by E=eV where ‘e’ is the charge on electron with

value 1.6 x 10-19C

Hence the expression for the de Broglie wavelength

Substituting the values for the constants h, m and e we get

Davisson-Germer’s experiment:

The De-Broglie’s hypothesis of possibility of wave nature of material particles under

appropriate conditions was first experimentally verified by Davisson and Germer. In order to

show that particles can also exhibit wave nature, it needs to be proved that material

particles can also produce effects such as interference, diffraction, etc which are

characteristic phenomena associated exclusively with waves.



Davisson and Germer were studying the phenomenon of scattering of electrons from

material targets and they observed diffraction of electrons in a crystal of nickel, similar to X-

ray waves undergoing diffraction in crystals, thus proving the wave behavior of electrons.

FIGURE:

The experimental apparatus consists of an

electron gun to produce a beam of

electrons, a solid nickel crystal used as a target mounted on a rotatable stand and an

ionization chamber (detector) C which is connected to a galvanometer to collect and

measure the current due to the scattered electrons.

The electron gun (G) consists of a filament which upon heating by a low-tension

battery emits electrons. These emitted electrons are accelerated by applying a high

potential from a high-tension battery and using a series of metallic diaphagrams as slits, a

narrow beam of electron is obtained. This narrow beam of electrons is incident normally on

the Nickel crystal (N) mounted on a rotatable stand. The electrons incident on the Nickel

crystal undergo scattering in all directions inside the crystal, just as X-ray waves are

scattered in the phenomenon of X-ray diffraction by crystals. The scattered electrons are

collected by the ionization chamber (C) and the current due to these collected electrons is

measured by the galvanometer connected to the ionization chamber. The ionization

chamber can be rotated along a circular path S to collect the electrons at various scattering

angles φ .

In the Davisson-Germer experiment, the accelerating potential was kept constant in

G and ionization chamber C collects scattered electrons at various scattering angles φ . For

each scattering angle φ , the ionization current as measured by the galvanometer was noted

and the same procedure was repeated by applying different potentials to the electron gun

G.

Initially by applying a constant potential of 40V, the ionization current was noted as

a function of scattering angle φ and the same was repeated by applying 44,48,54,60 and 68

volts respectively.

A polar plot of ionization current and scattering angle φ , for various applied

potentials is obtained as shown in the figures. In the polar plot, for each data point, the

angle of inclination to Y-axis equals the scattering angle and the length of the arrow to the

data point gives the ionization current.



At the acceleration potential of 40V, as seen from figure, the variation is found to be

smooth without any maxima or minima. However, at V=44volts a distinct maxima was

observed as seen in figure and this maxima became more and more pronounced till 54V.

With further increase in applied voltage, this maximum declines and fades away as seen in

Fig. The values of applied voltage and the scattering angle φ at which the ionization current

was maximum was found to be V=54volts and φ =50 degrees (Fig ).

If electrons were to be behaving only as particles, then from classical theories, it is

expected that with increasing applied potential, the ionization current due to scattered

electrons would increase and therefore, the nature of polar plot at higher potentials should

be similar or a laterally pulled version to that of 40volts curve. But as can be seen from the

above plots, such behavior is not observed.

To explain the observation of distinct maxima of the ionization current over a certain

scattering angle φ , Davisson & Germer proposed that the incident electron beam is

scattered from the nickel crystal as a beam of monochromatic waves associated with

electrons (i.e., De-Broglie waves or Electron waves associated with the electron) similar to

X-ray waves undergoing diffraction in crystals.

According to de Broglie’s hypothesis for an electron accelerated by potential difference of 54

V the de Broglie wavelength is given by

mAV

o 109

1066.154

10226.126.12 −−

×=×

==λ

According to Bragg’s law the diffracted waves from a crystal undergo constructive

interference only for that angle of incidence θ which satisfies the equation

θλ sin2dn =

In the Davisson-Germer experiment the constructive interference was observed at a

glancing angle of 065=θ and it would occur only for those waves with their wavelength λ

given by θλ sin2d= (assuming the order of diffraction n to be equal to one).

We therefore have mSin 10010 1065.1)65(1091.02 −− ×=×××=λ

In the above evaluation we have taken the value of md101091.0 −×= for the lattice

spacing in a nickel crystal.

The experimentally determined value is in good agreement with the value calculated

according to de Broglie’s hypothesis. Thus Davisson and Germer experiment not only

confirms the wave associated with moving particle it also verifies the de Broglie’s

hypothesis.



Phase velocity and Group velocity

Phase velocity (vp)

The velocity with which a wave travels is called phase velocity and is also called

wave velocity. If a point is marked on the wave representing the phase of the particle then

the velocity with which the phase propagates from one point to another is called phase

velocity. It is given by Where ‘ω’ is the angular frequency and ‘k’ is wave number.

Substituting for and . We get ,

Therefore Where ‘E’ is the energy and ‘p’ is momentum

Where ‘c’ is the velocity of light and ‘v’ is the velocity of the article.

From the above expression it is evident that the phase velocity is not only greater

than the particle velocity it is also greater than the velocity of light. Hence there is no

physical meaning for phase velocity of matter waves.

Properties of Matter waves

The following are the properties associated with the matter waves

1) Matter waves are associated only with particles in motion

2) They are not electromagnetic in nature

3) Group velocity is associated with matter waves

4) The phase velocity has no physical meaning for matter waves

5) The amplitude of the matter wave at a given point is associated with the probability of

finding the particle at that point.

6) The wave length of matter waves is given by

Group velocity (vg)

Since the velocity of matter waves must be equal to that of the particle velocity and

since no physical can be associated with phase velocity the concept of group velocity is

introduced.

Matter wave can be considered as a resultant wave due to the superposition of many

component waves whose velocities differ slightly. Thus a wave group or wave packet is

formed. The velocity with which the wave group travels is called group velocity which is

same as particle velocity. It is denoted by vg.



Expression for Group velocity using the concept of superposition of waves

Consider a wave group formed by the superposition of a minimum of two waves

which slightly differ in their velocities with amplitudes ‘A” traveling in the same direction and

are represented mathematically as below

………….(1)

………….(2)

According to the principle of superposition the resultant wave is given by

We know that

………..(3)

But and since very small

Therefore equation (3) could be written as

……..(4)

Compare the eqn.(4) with eqn.(1)

Eqn.(4) represents the resultant wave whose amplitude varies as

Which is a constant but varies as a wave

As by definition of group velocity with which the variation in amplitude is transmitted in the

resultant wave is the group velocity and is given by,

or

Relation between group velocity and phase velocity

The phase velocity is given by

The group velocity of matter waves is given by

………….. (1)

The wave number ‘k’ is given by

Differentiate the above equation we get,

⇒ …………. (2)

Substituting eqn.(2) in eqn.(1) we get



The relation between group velocity and phase velocity is given by

Relation between group velocity and particle velocity

Consider a particle of mass ‘m’ moving with a velocity ‘v’. We know that the group

velocity is given by

But the angular velocity is given by E

Differentiating we get

The propagation constant is given by ⇒

Differentiating we get

= …………… (1)

The total energy of the particle is given by E= Kinetic energy + Potential energy

Differentiate the above equation we get

⇒ The particle velocity

Hence the group velocity and particle velocity are equal.

Heisenberg’s Uncertainty Principle

Statement: “The simultaneous determination of the exact position and momentum

of a moving particle is impossible”

Explanation: According to this principle if ∆x is the error involved in the

measurement of position and ∆p is the error involved in the measurement of momentum

during their simultaneous measurement, then the product of the corresponding

uncertainties is given by

The product of the errors is of the order of Planck’s constant. If one quantity is measured

with high accuracy then the simultaneous measurement of the other quantity becomes less

accurate.

Physical significance: According to Newtonian physics the simultaneous

measurement of position and momentum are “exact”. But the existence of matter waves

induces serious problems due to the limit to accuracy associated with the simultaneous

measurement. Hence the “Exactness” in Newtonian physics is replaced by “Probability” in

quantum mechanics.



-ray microscope

Consider an imaginary experiment in which an electron is tried to be spotted using a

high resolution -ray microscope.

The limit of resolution of the microscope is given by

Here ‘λ’ is the wavelength of the scattered -ray photon, θ is the semi vertical

angle.

According to the definition of limit of resolution becomes the uncertainty in the

determination of position of the electron. In order to observe the electron, the scattered

photon from the electron must enter the microscope anywhere within angle of . The x

component of momentum ∆px may lie between and . Here p is the

momentum of the photon is given by . Since the momentum is conserved during the

collision, the uncertainty in the x component of momentum is given by

Thus the product of the uncertainties is of the order of ‘ h ’. More rigorous calculation results

in the value

Diffraction by single slit

Consider a narrow beam of electrons passes through a single narrow slit and

produces a diffraction pattern on the screen as shown in the Fig. The first minimum of the

pattern is obtained for n=1, in the equation describing the behavior of diffraction pattern

due to a single slit. Hence,

)1........(sin λθ =∆y

Where y∆ is the width of the slit and θ is the angle of diffraction corresponding to

first minimum.



In producing the diffraction pattern on the screen all the electrons have passed

through the slit but we can not say definitely at what place of the slit. Hence the uncertainty

in determining the position of the electron is equal to the width y∆ of the slit. From equation

(1) we have

θ

λ

sin=∆y ………..(2)

Initially the electrons are moving along the x-axis and hence they have no component of

momentum along y-axis. After diffraction at the slit, they are deviated from their initial path

to form the pattern and have a component θsinp . As y component of momentum may lie

anywhere between θsinp and θsinp− . Uncertainty in y component of momentum is

( ) θλ

θθθ sin2sin2sinsinh

ppppy ≈≈−−≈∆ ……….. (3)

Hence from equations (5) and (6)

hh

py y 2sin2

sin≈×≈∆×∆

λ

θ

θ

λ

hpy y ≈∆×∆

Thus the product of the uncertainties is of the order of ‘ h ’. More rigorous calculation results

in the value

Wave function

A wave is constituted by a periodic oscillation of a particular physical quantity. For

ex, in case of water waves, the quantity that varies is the height surface, in sound waves it

is the pressure variation and in case of electromagnetic waves it is the variation of electric

and magnetic fields that constitutes the electromagnetic wave.

In case of waves associated with material particles (matter waves) the quantity

whose variations make up the matter waves is called the wave function and is denoted

byψ . The value of the wave function ),,,( tzyxψ of a body at the point ),,( zyx in space and

time t , determines the likelihood of finding the body at the location ),,( zyx at that instant of

time ‘ t ’



The wave function for a wave moving along x-axis in complex notation is given by

Where -angular frequency, k- wave number or propagation constant,

- Amplitude of the wave

Physical significance of wave function

The wave function just as itself has no direct physical meaning. It is more difficult

to give a physical interpretation to the amplitude of the wave. The amplitude of the wave

function is certainly not like displacement in water wave or the pressure wave nor the

waves in stretched string. It is a very different kind of wave. But the quantity, the squared

Absolute amplitude gives the probability for finding the particle at given location in space

and is referred to as probability density. It is given by

Thus, in one dimension the probability of finding a particle in the width ‘dx’ of length ‘x’

Similarly, for three dimension, the probability of finding a particle in a given small volume

dV of volume V is given by

here

Here ‘P’ Probability of finding the particle at given location per unit volume and is called

Probability Density.

According to Max Born’s interpretation

The wave function is complex the probability density is given by

Where * is the complex conjugate of and the above product results in real number.

Normalization and Normalized wave function

Since the particle exists somewhere in volume V then the probability of finding the

particle in the given volume V is equal 1.

Thus

If we are unable to locate the particle in volume V then the notion can be extended

to the whole space with

But, normally, the value of the above integral will not be unity but contains an indefinite

constant which can be determined along with sign using above considerations. The process



is called normalization and the wave function which satisfies the above condition is called

normalized wave function.

Eigen functions and Eigen values

The Schrödinger wave equation is a second order differential equation. Thus solving

the Schrodinger wave equation to a particular system we get many expressions for wave

function ( ). However, all wave functions are not acceptable. Only those wave functions

which satisfy certain conditions are acceptable. Such wave functions are called Eigen

functions for the system. The energy values corresponding to the Eigen functions are called

Eigen values. The wave functions are acceptable if they satisfy the following conditions

1) must be finite everywhere (not zero everywhere)

2) must be single valued which implies that solution is unique for a given position in

space

3) and its first derivatives with respect to its variables must be continuous everywhere.

Schrödinger Wave Equation

In quantum mechanics, the basic fundamental governing equation which describes

the state of the system is the Schrödinger Equation

We can determine the motion of an atomic particle using Schrödinger Equation just

as we determine the motion of the classical particle using Newton’s law. The solution of the

Schrödinger Equation gives the wave function of the particle that carries information about

the wave behavior associated with the particle.

The Schrödinger Equation can be set up in two different contexts. One, which is

general and takes care both position and time variations of the wave function, It is called

Time dependent Schrödinger Wave Equation.

The other one is applicable only to steady state conditions, in which case the wave

function can have variation with position not with time. It is called Time independent

Schrödinger Wave Equation.

Time dependent Schrödinger Wave Equation

Consider a particle of mass ‘m’ moving with velocity ‘ v ’ along +ve X-axis. The deBroglie wave length ‘λλλλ’ is given by

)1(..................mv

h=λ

The wave equation for one dimensional propagation of waves is given by

VelocityWaveis'v'Where

axisXvealongtv

1

x 2

2

22

2

−+∂

ψ∂=

∂

ψ∂

Here )2........(..........e )kxt(i

0

−ϖ−ψ=ψ

Where ψψψψ0 is the amplitude at the point of consideration ωωωω is angular frequency and k is Wave Number.

Differentiating equation (2) with respect to time (t), we get



)3....(..........it

ωψ−=∂

ψ∂

Differentiating equation (2) twice with respect to x, we get

)4....(..........kx

2

2

2

ψ−=∂

ψ∂

Using Einstein and deBroglie equations

h

2pkand

h

2E π=

π=ω

Substitute the above in the equations (3) and (4)

)5........(Et2i

hor

h

2iE

t)3(

ψ=∂

ψ∂

π

−

ψπ

−=∂

ψ∂⇒

)6.........(px4

hor

h

4p

x)4(

2

2

2

2

2

2

22

2

2

ψ=∂

ψ∂

π−

ψπ

−=∂

ψ∂⇒

The total energy of the moving particle is given by

)VE(m2

por

E.PE.KE

2

−=

+=

Multiply above equation throughout by Ψ, we get

ψ−ψ=ψ VEm2

p 2

Substituting, for ψψ 2pandE from equations (5) and (6) resp., we get,

2

2

2

2

2

2

2

2

2

i1)7.......(t2

ihV

xm8

hor

t2

h

i

1V

xm2

1

4

h

=−∂

ψ∂

π=ψ+

∂

ψ∂

π−

∂

ψ∂

π−=ψ+

∂

ψ∂

π−

Q

Equation (7) is the one-dimensional time-dependent Schrödinger equation.

Time-Independent Schrödinger Wave Equation

The wave equation which has variations only with respect to position and describes the steady state is called Time-Independent Schrodinger wave equation.

Consider a particle of mass ‘m’ moving with velocity ‘ v ’ along +ve X-axis. The deBroglie wave length ‘λλλλ’ is given by



)1(..................mv

h=λ

The wave equation for one dimensional propagation of waves is given by

VelocityWaveisWhere

axisXvealongtx

'v'

)2........(..........v

12

2

22

2

−+∂

∂=

∂

∂ ψψ

Here )3........(..........)(

0

kxtie

−−= ϖψψ

Where ψψψψ0 is the amplitude at the point of consideration ωωωω is angular frequency and k is Wave Number.

Differentiating ψ twice with respect to time (t), we get

)4........(..........)(

0

2

2

2kxti

et

−−−=∂

∂ ϖψωψ

Substituting equation (4) in equation (2)

Substituting for λ from equation (1) we get

)5........(..........02

18

02

24

04

2

2

2

2

2

22

2

2

2

2

2

2

2

2

=

+

∂

∂

=

+

∂

∂

=

+

∂

∂

ψπψ

ψπψ

ψπψ

mvh

m

x

vm

hx

mv

hx

The kinetic energy of the particle 2

2

1mv is given by

( )( )

( )

04

4

21

v

1

2

2

2

2

2

2

2

2

2

2

2

22

2

=

+

∂

∂⇒

−=

∂

∂∴

−=−=∂

∂

ψλ

πψ

ψλ

πψ

λψπλ

ψωψ

x

x

lengthwavetheisandwavetheoffrequencytheisfHereffx



EnergyPotentialtheisVandparticletheofEnergyTotaltheisEhereVEmv −=2

2

1

Therefore equation (5) becomes

0)(8

2

2

2

2

=−

+∂

∂ψ

πψ

h

VEm

x

Generalizing the equation for three dimensions we get

0)(8

2

2

2

2

2

2

2

2

=−

+∂

∂+

∂

∂+

∂

∂ψ

πψψψ

h

VEm

zyx

0)(8

2

22 =

−+∆ ψ

πψ

h

VEm

Here2

2

2

2

2

22

zyx ∂

∂+

∂

∂+

∂

∂=∆

Hence the Time-Independent Schrodinger Wave equation for three dimensions.

Applications of Schrödinger wave equation

Particle in a one dimensional box or one dimensional potential well of infinite

height

Consider a particle of mass ‘m’ bouncing back and forth between the walls of one

dimensional potential well. The particle is said to be under bound state. Let the motion of

the particle be confined along the X-axis in between two infinitely hard walls at x=0 and

x=a. Since the walls are infinitely hard, no energy is lost by the particle during the collision

with walls and the total energy remains constant.

In between walls i.e. 0 < x < a, the potential

energy V=0.

Beyond the walls i.e. x ≤ 0 and x ≥ a, the potential

energy V=∞.

Since the particle is unable to penetrate the hard

walls it exists only inside the potential well. Hence

ψ=0 and the probability of finding the particle

outside the potential well is also zero.



Inside the potential well

Since the potential inside the well is V=0, the Schrodinger wave equation is given by

0)0(8

2

2

2

2

=−

+∂

∂ψ

πψ

h

Em

x

∴ 08

2

2

2

2

=+∂

∂ψ

πψ

h

Em

x

)1.......(..........02

2

2

=+∂

∂ψ

ψk

x Here )2.(..........

82

22

h

Emk

π=

For the given value of E, k is constant. The general solution for the equation (1) is given by

)3.....(..........cossin)( kxBkxAx +=ψ Where A and B are arbitrary constants. The values of

these constants can be obtained by applying the boundary conditions

I) At x=0 , ψ(x)=0 . Substituting the values in equation (3) we get

)4..(..........sin)()3(

0

0cos0sin0

kxAxbecomesequationHence

B

BA

=

=∴

+=

ψ

II) At x=a, ψ(x)=0. Substituting the values in equation (4) we get

)5.........(..........

),(0

sin0

a

nk

nkaSolutionnoOtherwiseASince

kaA

π

π

=⇒

=≠

=

Where n= 1, 2, 3,……

Thus the wave function becomes )6.....(..........sin)(a

xnAx

πψ =

Also substituting the value of ‘k’ from eq (5) into eq (2) we get

)7.....(..........8

82

22

2

22

ma

hnE

h

Em

a

n=⇒=

ππ hence the energy Eigen

values.



Thus substituting n=1 in the equation (7) we get

2

2

18ma

hE = is the ground state energy of the particle and is also called zero point

energy.

Hence 12 EnEn = E2 and E3 are energies of the first and second excited states respectively

and so on. Hence for a particle in the bound state, the energy values are discrete.

Normalization of wave function

The wave function for a particle in a box is given by equation (6)

a

xnAx

πψ sin)( =

The value of the arbitrary constant ‘A’ can be determined by the process of

normalization. Since the particle has to exist somewhere inside the box we have

∫∫ ==aa

dxxdxxP0

2

0

1)()( ψ Substituting the wave function from equation (6)

∫ =

a

dxa

xnA

0

22 1sinπ

Since [ ]θθ 2cos12

1sin 2 −= we have

∫ =

−

a

dxa

xnA

0

2

12

cos12

π Integrating the equation we get

12

sin22

0

2

=

−

a

a

xn

n

ax

A π

π The second term takes the value zero for both the limits

∴ [ ]a

AaA 2

102

2

=⇒=−

Thus the Eigen function is given by



a

xn

axn

πψ sin

2)(

=

The wave functions and the probability densities for the first three values of ’n’ are as

shown in fig

Thus for ground state (n=1) The probability of finding the particle at the walls is zero and at

the centre (a/2) is maximum. The first excited state has three nodes and the second excited

state has four nodes.



MODULE II

NON LINEAR OPTICS

LASERS Laser is an acronym for Light Amplification by Stimulated Emission of Radiation. Laser is

a highly “monochromatic coherent beam of light of very high intensity”. In 1960 Mainmann

built the first “LASER” using Ruby as active medium.

Interaction of Radiation with matter:

1. Stimulated Absorption: -

When an atom in the ground state say E1 absorbs a photon of energy (E2 - E1) it makes transition into exited state E2. This is called Stimulated or Induced absorption. It is represented as follows,

Photon + Atom = Atom*.

2. Spontaneous Emission: -Spontaneous emission is one in which atom in the excited state emits a photon when it returns to its lower energy state without the influence of any external energy.

Consider an atom in the excited state E2. Excited state of an atom is highly unstable. With in a short interval time, of the order of 10-8 sec,atom returns to one of its lower energy state say E1 and emits difference in energy in the form of photon of energy hv = E2 - E1 spontaneously.

If the two atoms are in the same excited state and returns to some lower energy states

two photons of having same energy are emitted. These Two photons may not travel in the

same direction. They produce in-coherent beam of light. Spontaneous emission is

represented as follows,

Atom* = Atom + Photon.

3. Stimulated Emission: -Consider an atom in the excited state E2. If a photon of energy E2 - E1 is made to incident on the atom in the excited state E2.

The incident photon forces (stimulates) the atom in the excited

state to make transition in to ground state E1 by emitting

difference in energy in the form of a photon. This type of

emission in which atom in the excited state is forced to emit a

photon by the influence of another photon of right energy is

called stimulated emission. Stimulated emission can be

represented as follows.

Photon + atom* = Atom + (photon + photon).

When stimulated emission takes place, incident photon and the emitted photon are in phase

with each other and travel along the same direction. Therefore they are coherent.

EEEE2222 EEEE1111

EEEE2222 EEEE1111

Photon Photon Photon Photon hhhhνννν hhhhνννν

hhhhνννν

EEEE2222 EEEE1111 Photon Photon Photon Photon



Lasing Action (Laser Action) : -

Let an atom in the excited state is stimulated by a photon of right energy so that atom makes stimulated emission. Two coherent photons are obtained. These two

coherent photons if stimulate two atoms in the exited state to make emission then four

coherent photons are produced. These four coherent photons so that stimulates 4 atoms in

the excited state, 8 coherent photons are produced and so on. As the process continues number of coherent photons increases. These coherent photons constitute an intense beam

of laser. This phenomenon of building up of number of coherent photons so as to get an intense laser beam is called lasing action.

Population inversion and optical pumping: -

In an order to produce laser beam there should be more number of stimulated emissions when compared to spontaneous emission. It is possible only if number of atoms in the exited stats is grater than that is the ground state. When system is in

thermal equilibrium, then number of atoms in the higher energy level is always less than the number of atoms in the lower energy level. If by some means number of atoms in the

exited slate is made to exceed number of atoms in the ground state then population inversion is said to have established between excited state and ground state. The method of achieving the population inversion is called pumping. If light is used to pump electrons

to the higher level then the method is called Optical Pumping. If the electric field is used to pump electrons to the higher level then the method is called Electrical Pumping.

Metastable State: - Population inversion can be created with the help of three energy levels as follows.

Let E1 is the ground state of an atom. Let E2 and E3

are the two excited states. If an atom is excited into the energy state, within a short inter of time of 10-8 sec, atom makes a transition into the energy state E2. Let

lifetime of the atom in the energy level E2 is of the order of 10-2 to 10-3 sec. Then atoms stay in the excited state

E2 for sufficiently long time without making any spontaneous emission.

As more and mare atoms are excited from the ground state to E1 more and more atoms are transferred from E3 to E2. As a result, within a short interval of time population inversion

is established between energy level E2 and E1. The energy level E2 in which atoms remain for unusually longer time is called Metastabte state. When transition from E3 to E2 takes place

Excited

EEEE1111 EEEE3333 EEEE2222

Ground

Metastable

Radiation less

Laser



excited atom looses energy in the form of heat without emitting any radiation. Such transitions are called radiation less transition (Non-radiative transitions).

Requisites of a Laser System: - The Three requisites of a Laser system are

1) Energy Source or Excitation Source for Pumping action 2) Medium Supporting population inversion called Active Medium 3) The Laser Cavity

Appropriate amount of energy is to be supplied for the atoms in order excite them to

higher energy levels. If the Input energy is in the form of light energy then pumping is

called optical pumping. If it is in the form of electrical energy then pumping is called

electrical pumping.

Population inversion occurs at certain stage in the Active medium due to the absorption of energy. After this stage the Active medium is capable of Emitting laser light. The Laser Cavity consists of an active medium bound between two mirrors. The Mirrors

reflect the light two and fro through the active medium. This also helps to tap certain

permissible part of laser energy from the active medium.

The Ruby Laser

Construction

• The ruby is a crystal of Al2O3 (Corundum) with some of the Al3+ ions replaced by Cr3+ ions. The chromium ions give the characteristic colour (red) to the ruby crystal. For the purpose of laser production, the doping concentration of chromium ions is 0.05%.

• A single crystal of ruby in the form of a cylindrical rod is chosen. The length of the rods can vary from 5 to 20 cm while their diameter can vary from 0.5 to 2cm.

• The end faces of the rod are made optically flat and parallel to each other. One face of the rod is fully silvered while the other face is partially silvered

• The ruby crystal is placed along the axis of a helical Xenon flash tube. The xenon flash tube is connected to a high voltage pulse generator. For each single voltage pulse, the Xenon tube gives out flashes of powerful light which last for several milliseconds.

• Surrounding the flash tube is a cylindrical mirror whose function is to reflect light on to the ruby crystal.

• During the working of the laser a lot of heat is generated. This heat is dissipated by circulating cold water in thin tubes which surround the crystal.

• The ruby laser satisfies the four requisites needed for any laser system. The crystal rod along with the mirrored faces functions as the resonant cavity. Optical pumping is achieved using light from xenon flash tube. Chromium ions are the active medium, which support population inversion.



Working

Shown here is the energy level diagram

of chromium ions.

• Light from the flash tube excites the ions from the ground state (4A1) to the two higher energy bands (4F1 and 4F2). Remember when chromium ions interacts the outermost energy levels split into many levels forming a band.

• As there are many levels present in this band, the number of photons available, for exciting the ions, are many. Therefore numerous ions are able to absorb the photons and make transition from the ground state to one of the levels of the band.

• The atoms in the 4F1 and 4F2 band reside there for a period of 10

-8s and then make a transition to the metastable levels 2Ā and Ē. The energy difference between the energy bands and the metastable levels is not released as electromagnetic energy but is taken up by the vibrating atoms of the lattice and is dissipated as heat. These kind of transitions are non- radiative in nature.

• Therefore 4F1 ⇒ 2Ā 4F1 ⇒ Ē

4F1 ⇒ 2Ā Non-radiative transitions

4F2 ⇒ Ē

• The 2Ā and Ē levels being metastable, the atoms reside in them for an unusually long period of time. In a short while the number of atoms in the two metastable levels is more than the ground level. Thus population inversion is established. Induced transitions between these metastable levels and the ground state give rise to the needed laser radiation. Transition from 2Ā to the ground state gives rise to photons of wavelength 692.8nm and the transition from Ē to the ground state gives rise to photons of wavelength 694.3nm.

• The output of the laser is taken from the partially silvered mirror. In the output the intensity of radiation of wavelength 694.3nm is more than the 692.8nm radiation. One of the probable reasons for this could be that the population of the Ē level is more than the 2Ā level. That is why more photons of wavelength 694.3nm would be released per second. The second probable reason (although not significant) is that the probability of transition from the Ē to the ground level is more than the probability of a transition from 2Ā level to the ground level. Why it is not significant is that the two levels have an energy difference of only 0.004eV. This is not too great to cause a significant difference in the probabilities.

Application of ruby laser

• It is used in holography • As drilling requires pulsed laser, ruby lasers are the most suitable The biggest disadvantage of this laser is that since the output is discontinuous, its use is

limited to only special applications. Wherever continuous laser beam is required the helium -

neon laser is more suitable.



Helium-Neon Laser (Gaseous state laser): -

Construction: -It consists of quartz discharge tube of length 1m and diameter 1.5cm fitted

with Brewster’s windows on either side and filled with the mixture of He and Ne gas in the

ratio of 10:1. It is placed between two highly parallel plane mirrors one of which is

completely silvered while the other is partially silvered. The ends of the tube have two

electrodes which are connected to a high power voltage source.

Working: The energy level diagram for He and Ne atoms are as shown in the fig. When

discharge is produced in the tube large numbers of electrons are produced. These highly

energetic electrons collide with He atoms, which are abundant and excite them to energy

levels 21s or 23s of He system. This type of collision is called collision of first kind and

represented as follows,

e1+He →→→→ He + e2 Where el and e2 are energies of electrons before and after collision.

When the helium atoms collide with the neon atoms in the ground state, because of close

coincidence in the energy values SSEE 321 ≅

; SSEE 223 ≅

, resonant energy transfer takes

place from helium to neon atoms. As a result, the neon atoms get excited to 2S and 3S

levels, whereas the helium atoms return to the ground state. This is called 2nd kind of

collision and can be represented as

He* + Ne Ne* + He



Here, the states 3S and 2S are called as virtual metastable states because the energy

values of 3S and 2S of Ne are equal to the 21S and 23S metastable state of He. Thus

population inversion built up between 2S and 3S levels with the lower energy level 2p which

leads the laser transitions. [3S to 2p transition gives laser light of wavelength 6328Ao and

2S and 2p transition giving rise to 11523Ao radiation which is in the Infrared region]

Applications of Laser

Because of high intensity, high degree of monochromaticity and coherence, lasers find

remarkable applications in medicine, communication, defence, photography, material

processing etc.

Laser Welding

In performing the task of welding, laser welding is superior to other welding such as arc

welding, gas welding, electron welding, etc.



• Focus the laser beam on to the spot to be welded. • Due to the excess of heat generated, only focused portion melts. • The heat produced by the beam is so intense that, impurities in the material such

as oxides float up on the surface and upon cooling the material becomes homogeneous solid structure and it makes a strong joint.

Advantages:

• Laser welding is a contact less process and thus no foreign materials can enter into the welded joint.

• In this type of welding, no destruction occurs in the shape of work piece and the heat is dissipated immediately ( since the total amount of heat supplied is very small compared to the regular welding)

• The laser beam can be controlled to a great precision, so that we can focus the laser beam precisely to the welding spot. Even we can weld difficult to reach the locations in the material.

• Since the heat affected zones are very small, laser welding is ideal at places which are surrounded by heat sensitive components.

Laser Cutting

Laser cutting of metals is generally associated by gas blowing. The oxygen gas is

passed through the nozzle and the tip of the nozzle is pointed at the spot, where the

laser beam is focused.

• The combustion of the gas burns the metal thus reducing the laser power required for cutting.

• Also the tinny splinters along with the molten part of the metal will blow away by the oxygen jet.

• The blowing action increases the depth and also the speed of cutting. • The laser, which controls the accuracy of the cutting thus, the cut edges will be high

quality. Advantages:

• The quality of cutting is very high • There will be no thermal damage and chemical change when cutting is done in inert

atmosphere. • 3-d cutting can also be done very easy.

Drilling:

• Drilling of holes is achieved by subjecting the material to 10-4 to 10-3S duration pulse.

• The intense heat generated over a short duration by the pulses evaporates the material locally, thus leaving a hole.

Advantages:

Conventional Method Laser

The tools wear out while drilling This problem doesn’t exist in laser

setup



Whereas it could be done only to a

limited extent

Drilling can be achieved at any

oblique angle

It is difficult Very fine holes 0.2 to 0.5mm

diameter can be drilled. The holes

may be even adjacent to each other

Large force has to apply to drill the

hard materials or brittle materials.

Very hard material or brittle

materials can be drilled. There is no

mechanical stress with a laser beam.

Measurement of pollutants in atmosphere:

The concentration of pollutants in the atmosphere such as carbon monoxide,

sulphur dioxide, nitrous oxide, etc, can be measured using laser the way RADAR

system is used. Hence it is called LIDAR i.e Light Detection and Ranging. The laser

technique consists of a Laser source, retro reflector, optical detector, signal processing

unit and analyzer.

Project the pulses of laser beam to the atmosphere, the area where the

pollutants are to be measured. The back scattered light by the congestion of matter is

detected by the photo detector. The reflected laser beam undergoes attenuation due to

the absorption by the pollutants in the atmosphere. Since different gases in

atmosphere absorb laser energy at different wavelength, the amount of absorbance by

each wavelength indicates the amount of pollutants in the atmosphere. The energy of

the attenuated beam received at the detector is integrated and compared with the

reference laser energy source. The difference in energy called error signal is analyzed

and convert into a readout signal by the computer. The reading indicates the

concentration and distribution of pollutants at different section in the atmosphere. But

it does not give any information about the nature of the scattered particles. However it

can be obtained by Raman back scattering experiment.

Raman Back Scattering:

Laser light is passed through the sample and the spectrum of the transmitted light is

obtained.

Since laser is a monochromatic, hence, we expect only one line in the spectrum. But

due to Raman scattering, we are observing several lines along with the expected line.

The other lines of low intensity lie symmetrically above and below to this line. These

additional lines are called side bands and their frequencies result when the oscillating

frequencies of the gas molecules are added or subtracted from the incident light’s

frequency.

Since different types of gas molecules will have different oscillating frequencies and

produce different side band.



Thus by observing Raman spectra of the back scattered light in the gas sample, the

nature of the scattering particles and their compositions can be measured.

HOLOGRAPHY

HOLO - COMPLETE GRAPHY – RECORDING (WRITING)

Holography was discovered by Dennis Gabor in 1948.

Defn: Holography is a technique of capturing pictorial details of 3-d on 2-d recording

aid, by using the phenomenon of interference.

When an object is illuminated with the light source, the light gets reflected and

scattered from the various parts of the object and they carry the information of the

object in the form of intensity and phase.

** In Photography only the intensity is recorded and the phase information is lost

** In holography both intensity and phase distribution is recorded simultaneously using

interference technique. The holographic picture provides 3-d effects even though the

recording is of 2-d.

Principle of Hologram construction:

The interference pattern, which is formed due to the superposition of reference and

object beams, has the ability to produce the transmitted effect of the object beam,

without the presence of the object, by diffracting the reference beam.

** The photographic plate on which the interference fringes are recorded is the

hologram.

The holography consist of two steps process called recording and reconstructions of the

image.

Recording of the image of an object:

There are two methods of recording the image, they are:

(1) Wavefront division technique: [Wavefront: It is the locus of points where the

particles vibrate in the same phase simultaneously.]

-> The given object and a mirror are placed one below the other such that a

part of the expanded laser beam is incident on the mirror and remaining part

falls on the object.

->The part of light reflected from the mirror (plane Wavefront) called reference

beam is incident on the photographic plate.

� When the light incident on the object, every point on the object scatters the incident light.

� Hence spherical wave fronts generates from each points on the object. The reflected beam is called object beam which incident on the photographic plate.



� The photosensitive surface responds to the resultant effect of interference between the spherical wavelets of the object beam, and the plane waves of the reference beam. Thus the interference effects are recorded on the plate.

� The interference pattern consists of concentric circular rings pattern that mark successive regions of constructive and distructive interference. The ring pattern is called Gabor Zone Plate. Every set of spherical wavelets that start from each point on the object generates its own zone plates. Thus recording consists of number of zone plates.

� Such a developed photographic film is called hologram.

(2) Amplitude division technique: In this method an expanded laser source is

incident on a beam splitter. The beam splitter reflects a part of the incident light and

remaining part will be transmitted.

The beam splitter is oriented such that the reflected light incident on the mirror

and transmitted light incident on the object.

The mirror in-turn reflects the beam called reference beam, directly on to a

photographic plate kept at a suitable position for recording the image of the object.

The transmitted light, which is incident on the object, gets scattered. The

spherical wave fronts associated with the scattered rays serve as object beam and

interfere with the reference beam at the photographic plate and the resultant pattern is

recorded in it.

The photographic plate after the development becomes the hologram of the object.



Reconstruction Process:

It is a second step in holography. For the reconstruction of the image, the same

laser beam is directed at the hologram in the same direction as the reference beam

was incident on it at the stage of recording.

When the light is incident on the hologram, diffraction takes place and secondary

waves originates from each constituent zone plate, which interfere constructively in

certain directions and generate both a real and a virtual image of the corresponding

point of the object on the transmission side of the hologram.

A real image will be formed infront of the hologram at the same distance as that

of the virtual image behind from the hologram.

By seeing through the hologram (like seeing through a window) from the

transmission side, it appears as though the original object is lying on the other side of

it at the same place. This is virtual image due to regeneration. By switching the

direction of view, different set of points which corresponds to constructive interference

are observed, which regenerate a different prospective of the object. Therefore, it gives

three dimensional effects.

Applications of Holography:

1. Holographic interferometry:

Interferometry is This is used to study the small distortions of an object that take

place such as due to stress or vibration etc. An object beam from an object and a

reference beam are made to superpose on a photographic film forming an interference

pattern. After the film is developed, it is put back in the same place and the reference

beam is now sent again as before. Also the object now is put under stress, so that it

undergoes deformation. The object beam from the deformed object superposes on the

diffracted reference beam. The diffracted reference beam, which imitates the object

beam, forms an interference pattern with the object beam from the stressed object.

This interference pattern gives us the information about the kind and the extent of

deformation. This is very useful when the deformation is extremely small and as such

cannot be determined by other conventional techniques.



2. Diffraction grating:

When two parallel beams superpose on the photographic film, the interference

pattern consists of parallel straight fringes. The film when developed appears like a

grating. The quality of such a grating is much superior when compared to the

conventional grating in the sense that the grating constant in this case is truly a

constant .

3. Acoustic grating:

In this case two coherent ultrasonic breams, one reference and the other reflected

from an object, are made incident on a medium. The resulting interference pattern

serves as a grating for laser light, which forms an optical image of the object. This is

useful for imaging the human body parts and studying physical changes.

4. Encoding : If the reference beam is sent through a mask, then the interference pattern

becomes very unique. If an attempt is made to read the hologram and if the person

reading the hologram is not aware of the masking used, then he will not be able to

decipher the image of the object. This procedure ensures further secrecy in recording

information.

OPTICAL FIBERS

Optical fiber is a device used to transmit light signals through the transparent

medium made up of dielectric materials like glass from end to other end over a long

distance.

Construction:-

1) The innermost Light guiding region called Core.

2) The middle region-covering core made of material

similar to Core is called Cladding.

The RI of Cladding is less than that of Core.

3) The outermost protecting layer for Core and Cladding

from moisture, crushing and chemical reaction etc., is

called Sheath.

The Optical Fibers are either made as a single fiber or a

flexible bundle or Cables.

A Bundle fiber is a number of fibers in single jacket.

Principles of Optical fibres:-

It is based on the principle of Total Internal reflection.

Consider a ray of light passing from denser medium to

rarer medium. As the angle of incidence increases the



angle of refraction also increases. For a particular angle of incidence called Critical

Angle the refracted ray just grazes the interface (Angle of refraction is 90o). If the

angle of incidence is greater than Critical Angle then the ray reflected back to the

denser medium. This phenomenon is called Total Internal Reflection.

TIR is not just one kind of reflections. It may be noted that, some light energy is

always lost during reflections occur at the surface of mirror, polished metallic surface.

But in case of TIR, there is no loss of light energy at the reflecting surface. The entire

incident energy is returned along the reflected light. Hence, it is called TIR. Because

of no loss of energy during reflection, the optical fibers are able to sustain the light

signal transmission over long distance inspite of infinite number of reflections that

occur within the optical fiber.

Propagation of light through fiber Optical fiber as a light guide): -

The incident light enters the core and strikes the

interface of the Core and Cladding at large angles as

shown in fig. Since the Cladding has lower RI than

Core the light suffers multiple Total Internal

Reflections. This is possible since by geometry the

angle of incidence at the interface is greater than the

Critical angle. Since the Total internal reflection is

the reflection at the rarer medium there is no energy

loss. Entire energy is transmitted through the fiber.

The propagation continues even the fiber is bent but

not too sharply. Since the fiber guides light it is

called as fiber light guide or fiber waveguide.

Numerical Aperture:

Consider an optical fiber consisting of inner cylindrical core made of glass of refractive

index n1 and is surrounded by another material called cladding of refractive index n2

such that n2 < n1.

Consider a ray of light AO incident on the core at ‘O’ at an angle θo with the fiber

axis. Then it refracts along OB at an angle of θ1 in the core.



The refracted ray is incident on the interface between core and cladding at B an

angle of incidence (90 - θ1). Assuming this angle (90 – θ1) is equal to critical angle,

then the ray is refracted at 900 to the normal drawn to the interface. i.e. it grazes along

BC.

Now, it is clear from the figure that any ray that enters into the core at angle θi < θo will have refractive angle less than θ1 because of which its angle of incidence at

the interface (=90 - θ1) will become greater than critical angle of incidence and thus

undergoes total internal reflection.

If the angle of incidence at ‘O’ is greater than θo, then the refracted ray pass

through the cladding and it will be lost.

If AO is rotated around the fiber axis keeping θo as constant, then it forms a

conical surface called acceptance cone. One those rays which enter within this

acceptance cone will undergo total internal reflection and propagates through the fiber.

“The angle θ0 is called waveguide acceptance angle or acceptance cone half

angle and Sinθ0 is called Numerical Aperture of the fiber”. The N.A. represents the

amount of light rays that can be transmitted along the optical fiber.

Expression for N.A.:

Let n0, n1 and n2 be the refractive indices of surrounding medium, core and cladding of

the fiber respectively.

Apply Snell’s law to the surface PQ, which separates surrounding medium and

core:

n0 0θSin = n1 1θSin

0θSin = 0

1

n

n 1θSin …(1)

Apply the Snell’s law to surface RS which separates core and cladding:

1n )90( 1θ−Sin = 2n 90Sin

1n 1θCos = 2n

1θCos = 1

2

n

n …(2)

Rewrite the equation (1) => 0θSin = 0

1

n

n1

2cos1 θ−



= 0

1

n

n

2

1

2

21n

n−

= 0

2

2

2

1

n

nn −

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

0θSin = N.A. = 2

2

2

1 nn −

The condition for propagation is the angle of incidence θi should be less than

acceptance

angle θ0.

i.e. iθ < 0θ

i.e. iSinθ < 0θSin

iSinθ < N.A.

iSinθ < 2

2

2

1 nn −

sine of the angle of incidence must be less than numerical aperture.

Fractional RI Change (∆):-

It is the Ratio of RI difference between Core and Cladding

to the RI of core.



Relation between N A and ∆: -

Modes of Propagation

Light propagates as an electromagnetic wave through an optical fiber. It is true that all

waves, having directions above the critical angle, will be trapped within the fiber due to

TIR. But is not true that all such waves propagate along the fiber and only certain ray

directions are allowed to propagate. The allowed directions correspond to the modes of

the fiber i.e. Mode refers to the number of paths for the light rays to propagate in the

fiber. The number of modes that a fiber will support depends on d/λ . Where d –

diameter of the core and λ is the wavelength of the wave transmitted.

[Note: As a ray gets repeatedly reflected at the walls of the fiber, phase shift occurs.

Consequently, the waves traveling along certain zigzag paths will be in phase and

intensified, however, some other paths will be out of phase and hence the signal

strength diminishes due to destructive interference. The light ray paths along which the

waves are in phase inside the fiber are known as modes.]

V- Number:

The number of modes supported for propagation in the fiber is determined by a

parameter called V-number and is given by

V = λ

πd

on

nn 2

2

2

1− d – diameter of the core, λ - wavelength of

the light, n0 – R.I. of the surrounding medium, n1 – R.I. of the core, n2 – R.I. of the

cladding

Number of modes ≅ 2

2V



Types of Optical Fiber:

Optical fibers are classified into 3-types based on their R.I. of core and cladding

and number of modes of propagation in the fiber.

(1) Step index single mode fiber (2) Step index multi mode fiber (3) Graded Index multi mode fiber.

(1) Step Index Single Mode optical fiber:

(2)Step index multi mode fiber

(** Construction is similar to that of a single mode fiber)

• It consists of a core of uniform refractive index n1

• The diameter of the core is about 50 - 200 mµ

• The core is surrounded by a material of uniform R.I. n2 called cladding such that n2 < n1. The external diameter of the cladding is 100 – 250 mµ

• The variation of R.I.s of core and cladding takes the shape of step as shown in fig.

• Since the core diameter is very large, therefore, it will be able to support propagation of large number of modes.

• LED or Laser can be used as a source. • Applications: It can be used in data links

which has lower band width requirements.

• It consists of a core of uniform refractive index n1

• The diameter of the core is about 10 mµ


• The variation of R.I.s of core and cladding takes the shape of step as shown in fig.

• Since the core diameter is very small, therefore, it can guide a single mode as shown in figure.



(3)Graded Index multi mode fiber (GRIN)

Attenuation (Fiber Loss)

The loss of light energy of the optical signal as it propagates through the fiber is called

attenuation or fiber loss.

The main reasons for the loss of light intensity over the length of the cable is

due to

(i) absorption (ii) Scattering (iii) Radiation loss

(i) Absorption Losses: In this case, the loss of signal power occurs due to absorption

of photons associated with the signal. Photons are absorbed by (a) impurities in the

silica glass (b) Intrinsic absorption by the glass material.

(a) Absorption by impurities: During the light propagation the electrons of the impurity atoms like copper, chromium, iron etc, present in the fiber glass absorb the photons and get excited to higher energy level. Later these electrons give up the absorbed energy either as heat or light energy. But the emitted light will have different wavelength with respect to the signal. Hence it is loss.

• The core material has a special feature that its R.I. value decreases in the radially outward direction from the axis and becomes equal to that of the cladding at the interface.

• It is obvious from the figer that a ray is continuously bent and travels a periodic path along the axis. The ray entering at different angles follow different paths with the same period.

• The diameter of the core is about 50 - 200 mµ


• Since the core diameter is very large, therefore, it will be able to support propagation of large number of modes.

• LED or Laser can be used as a source.

• Applications: It can be used in



(b) Intrinsic Absorption: some times even if the fiber material has no impurities, but the material itself may absorb the light energy of the signal. This is called intrinsic absorption.

(ii) Scattering Loss: (Rayleigh scattering) Since, the glass is heterogeneous mixture of many oxides like SiO2, P2O5, etc, the compositions of the molecular distribution varies from point to point. In addition to it, glass is a non-crystalline and molecules are distributed randomly. Hence, due to the randomness in the molecular distribution and inhomogeneties in the material, there will be sharp variation in the density (refractive index value) inside the glass over distance and it is very small compared to the wavelength of light. Therefore, when the light travels in the fiber, the photons may be scattered. (This type of scattering occurs when the dimensions of the

object are smaller than the wavelength of the light. Raleigh scattering4

1

λα ). Due to

the scattering, photons moves in random direction and fails to undergo total internal reflection and escapes from the fiber through cladding and it becomes loss.

(iii) Radiation loss: Radiation losses occur due to bending of fiber. There are two types of bends:

(a) Macroscopic bends: When optical fiber is curved extensively such that incident angle of the ray falls below the critical angle, then no total internal reflection occurs.

(b) Microscopic bends:

The microscopic bending is occur due to non-uniformities in the manufacturing of the

fiber or by non-uniform lateral pressures created during the cabling of the fiber. At

these bends some of the radiations leak through the fiber due to the absence of total

internal reflection and leads to loss in intensity.

Hence, some of the light rays escape through the

cladding and leads to loss in intensity of light



Attenuation co-efficient (α ):

The net attenuation can be determined by a factor called attenuation co-

efficient (α ).

When light travels in a material medium, there will always be loss in its intensity

with distance traveled. The rate of decrease of intensity of light with distance traveled

in the homogeneous medium is proportional to the initial intensity called Lamberts’s

law. i.e. if P is the initial intensity and L is the distance propagated in the medium,

then,

dL

dP− α P ( negative sign indicates that it is a decrement)

Or dL

dP− = α P (1)

where α is a constant called attenuation coefficient, or simply as attenuation.

Equation (i) can be rewritten as P

dP = - α dL

By integrating on both side ∫ P

dP = α− ∫ dL (2)

If Pin is the initial intensity with which light is entering into the fiber and Pout be the

intensity of light at the end of the fiber then equation (2) becomes,

∫out

in

P

PP

dP = α− ∫

L

dL0

The unit of attenuation for light in optical fiber is Bel.

Or α = - L

1

in

out

P

P10log Bel / unitlength

In optical fiber technology, it is customary to express α in terms of decibel/kilometer.

P is in watt and 1B = 10decibel

Therefore, α = - L

10

in

out

P

P10log dB / km



Applications of Optical Fiber:

A typical point to point communication system is shown in figure.

The analog information such as voice of telephone user is converted into electrical

signals in analog form and is coming out from the transmitter section of telephone.

The analog signal is converted into binary electrical signal using coder. The binary data

comes out as a stream of electrical pulses from the coder.

These electrical pulses are converted into pulses of optical power by modulating the

light emitted from an optical source like LED. This unit is called an Optical transmitter.

Then optical signals are fed into the optical fiber. Only those modes of light signals,

which are funneled into the core within the acceptance angle, are sustained for

propagation through the fiber by means of TIR.

The optical signals from the other end of the fiber are fed to the phtodetector, where

the signals are converted into binary electrical signals.

Which are directed to decoder to convert the stream of binary electrical signals into

analog signal which will be the same information such as voice received by another

telephone user.

Note: As the optical signals propagating in the optical fiber are subjected to two types

of degradation – attenuation and delay distortion. Attenuation is the reduction in the

strength of the signal because of loss of optical power due to absorption, scattering of

photons and leakage of light due to fiber bends. Delay distortion is the reduction in the

quality of signal because of spreading of pulses with time.

These effects cause continuous degradation of signal as light propagates and hence it

may not possible to retrieve the information from the light signal. Therefore, a

repeater is needed in the transmission path. An optical repeater consists of receiver,

amplifier and transmitter.

ADVANTAGES of Optical Fiber:



1. Optical fibers can carry very large amounts information. 2. The materials used for making optical fibers are silicon oxide and plastic,

both are available at low cost. 3. Because of the greater information carrying capacity by the fibers, the

cost, length, channel for the fiber would be lesser than that for the metallic cable.

4. Because of their compactness, and light weight, fibers are much easier to transport.

5. There is a possibility of interference between one communication channel and the other in case of metallic cables. However, the optical fiber are totally protected from interference between different communication signals, since, no light can enter a fiber from its sides. Because of which no cross talk takes place.

6. The radiation from lightning or sparking causes the disturbance in the signals which are transmitting in the metallic cable but cannot do for the fiber cable.

7. The information cannot be tapped from the optical fiber. 8. Since signal is optical, no sparks are generated as it could in case of

electrical signal. 9. Because of it superior attenuation characteristics, optical fibers support

signal transmission over long distances.’

Limitations of Optical fiber communications system:

1. Splicing is skilful task, which if not done precisely, the signal loss will be so much. The optic connectors, which are used to connect (splicing) two fibers are highly expensive.

2. While system modifications or because of accidents, a fiber may suffer line break. To establish the connections, it requires highly skilful and time consuming. Hence, maintenance cost is high.

3. Though fibers could be bent to circles of few centimeters radius, they may break when bent to still smaller curvatures. Also for small curvature bends, the loss becomes considerable.

4. Fibers undergo expansion and contraction with temperature that upset some critical alignments which lead to loss in signal power.



MODULE III SOLID STATE PHYSICS Crystal Structure

A Crystal is a solid composed of atoms or other microscopic particles arranged in an

orderly repetitive array. The study of Crystal physics aims to interpret the macroscopic properties in terms of properties of the microscopic particles of which the solid is composed. The arrangement of atoms in a Crystal is called Crystal Structure.

Lattice points and Space Lattice: -

Points can be imagined in space about which atoms or molecules are located. Such points are called Lattice Points. The totality of such points is called Space Lattice or Crystal Lattice. A Three-Dimensional space lattice (3-D space lattice) may be defined as a finite array of lattice points in three-dimension such that each and every lattice point has identical surrounding in the array.

(2-D Space Lattice)

(Lattice point)

Lattice + Basis = Crystal Structure

Basis and Crystal structure: -Every lattice point can be associated with one or unit assembly of atoms or molecules identical in composition called Basis or Pattern. The regular periodic three-dimensional arrangement of Basis is called Crystal Structure. Space lattice is imaginary. Crystal structure is real.

Bravais and Non-Bravais lattice: - A Bravais lattice is one in which all lattice points are identical in composition. If the lattice points are not identical then lattice is called Non -Bravais lattice.

The set of lattice points ‘s together constitutes a Bravais lattice. Similarly the set of lattice points ‘s together constitutes a Bravais lattice. But set of all lattice points ‘s and ‘s together constitute a Non – Bravais lattice. Hence a Non-Bravais lattice could be considered as the superposed pattern of two or more interpenetrating Bravais lattices.



Unit cell and Lattice parameters: - In every crystal some fundamental grouping of particles is repeated. Such fundamental grouping particles is called unit cell. A unit cell

is chosen to represent the symmetry of the crystal. Hence the unit cell with maximum

symmetry is chosen. They are the basic building blocks of the crystal. When these unit

cells are translated in three dimensions that will generate the crystal.

Each crystal lattice is described by type of unit cell. But each unit cell is described three vectors a, b and c when the length of the vectors and the angles (α,β,γ) between them are specified. They are nothing but the intercepts of the faces and the interfacial angles. All together they constitute lattice parameters.

Primitive Cell :-Some times reference is made to a primitive cell. Primitive cell

may be defined as a geometrical shape which, when repeated indefinitely in three dimensions, will fill all space and it consists of lattice points only at corners.

It consists of only one atom per cell. There fore unit cells may be primitive

(simple) or Non-primitive.



Crystal systems and Bravais space lattices:- Based on lattice parameters crystals are classified into seven basic systems. If atoms are placed only at corners seven crystal systems yield seven lattices. But few more lattices could be constructed by placing atoms at face center, body center etc., Bravais showed that there are 14 such lattices exist in nature. Hence the name Bravais space lattices. Each crystal system differs from the other in lattice parameters.

1) Cubic Crystal system (Isometric) a = b = c and α = β = γ = 90°

(Simple or primitive, Face centered (FCC) and Body centered (BCC)),

2) Tetragonal Crystal system a = b ≠ c and α = β = γ = 90°

(Simple and Body centered)

3) Orthorhombic Crystal system a ≠ b ≠ c and α = β = γ = 90°

(Simple, Face centered (FCC), Body centered (BCC) and Base centered),

4) Monoclinic Crystal system a ≠ b ≠ c and α = β = 90° ≠ γ

(Simple and Base centered).

5) Triclinic Crystal system a ≠ b ≠ c and α ≠β ≠ γ ≠ 90°

(Simple).

6) Trigonal Crystal system (Rhombohedral) a = b = c and α = β = γ ≠ 90°

(Simple).

7) Hexagonal crystal system a = b ≠ c and α = β = 90°, γ =120°

(Simple).



Directions and Planes: -

Directions: -In crystals there exists directions and planes in which contain concentration

of atoms. It is necessary to locate these directions and planes for crystal analysis. Arrows in two dimensions show directions. “The directions are described by giving

the coordinates of the first whole numbered point ((x, y) in two dimension,(x,y,z) in

three dimension) through which each of the direction passes”. Directions are enclosed within square brackets.

Planes: - The crystal may be regarded as made up of an aggregate of a set of parallel

equidistant planes, passing through the lattice points, which are known as lattice planes. These lattice planes can be chosen in different ways in a crystal. The problem in

designating these planes was solved by Miller who evolved a method to designate a set

of parallel planes in a crystal by three numbers (h k l) called Miller Indeces.

Different lattice planes



Steps to determine miller Indeces of given set of parallel planes: - “Miller indices

may be defined as the reciprocals of the intercepts made by the plane on the crystallog4raphic axes when reduced to smallest numbers.

Consider a plane ABC which is one of the planes belonging to the set of parallel planes with miller indices (h k l). Let x, y and z be the intercepts made by the plane along the Three crystallographic axes X, Yand Z respectively.

1) Determine the coordinates of the intercepts made by the plane along the three crystallographic axes.

2) Express the intercepts as multiples of the unit cell dimensions, or lattice parameters along the axes

3) Determine the reciprocals of these numbers

4) Reduce them into the smallest set of integral numbers and enclose them in simple brackets. (No commas to be placed between indeces)

Eg,

1) The intercepts x=2a, y=2b & z=5c Generally x=pa, y=qb, z=rc.

2) The multiples of lattice parameters are

x 2a y

= = 2, = 2, &

z

= 5

a a b c

3) Taking the reciprocals

a 1

= ,

b 1

= ,

c 1

& =

x 2 y 2 z 5

4) Reducing the reciprocals to smallest set of integral numbers by taking LCM.

10 × 1 , 10 ×

1 , 10 ×

1

2 2 5

5 5 2

Miller indices of plane ABC = (h k l) =(5 5 2).



Note: - a) All parallel equidistant planes have the same Miller indices.

b) If the Miller indices have the same ratio, then the planes are parallel.

c) If the plane is parallel to any of the axes, then the corresponding

Intercept is taken to be ∞.

Expression for Interplanar spacing in terms of Miller Indeces:

Consider a Lattice plane ABC, which is one of the planes belonging to the set of

planes with Miller indeces )( lkh . Let x, y and z be the intercepts made by the plane along the Three crystallographic axes X, Y and Z respectively.

Let OP be the perpendicular drawn form the origin to the plane. Let α|,β| and γ| be the angles made by OP with the crystallographic axes X, Y and Z respectively. Let another consecutive plane parallel to ABC pass through the origin. Let a, b and c be the lattice parameters. OP is called interplanar spacing and is denoted by dhkl.

From right angled triangle OCP

cos α|

cos β|

cos γ |

= OP

=

OC

= OP

=

OB

= OP

=

OA

8dhkl

x

8dhkl

y

8dhkl

z

but we know that

h = a , k = b & l =

c O

x y z

⇒ x = a , y =

b & z =

c

h k z

There fore

cos α| = h d



hkl hkl

d

hkl

a hkl

cos β| = k d

b

cos γ| = h d

hkl

c hkl

for the rectangular Cartesian coordinate system we have (cos α|)2 + ( cos β|)2 + ( cos γ|)2 = 1

h2

d2 + d2

k 2

+ d2

l2

= 1

a2 b

2 c

2

⇒ 2

hkl

=

h2

1

k 2

l2

+ +

a2

d =

b2

1

c2

hkl h2 k 2 l2

+ +

a2

b2

c2

is the expression for Interplanar spacing. For a cubic lattice a=b=c, There fore

dhkil =

a

h2 + k 2 + l 2



Definitions

1) Coordination number: - It is the number of equidistant nearest neighbors that an

atom has in a crystal structure.

2) Nearest neighbor distance: - It is the distance between two nearest neighbors in a crystal structure.

3) Atomic packing factor (APF) or Packing fraction: - It is the fraction of space occupied by atoms in a unit cell. It is defined as the ratio of volume occupied by atoms in unit cell to the volume of the unit cell. If the number of atoms per unit cell are ‘n’ and if Va is the volume of atoms in the unit cell and V is the volume of the unit cell then,

n × V

APF = a

V

4) Lattice Constant: - In a cubic lattice the distance between atoms remains constant

along crystallographic axes and is called Lattice Constant.

Simple Cubic Structure:

In simple cubic structure each atom consists of 6 equidistant nearest neighbors. Hence

its co-ordination number is 6.

Eight unit cells share each atom at the corner. Hence only 1/8th of the volume of the atom lies in each cell. Since the atoms are present only at corners, the No. of atoms per unit cell is given by

n = 1

× 8 = 1 atom

8

We know that the APF is given by



n × V

APF = a

V

4πR 3

1 ×

APF = 3

a3

In this structure the atoms touch each other along the sides of the cube. There fore a = 2

R, Where R is the radius of each atom.

4πR 3

APF =

3(2R)3

APF =

4πR 3

3 (8 R 3 )

APF = 0.5235 Hence atoms occupy 52.35% off the volume of the unit cell.

Body Centered Cubic (BCC) Structure:

Each atom has 8 equidistant nearest neighbors. Hence the co-ordination number is 8.

Since there are eight atoms at corners and 1 atom at the body center, the no. of



3

atoms per unit cell is given by

n = 1 + 1

× 8 = 2 atoms

8

Also in this structure the atoms touch each other along the body diagonal. There fore

(4R)2 = ( 2 a)2 + a2

Where R is the radius of the atom

16R 2 = 2 a2 + a2 = 3a2

2

a2 = 16R

3

a = 4R

3

Now the APF is given by

n × V

APF = a

V

4πR 3

1 ×

APF = 3

a3

4πR 3

1 ×

APF = 3

4R

3

APF = 3π

8

APF = 0.6802 Hence atoms occupy 68.02% of the volume of the unit



Face Centered Cubic (FCC) Structure:

In FCC structure in addition to atoms at corners, atoms are present at face centers.

Each atom consists of 12 equidistant nearest neighbors. Hence the coordination number

is 12.

The number of atoms per unit cell is

n = 1

× 6 + 1

× 8 = 4 atoms

2 8

In this structure atoms touch each other along the face diagonal. There fore

(4R)2

16R 2

= a2 + a2

= a2 + a2

2

= 2a2

Where R is the atomic radius.



a =

3

a2 = 16R

2

4R

= 2 2R

2

The APF is given by

APF = n×Va

V

4πR 3

1 ×

APF = 3

a3

1 × 4πR

APF = 3

(2 2R )3

APF = 2π

6

APF = 0.7405 Hence atoms occupy 74.05% of the volume of the unit cell.

X-Ray Diffraction

The wavelength of x-ray is of the order of Angstrom (Å). Hence optical grating

cannot be used to diffract X-rays. But the dimension of atoms is of the order of few angstroms and also atoms are arranged perfectly and regularly in the crystal. Hence

crystals provide an excellent facility to diffract x-rays.

Bragg’s X-Ray Diffraction and Bragg’s Law: -

Bragg considered crystal in terms of equidistant parallel planes in which there is regularity in arrangement of atoms. These are called as Bragg planes. There are different families of such planes exist in the crystal and are inclined to each other with certain angle.

In Bragg’s Diffraction the crystal is mounted such that an X-ray beam is inclined

on to the crystal at an angle θ. A detector scans through various angles for the diffracted



X-rays. It shows peaks for (maximum current) for those angles at which constructive interference takes place. Bragg’s law gives the condition for constructive interference.

Derivation of Bragg’s Law:

Consider Monochromatic beam of X-Rays. It is incident on the crystal with glancing

angle ‘θ’. Ray AB, which is a part of the incident beam, is scattered by an atom at ‘B’ along BC. Similarly the ray DE is scattered by an atom in the next plane at ‘E’ along EF. The two scattered rays undergo constructive interference if path difference between the rays is equal to integral multiple of wavelength.

Construction: -Bp and BQ are the perpendiculars drawn as shown in the fig. The path difference δ = PE + EQ = nλ ----------- (1)

From Right angled triangle PBE

Sinθ = PE

BE



Where BE = d (Interplanar spacing)=dhkl

Therefore PE= BE Sinθ = d Sinθ

Similarly From Right angled triangle QBE

QE = BE Sinθ = d Sinθ

Substituting in (1) δ = d Sinθ +d Sinθ = nλ δ = 2 d Sinθ = nλ

Therefore the condition for constructive interference is integral multiple of Wavelength of

X- Rays is

2 d Sinθ = n λ Hence Bragg’s Law.

Since Bragg diffraction satisfies the laws of reflection it is also called Bragg reflection.

Bragg’s X-ray Spectrometer(Determination of wavelength and Interplanar

spacing): -

It is an instrument devised by Bragg to study the diffraction of X-Rays using a crystal as Grating. It is based on the principle of Bragg Reflection.



Construction: - Monochromatic X-Ray Beam from an X-Ray tube is collimated by slits

s1 and s2 and is incident on the crystal mounted on the turntable at a glancing angle θ. The crystal can be rotated using the turntable. The reflected X-Ray beam is again collimated by slits s3 and s4 and allowed to pass through ionization chamber fixed on the Mechanical Arm. Due to ionization in the medium current flows through the external circuit, which is recorded by the Quadrant Electro Meter (E). In order to satisfy the laws

of reflection the coupling between the turntable and the mechanical arm is so made that,

if the turntable is rotated through an angle θ then mechanical arm rotates through an angle 2θ.

Experiment: Rotating the turntable increases glancing angle. Ionization current is measured as a function of glancing angle. The Ionization current is plotted versus glancing angle. It is as shown below.

The angles corresponding to intensity maximum are noted . The lowest

angle θ, corresponding to maximum intensity corresponds to the path difference λ .

∴ 2d sinθ1=n1λ=λ

Similarly for next higher angles

2d sinθ2=n2λ=2λ

2d sinθ3=n3λ=3λ and so on…

⇒ sinθ1: sinθ2:sinθ3 = 1: 2 : 3 ………(1).

If equation (1) is satisfied for θ1,θ2, and θ3 etc. then the Bragg’s law is verified.

By determining θ using Bragg’s Spectrometer and by knowing the



Value of Interplanar separation (d), Wavelength (λ) of X-Ray beam can be calculated.

By determining θ using Bragg’s Spectrometer and by knowing the value of

Wavelength (λ) of X-Ray beam, Interplanar separation (d) can be calculated.

Crystal structure of Sodium Chloride (NaCl): - NaCl is an ionic compound. Hence both Na and Cl are in ionic state. The molecule is under equilibrium because; the attractive force due to ions is balanced by repulsive force due to electron clouds.

The Bravais lattice of NaCl is FCC with the basis containing one Na ion and one Cl ion.

The bond length is 2.813Å. For each atom there are 6 equidistant nearest neighbors of opposite kind. Hence the co ordination number is 6. There are 12 next nearest neighbors

of the same kind. The conventional cell which consists of four molecules of NaCl is as shown in the figure. The coordinates of the ions in the conventional cell is as given below. (taking Na ion as origin)



This structure could be considered as the superposed pattern of two interpenetrating

Bravais lattices each made of one type of ion.

Crystal structure of Diamond (Allotropic form of Carbon): - Diamond is an allotropic form of carbon. The Bravais lattice is an FCC similar to ZnS. There are 18 carbon atoms in the unit cell. 8 at corners, 6 at face centers and 4 at intermediate tetrahedral positions. The unit cell is as shown in the fig.

In the unit cell, each carbon atom bonds to four other carbon atoms in the form of a

tetrahedron. Since each atom has four equidistant nearest neighbors the coordination number is 4.This structure could be considered as the superposed pattern of two

interpenetrating Bravais FCC lattices each made of Carbon with one displaced from the other along 1/4th of the body diagonal. The interatomic distance is 1.54Å and the lattice constant 3.56Å.



MODULE IV DIELECTRIC PROPERTIES OF SOLIDS DielectricsDielectricsDielectricsDielectrics are electrically non-conducting materials such as glass, porcelain etc, which exhibit remarkable behaviour because of the ability of the electric field to polarize the material creating electric dipoles. Dielectric ConstantDielectric ConstantDielectric ConstantDielectric Constant Faraday discovered that the capacitance of the condenser increases when the region between the plates is filled with dielectric. If C0 is the capacitance of the capacitor without dielectric and C is the capacitance of the capacitor with dielectric then the ratio C / C0 gives εεεεr called relative permittivity or Dielectric constant. Also for a given isotropic material the electric flux density is related to the applied field strength by the equation D D D D = ε EEEE, Where εεεε is Absolute permittivity. In SI system of units the relative permittivity is given by the ratio of absolute permittivity to permittivity of free space. ε = ε0 εr ε0 is permittivity of free space. εr is relative permittivity or dielectric constant. For an isotropic material, under static field conditions, the relative permittivity is called static dielectric constant. It depends on the structure of the atom of which the material is composed. Polarization of dielectricsPolarization of dielectricsPolarization of dielectricsPolarization of dielectrics: - “The displacement of charged particles in atoms or molecules of dielectric material so th“The displacement of charged particles in atoms or molecules of dielectric material so th“The displacement of charged particles in atoms or molecules of dielectric material so th“The displacement of charged particles in atoms or molecules of dielectric material so that at at at net dipole moment is developed net dipole moment is developed net dipole moment is developed net dipole moment is developed inininin the material along the applied field the material along the applied field the material along the applied field the material along the applied field direction direction direction direction is called polarization is called polarization is called polarization is called polarization of dielectric.” of dielectric.” of dielectric.” of dielectric.” Polarization is measured as dipole moment per unit volume and is a vector quantity. µ

rrNP= Where µ

r is average dipole moment per molecule and N is number of molecules per unit volume. Also Err

αµ = where α is a constant of proportionality called polarizability.



In Polar dielectric materials, When the external electric field is applied all dipoles tend to align in the field direction and hence net dipole moment develops cross dielectric material. This is the polarization of polar dielectric materialspolarization of polar dielectric materialspolarization of polar dielectric materialspolarization of polar dielectric materials. In non polar dielectric materials dipoles are induced due to the applied electric field which results in the net dipole moment in the dielectric material in the direction of the applied field. This is the polarization of nonpolarization of nonpolarization of nonpolarization of non----polar dielectric materialspolar dielectric materialspolar dielectric materialspolar dielectric materials. As the polarization measures the additional flux density arising from the presence of the material as compared to free space it has the same unit as D and is related to it as litysusceptibiElectricisWhere

E

p

Also

EP

PEE

EDSince

PED

r

r

r

r

χ

χεε

εε

εεε

εε

ε

=−=

−=∴

+=

=

+=

)1(

)1(

0

0

00

0

0 Electrical Polarization mechanismsElectrical Polarization mechanismsElectrical Polarization mechanismsElectrical Polarization mechanisms The electrical polarization takes place through four different mechanisms. They are 1. Electronic polarization. 2. Ionic polarization. 3. Orientation polarization. 4. Space charge polarization. The net polarization of the material is due to the contribution of all four polarization mechanisms. PPPP = P = P = P = Peeee + P + P + P + Piiii + P + P + P + Poooo + P + P + P + Pssss 1) 1) 1) 1) Electronic polarizationElectronic polarizationElectronic polarizationElectronic polarization: - This occurs through out the dielectric material and is due to the separation of effective centers of positive charges from the effective center of negative charges in atoms or molecules of dielectric material due to applied electric field. Hence dipoles are induced within the material. This leads to the development of net dipole moment in the material and is the vector sum of dipole moments of individual dipoles.



2)2)2)2) Ionic polari Ionic polari Ionic polari Ionic polarizationzationzationzation: - This occurs in ionic solids such as sodium chloride etc. Ionic solids possess net dipole moment even in the absence of external electric field. But when the external electric field is applied the separation between the ions further increases. Hence the net dipole moment of the material also increases. It is found that the ionic dipole moment also proportional to the applied field strength. .litypolarisabiioniciswhere

Ehence

i

ii

α

α=µ Ionic Polarization is given by ionic dipole moment per unit volume. .volumeunitperatomsof.noisNwhere

Nphence ii µ= 3)3)3)3) Orientation Polarization Orientation Polarization Orientation Polarization Orientation Polarization: - This occurs in polar dielectric material, which possesses permanent electric dipoles. In polar dielectrics the dipoles are randomly oriented due thermal agitation.

-

+

-

-

-

+

+

+

+

+

+ +

-

- -

-

+ +

E = 0. E > 0.

( )N

bygivenislitypolarizabielectronicThe

mperatomsofnumberNiswhereENNP

litypolarizabielectronicwhere

EE

ERmomentdipoleelectronic

re

eee

e

eee

e

1

.

4

0

3

3

0

−=

==

=⇒∝⇒

==⇒

εεα

αµ

α

αµµ

πεµ

E<0 E>0



Therefore net dipole moment of the material is zero. But when the external electric field is applied all dipoles tend to align in the field direction. There fore dipole moment develops across the material. This is referred to as orientation polarization (PPPPoooo). Orientation polarization depends on temperature. Higher the temperature more the randomness in dipole orientation smaller will be the dipole moment. The orientation polarizability is given by Tk3

2

0

µα =

4)4)4)4) Space charge polarization Space charge polarization Space charge polarization Space charge polarization: - This occurs in materials in which only a few charge carriers are capable of moving through small distances. When the external electric field is applied these charge carriers move. During their motion they get trapped or pile up against lattice defects. These are called localized charges. These localized charges induce their image charge on the surface of the dielectric material. This leads to the development of net dipole moment across the material. Since this is very small it can be neglected. It is denoted by PPPPssss. Internal Field or Local FieldInternal Field or Local FieldInternal Field or Local FieldInternal Field or Local Field:- When dielectric material is placed in the external electric field, it is polarized creating electric dipoles. Each dipole sets electric field in the vicinity. Hence the net electric field at any point within the dielectric material is given by “The sum of external field and the “The sum of external field and the “The sum of external field and the “The sum of external field and the field due to all dipoles surrounding that point”field due to all dipoles surrounding that point”field due to all dipoles surrounding that point”field due to all dipoles surrounding that point”. This net field is called internal fieldinternal fieldinternal fieldinternal field or Local fieldLocal fieldLocal fieldLocal field. Expression for Internal field in case of Solids and LiquidsExpression for Internal field in case of Solids and LiquidsExpression for Internal field in case of Solids and LiquidsExpression for Internal field in case of Solids and Liquids ( ( ( (One dimensional One dimensional One dimensional One dimensional )))): Consider a dipole with charges ‘+q ‘and ‘–q’ separated by a small distance ‘dx’ as shown in fig. The dipole moment is given by µ = q dx. Consider a point ‘P’ at a distance ‘r’ from the center of dipole.

E = 0. E > 0.

Where ‘kkkk’ is Boltzman constant, T T T T is absolute temperature and µµµµ is permanent dipole moment.



The electric field ‘E’ at ‘P’ can be resolved into two components. 1) The Radial Component along the line joining the dipole and the point. It is given by 3

0

r r4

Cos2E

επ

θµ=

r 2) The Tangential component or Transverse component perpendicular to the Radial component is given by 3

0 r4

SinE

επ

θµ=

θ

r Where ‘θ’ is the angle between the dipole and the line joining the dipole with the point ’P’, ‘ε0’ is permitivity of free space and ‘r’ is the distance between the point and dipole. Consider a dielectric material placed in external electric field of strength ‘E’. Consider an array of equidistant dipoles within the dielectric material, which are aligned in the field direction as shown in the figure. E C A X B D F

a a a a a

2a 2a

3a 3a

θ

r

P

Err

Eθ

E

dx +q -q



Let us find the local field at ‘X’ due all dipoles in the Array. The field at ‘X’ due to dipole ‘A’ is given by θ

+= EEE rXA

3

0

XA

3

0

r

a2E

.0E,a4

0Cos2EHence

0andarHere

επ

µ=⇒

=επ

µ=

=θ=

θ

o

o The field at ‘X’ due to dipole ‘B’ is given by

θEEErXB

+= ( )

30

XB

3

0

r

a2E

.0E,a4

180Cos2EHence

180andarHere

επ

µ=⇒

=−επ

µ=

=θ−=

θ

o

o Hence the Total field at ‘X’ due to equidistant dipoles ‘A’ and ‘B’ is given by

XBXA1 EEE += 3

0

1

3

0

3

0

1

aE

a2a2E

επ

µ=⇒

επ

µ+

επ

µ=

Similarly, the total field at ‘X’ due to equidistant dipoles ‘C’ and ‘E’ is given by XDXC2 EEE += ( )a2r

)a2(E

)a2(2)a2(2E

3

0

2

3

0

3

0

2

=επ

µ=⇒

επ

µ+

επ

µ=

Q

Similarly, the total field at ‘X’ due to equidistant dipoles ‘D’ and ‘F’ is given by XFXE3 EEE +=

( )a3r)a3(

E

)a3(2)a3(2E

3

0

3

3

0

3

0

3

=επ

µ=⇒

επ

µ+

επ

µ=

Q



The net field at ‘X’ due to all dipoles in the array is given by

3

0

|

1n3

1n33

0

333

0

3

0

3

0

3

0

|

4321

|

a

2.1E

2.1n

1but

n

1

a

3

1

2

11

a

)a3()a2(aE

EEEEE

επ

µ=∴

≅

επ

µ=

+++

επ

µ=

+επ

µ+

επ

µ+

επ

µ=

++++=

∑

∑

∞

=

∞

=

LLL

LLL

LLL

3

0

i

3

0

|

i

a

E2.1EE

.EWkt

a

2.1EEEE

bygivenis'X'atFieldLocalThe

επ

α+=∴

α=µ

επ

µ+=+=

0

0

ε

γα

ε

αγ

PEEENPonpolarizatibut

ENEE

i

i

+=∴=

+=∴

For Three-Dimension the above equation can be generalized by replacing 1/a3 by ‘N’ (where ‘N’ the number of atoms per unit volume), and 1.2/π by γ called Internal Field Constant.

Since γ, Ρ and ε0 are positive quantities Ei > E. For a Cubic Lattice γ = 1/3 and the Local field is called Lorentz field. It is given by 03ε

PEE L +=



ClausiusClausiusClausiusClausius----Mosotti Relation: Mosotti Relation: Mosotti Relation: Mosotti Relation: Consider an Elemental solid dielectric material. Since they don’t posses permanent, dipoles, for such materials, the ionic and orientation polarizabilities are zero. Hence the polarization P is given by )1(...............

31

31

3

3

0

0

0

0

eqnN

ENP

ENN

P

PNENP

PEN

FieldLorentzisEWhereENP

e

e

e

e

ee

e

LLe

−

=∴

=

−

+=

+=

=

ε

α

α

αε

α

εαα

εα

α Where ‘N’ is the no. of dipoles per unit volume, eα is electronic polarizability 0ε is permittivity of free space, and E is the Electric field strength. The polarization is related to the applied field strength as given below

)2(...................)1(0

00

0

0

eqnEP

PEE

EDSince

PED

r

r

r

−=∴

+=

=

+=

εε

εεε

εε

ε Where ‘D’ is Electric Flux Density and rε is Dielectric Constant. Equating equations (1) and (2) E

N

ENP r

e

e )1(

31

0

0

−=

−

=∴ εε

ε

α

α 1

3)1(

31

)1(

00

00

=+−

−=

−

ε

α

εε

α

ε

α

εε

α

e

r

e

e

r

e

N

E

EN

N

E

EN



+

−=∴

=

+

−

2

1

3

11)1(

3

3

0

0

r

re

r

e

N

N

ε

ε

ε

α

εε

α

The above equation is called Clausius – Mosotti relation. Using the above relation if the value of dielectric constant of the material is known then the electronic polarizability can be determined. Properties of Dielectrics under alternating field conditions( Frequency dependence of Dielectric Properties of Dielectrics under alternating field conditions( Frequency dependence of Dielectric Properties of Dielectrics under alternating field conditions( Frequency dependence of Dielectric Properties of Dielectrics under alternating field conditions( Frequency dependence of Dielectric constant, Dielectric loss & comconstant, Dielectric loss & comconstant, Dielectric loss & comconstant, Dielectric loss & complex dielectric constant)plex dielectric constant)plex dielectric constant)plex dielectric constant)::::- It is found under alternating field conditions of high frequency, the dielectric constant is a complex quantity. When dielectric materials are placed in alternating field the polarization tend to reverse as the polarity changes. If the frequency of the field is low (less than 1M Hz), then the polarization can follow the alternations of the field and hence the dielectric constant remains static. Under alternating field conditions of high frequency (Greater than 1MHz) the oscillations of dipoles lag behind those of field. If the frequency is increased further they are completely unable to follow the alternations in the field and hence the molecular process Orientation polarization ceases due to dielectric relaxation. This occurs in the frequency range 106 Hz to 1011Hz.As the frequency is increased further other polarizing mechanisms start to cease one after another. The ionic polarization ceases in the frequency range 1011 Hz to 1014Hz. Finally only electronic polarization Frequency

Polariza

tion

Space Charge

Orientation

Ionic

Electronic



remains. Hence as the frequency of the field increases the polarization decreases and hence the dielectric constant decreases. This is known as Anomalous Dielectric DispersionAnomalous Dielectric DispersionAnomalous Dielectric DispersionAnomalous Dielectric Dispersion. Dielectric LossDielectric LossDielectric LossDielectric Loss: - In the alternating field conditions during the rotation of dipoles they have to overcome some sort of internal friction, which is dissipated as heat by the material. This is called as dielectric loss. Complex Dielectric ConstantComplex Dielectric ConstantComplex Dielectric ConstantComplex Dielectric Constant: - The complex dielectric constant is given by εr* =εr’ - εr’’. Where εr’’ determines Dielectric Loss. εr’ determines the component of current out of phase by 90° with the field. Important applications of Dielectric Materials: Important applications of Dielectric Materials: Important applications of Dielectric Materials: Important applications of Dielectric Materials: Dielectric materials find a wide range of applications as insulating materials. 1) Plastic and Rubber dielectric are used for the insulations of electrical conductors 2) Ceramic beads are used for the prevention of electric short circuiting and also for the purpose of insulation. 3) Mica and Asbestos insulation is provided in electric Iron in order to prevent the flow of electric current to outer body. 4) Varnished cotton is used insulators in transformers. 5) Dielectric materials are used in the energy storage capacitors. 6) Piezoelectric crystals are used in oscillators and vibrators.



MODULE V ELECTRICAL CONDUCTIVITY IN METALS

Classical free electron theory of metals: In order to explain electrical conductivity in metals, Lorentz and Drude put forward a theory called free electron theory of metals. It is based on the following assumptions. 1. Free electrons in a metal resemble molecules of a gas. Therefore, Laws of kinetic theory of gasses are applicable to free electrons also. Thus free electrons can be assigned with “average velocity c”, ‘Mean free path λ” and “mean collision time τ”. 2. The motion of an electron is completely random. In the absence of electric field, number of electrons crossing any cross section of a conductor in one direction is equal to number of electrons crossing the same cross section in opposite direction. Therefore net electric current is Zero. 3. The random motion of the electron is due to thermal energy. Average kinetic energy of the electron is given by

Where c=average velocity or thermal velocity

4. Electric current in the conductor is due to the drift velocity acquired by the electrons in the presence of the applied electric field. 5. Electric field produced by lattice ions is assumed to be uniform throughout the solid. 6. The force of repulsion between the electrons and force of attraction between electrons and lattice ions is neglected. The Drift Velocity :

In the absence of the applied electric field, motion of free electron is completely random. During their motion electrons undergo collisions with the residual ions and during each collision direction and magnitude of their velocity changes in general. When electric field is applied, electrons experiences force in the direction opposite to the applied field. Therefore in addition to their random velocity, electron acquires velocity in the direction of the force. Since electrons continue to move in their random direction, with only a drift motion due to applied field, velocity acquired by the electrons in the direction opposite to the applied field is called Drift velocity and is denoted by vd. Note the vd is very small compared to c, the average thermal velocity. Electric current in a conductor is primarily due to the drift velocity of the electrons. Electronic Conduction in Solids

Relaxation Time, Mean collision time and Mean free path:

Mean free path (λ): The average distance traveled by electrons between two successive collisions during their random motion is called mean free path, it denoted by λ. Mean collision Time(ττττ): The average time taken by an electrons between two successive collisions during their random motion is called mean collision time, it is denoted by ττττ. The relationship between λ and ττττ is given by λ=cττττ.



Relaxation Time(ττττr): In the presence of an applied electric field, electrons acquire drift velocity vd in addition to the thermal velocity c. if electric field is switched off, vd reduces and becomes zero after some time. Let electric field is switched off at the instant t = 0, when drift velocity vd=v0. The drift velocity of the electron after the lapse of ‘t’ seconds is given by

Where ττττr is called relaxation time. Suppose t=ττττr , then vd=v0 , e

-1=1/ev0

Thus the relaxation time is defined as the time during which drift velocity

reduces to 1/e times its maximum value after the electric filed is switched off.



HALL EFFECT

Suppose a material carrying an electric current is placed in a magnetic field. Then an electric field is produced inside the material in a direction which is at right angles to both the current and magnetic field. This is called Hall effect discovered by Edwin H. Hall in 1879.

Let us consider a metal. Let the current J be in the X direction as shown in figure. The current carriers which are the free electrons in the metal have a drift velocity v in the negative X direction. Let a magnetic field B be applied along the Z direction. Then the electrons will experience a force of magnitude Bev. The direction of the force is by Fleming’s left hand rule in the negative Y direction. This force is called Lorentz force. Under the action of this force the free electrons will move in the negative Y direction and collect on the bottom surface of the material. There will be a deficit of electrons on the top surface. This is equivalent to a collection of positive charges on the top surface. The collection of positive



and negative charges on the top and bottom faces of the material results in an electric field in the negative Y direction. This electric field is called Hall field. If its magnitude is EH the force due to it on an electron is eEH in the positive Y direction. This force opposes the Lorentz force. Soon an equilibrium condition will set up when

BeveEH = …….(1)

If the current density in the X direction is J then,

nevJ = ……..(2)

Where n is the concentration of the free electrons. Then by combining (1) & (2) we get

Or JBRE HH =

Where ne

R H

1= is called the Hall coefficient.

ne

BJEH =



MODULE VI

MAGNETIC PROPERTIES OF SOLIDS

Classification of Diamagnetic, Paramagnetic and ferromagnetic materials

Diamagnetic Materials

The substances which tend to move from stronger to weaker region of the magnetic field

are called diamagnetic substances. These are weakly repelled by a strong magnet. In these

substances, there are equal numbers of electrons spinning and orbiting in opposite

directions so that the electrons of an atom remain paired in such away that their net

magnetic moment is zero in the absence of any external magnetic field.

Examples: Antimony, bismuth, copper, gold, lead, mercury, silver etc…

Properties

1. The relative permeability is less than unity

2. The susceptibility is negative

3. The susceptibility does not vary with temperature.

Paramagnetic Materials

The substances which tend to move from weaker to stronger regions of magnetic field are

called paramagnetic substances. These are feebly attracted by a strong magnet. In these

substances, the atoms or molecules with one or more unpaired electrons possess a

permanent magnetic moment. But they are oriented randomly in the absence of an external

magnetic field.

Examples: Aluminium, chromium, manganese, platinum, sodium etc…

Properties

1. Low magnetization, low susceptibility

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

3. Permeability is greater than one



Paramagnetic susceptibility χ = T

C Where C is curie constant

This result namely the magnetic susceptibility of atoms varies as T

1 is known as Curies law.

Ferromagnetic materials

The substances which are strongly attracted by a magnet are called ferromagnetic substances. 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.

Examples: iron, cobalt, nickel and their alloys.

Properties

1. The relative permeability is very high

2. The susceptibility is very high and it depends on the temperature.

3. They exhibit magnetostriction and hysteresis

Ferromagnetic susceptibility χ = ( )θ−T

C Where C is curie constant

B H Curve ( Hysterisis)

Bsat Br

-Hc



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.

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.

Soft magnetic materials:

Magnetic materials that have high permeability and low coercive force are called soft

magnetic materials.



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

possess small hysteresis loss.

Properties:

1. Low remnant magnetism

2. High permeability

3. High susceptibility

4. Low coercivity

5. Low hysteresis energy loss

6. Thin hysteresis loop

7. Low eddy current loss

Ex: Soft iron, mild Steel, Sendust, perm alloy, pure nickel

Uses:

1. They are used in the construction of cores of transformers.

2. Relatively pure iron is frequently used as the magnetic core for direct current

applications.

3. The magnetic mild steel is used for relays, reed switches and pole pieces for

electromagnets.

4. Iron-nickel alloys are used for audio frequency applications.

Hard magnetic materials :

Magnetic materials that have large energy product (BH) and high coercivity are called hard

magnetic materials.



Properties:

1. Large hysteresis loop

2. High coercivity

3. High remanent magnetization

4. High permeability

5. High hysteresis energy loss

6. High saturation flux density

Ex: Carbon, tungsten-steel alloy, alloy steel, alloys of aluminium

Uses:

1. The permanent magnets are used in instruments like galvanometers, ammeters,

voltmeters, speedometers and recorders.

2. Tungsten steel are used in chucks and latches, tool holders, magnetic bearings and

mixers.

3. They are used in electronic devices such as telephones and tape recorders, loud speakers

and hearing aids.



Superconductivity:

Temperature dependence of resistivity of metal

Thus net Resistivity of a metal can be written as

ρ = ρ 0+ ρ (T)

Thus net Resistivity of conductor is equal to sum temperature independent part and

temperature dependent part as shown in the graph

Superconductivity

Kamerlingh Onnes discovered the phenomenon of superconductivity in the year 1911. When

he was studying the temperature dependence of resistance of Mercury at very low

temperature he found that resistance of Mercury decreases with the decrease in

temperature up to a particular temperature Tc= 4.15K . Below this temperature the

resistance of mercury abruptly drops to zero. Between 4.15K and Zero degree Kelvin

Mercury offered no resistance for the flow of electric current. The phenomenon is reversible

and material becomes normal once again when temperature was increased above 4.15K. He

called this phenomenon as superconductivity and material which exhibited this property as

superconductors.

The variation of resistivity with temperature for a metal is as shown in the fig. Resistivity in the case of pure metal decreases with the decrease in temperature and becomes zero at absolute zero temperature. While in the case of impure metals the resistivity of metal will have some residual value even at absolute zero temperature. This residual resistance depends only on the amount of impurity present in the metal and is independent of the temperature.



Thus the phenomenon of super conductivity is defined as:

“The phenomenon in which resistance of certain metals, alloys and compounds

drops to zero abruptly, below certain temperature is called superconductivity

The temperature, below which materials exhibit superconducting property is called

critical temperature, denoted by Tc. Critical temperature Tc is different for different

substances. The materials, which exhibit superconducting property, are called

Superconductors.

Above critical temperature material is said to be in normal state and offers resistance

for the flow of electric current. Below critical temperature material is said to be in

superconducting state. Thus Tc is also called as transition temperature.

Meissner Effect

In 1933, Meissner and Ochsenfeld showed that when a superconducting material is

placed in a magnetic field, it allows magnetic lines of force to pass through, if it’s

temperature is above Tc and if temperature is reduced below the critical temperature Tc, it

expels all the lines of force completely out of the specimen to become a perfect diamagnetic

material. This is known as Meissner effect.

Since superconductor exhibits perfect diamagnetism below the critical temperature Tc,

magnetic flux density inside the material is zero.

Therefore B=0, for T< Tc



Relationship between flux density and the strength of the magnetizing field is given by

µµµµ0 = Absolute permeability of free space

M = Intensity of magnetization of the material &

H = Strength of the magnetizing field

Thus superconductor possesses negative magnetic moment when it is superconducting

state.

Consider a superconducting material above its critical temperature. A primary coil and

secondary coil are wound on the material. The primary coil is connected to battery and the

secondary coil is connected to a Ballistic Galvanometer. When the primary circuit is closed,

current flows through it, which sets up a magnetic field in it.

The magnetic flux immediately links with the secondary coil. This change in flux across

the secondary coil produces the momentary current and hence the galvanometer shows the

deflection.



After certain time, the primary current becomes steady; flux linkage with the secondary

coil becomes unchanging. As a result, no change in the flux linkage in the secondary coil

and hence no more current driven in the secondary circuit.

Now, decreases the temperature of the super conductor gradually. As soon as the

temperature crosses below the critical temperature, the B.G. suddenly shows a deflection,

indicating that the flux linkage with the secondary coil has changed.

The change in flux linkage is attributed to the expulsion of the magnetic flux

from the body of the superconducting material as shown in figure.

Critical field

We know that when superconductor is placed in a magnetic field it expels magnetic lines

of force completely out of the body and becomes a perfect diamagnet. But if the strength of

the magnetic field is further increased, it was found that for a particular value of the

magnetic field, material looses its superconducting property and becomes a normal

conductor. The value of the magnetic field at which superconductivity is destroyed is called

the Critical magnetic field, denoted by Hc. It was found that by reducing the temperature

of the material further superconducting property of the material could be restored. Thus,

critical field doesn’t destroy the superconducting property of the material completely but

only reduces the critical temperature of the material



Critical magnetic field Hc depends on the temperature of the material. The relationship

between the two is given by

Isotopic effect

The variation of critical temperature of an element with isotopic mass is called isotopic

effect.

The transition temperature is inversely proportional to the square root of the atomic mass of

the isotope of a single superconductor.

aC

MT

1α a= constant equal to ½, M= atomic weight

Critical current density

The application of a large value of electric current to superconducting material destroys the

superconducting property. Consider a coil of wire wound on superconductor .Let I be the

current Flowing through the wire. The application of the current induces a magnetic field.

Thus, induced Magnetic field in the conductor destroys the superconducting property.

The induced critical current is given by



Ic = 2πrHc Ic=critical current density

Hc= critical field

BCS theory of Superconductivity

Bardeen, Cooper and Schrieffer explained the phenomenon of superconductivity in the

year 1957. The essence of the BCS theory is as follows.

We know that resistance of the conductor is due to the scattering of electrons from the

lattice ions.

Consider an electron moving very close to a lattice ion. Due to coulomb interaction

between electron and ion, the ion core gets distorted from its mean position. It is called

lattice distortion. Now another electron moving close to this lattice ion interacts with it.

This results in the reduction in the energy of the electron. This interaction can be looked

upon as equivalent to the interaction between two electrons via lattice. During the

interaction exchange of phonon takes place between electron and the lattice. This

interaction is called electron-lattice –electron interaction via the phonon field.

Because of the reduction in energy between the two electrons, an attractive force comes

into effect between two electrons. It was shown by Cooper that, this attractive force

becomes maximum if two electrons have opposite spins and momentum. The attractive

force may exceed coulombs repulsive force between the two electrons below the critical

temperature, which results in the formation of bound pair of electrons called cooper pairs.

At temperatures below the critical temperature large number of electron lattice- electron

interaction takes place and all electrons form a cloud of cooper pairs.

Cooper pairs in turn move in a cohesive manner through the crystal, which results in an

ordered state of the conduction electrons without any scattering on the lattice ions. This

results in a state of zero resistance in the material.

Persistent current

Once a current is started in a closed loop of superconducting material, it will continue to

keep flowing of its own accord, around the loop as long as the loop is held below the critical

temperature. Such a steady current Which flows with undiminishing strength is called a

persistent current. The persistent current does not need external power to maintain it

because there does not exist I2R losses. In one instance, a current is maintained in a

superconducting loop for more than two years. Persistent current is one of the most



important properties of a superconductor. Superconductor coils with persistent currents

produce magnetic fields and can therefore serve as magnets. Such a superconducting

magnet does not require a power supply to maintain its magnetic field.

Types of Superconductors

Type I or Soft Superconductors:

Superconducting materials, which exhibit, complete Meissner effect are called Soft

superconductors.

We know that below critical temperature, superconductors exhibit perfect diamagnetism.

Therefore they possess negative magnetic moment.

The graph of magnetic moment Vs magnetic field is as shown in the Fig. As field strength

increases material becomes more and more diamagnetic until H becomes equal to Hc. At

Hc, material losses both diamagnetic and Superconducting properties to become normal

conductor. It allows magnetic flux to penetrate through its body. The value of Hc is very

small for soft superconductors. Therefore soft superconductors cannot withstand high

magnetic fields. Therefore they cannot be used for making superconducting magnets

Type II or Hard Superconductors

Superconducting materials, which can withstand high value of critical magnetic fields, are

called Hard Superconductors.



The graph of magnetic moment Vs magnetic field is as shown in the Fig. Hard

superconductors are characterized by two critical fields Hc1 and Hc2. When applied

magnetic field is less than Hc1 material exhibits perfect diamagnetism. Beyond Hc1 flux

penetrates and fills the body partially. As the strength of the field increases further, more

and more flux fills the body and thereby decreasing the diamagnetic property of the

material. At Hc2 flux fills the body completely and material losses its diamagnetic property

as well as superconducting property completely.

Between Hc1 and Hc2 material is said to be in vortex state. In this state though there is

flux penetration, material exhibits superconducting property. Thus flux penetration occurs

through small-channelised regions called filaments. In filament region material is in normal

state. As Hc2 the field strength increases width of the filament region increases at they

spread in to the entire body, and material becomes normal conductor as a whole. The value

of Hc2 is hundreds of times greater than Hc of soft superconductors. Therefore they are

used for making powerful superconducting magnets.



Applications of Superconductivity

1. Superconducting Magnets:

We know that in ordinary electromagnet strength of the magnetic field produced depends

on the number of turns (N) in the winding and the strength of the current (I) flowing

through the winding. To produce strong magnetic field either N or I should be increased. If

N is increased size of the magnet increases and if I is increased power loss (I2R) increases,

which results in production of heat. Therefore there are limitations to increase N and I. If

superconducting wires are used for winding in electromagnets, even with small number of

turns strong magnetic fields can be produced by passing large current through the winding,

because there is no loss of power in superconductors. The type II superconductors, which

have high Hc and Tc values, are commonly used in superconducting magnets. Ex: Niobium-

tin, Niobium-aluminum, niobium-germanium and vanadium-gallium alloys.

The superconducting magnets are used in Magnetic Resonance Imaging

(MRI) systems, for plasma confinement in fusion reactors, in magneto-hydrodynamic power

generation, in Maglev vehicles, etc.

2. Maglev (Magnetically Levitated vehicles)

1. In these vehicles transportation is by setting afloat the carriage above the guide

way.

2. Utility of such levitation is that, in the absence of contact between the moving and

stationary systems, the friction is eliminated.

3. Great speed is achieved with low energy consumption.

4. The phenomenon is based on Meissner effect.



5. The vehicle consists of superconducting magnets built into its base. There is an

Aluminium guide way over which the vehicle will be set afloat by magnetic

levitation. This is brought about by enormous repulsion between two powerful

magnetic fields, one produced by superconducting magnet inside the vehicle and

the other one by electric currents in the Aluminium guide way.

6. Guide way is divided into a number of segments provided with coils. The flow of

currents through the coils could be related to the position and instantaneous speed

of the vehicle.

7. The currents in the guide way not only produce the necessary magnetic field to

levitate the vehicle but also help in propelling the vehicle forward.

8. The vehicle is provided with retractable wheels. With these wheels the vehicle runs

on the guide way. Once levitated in air the wheels are retracted into the body,

while stopping, the wheels are drawn out and vehicle slowly settles on the guide

way by running over a distance.

3. SQUID (Superconducting Quantum Interference Device)

It is an ultra-sensitive measuring instrument used for the detection of very weak

magnetic fields. (~ 10-14T)

It is formed by incorporating two Josephson’s junctions (J1 and J2) in the loop of a

superconducting material.

An arrangement consisting of two superconductors separated by a thin oxide layer

(insulator) is known as Josephson’s junction. When the wavelength of the matter wave

generated by the cooper pair is greater than barrier width, cooper pairs from one

superconductor can tunnel through the barrier and reach the other superconductor.



When the magnetic field is applied to this superconducting circuit, it induces a

circulating current which produces just that much opposing magnetic field as to exclude the

flux from the loop.

The flux remains excluded so long as the junction currents do not exceed a critical

value. (Where the critical current IC, which is the maximum current across the junction

under zero voltage condition)

But the circuit switches to resistive phase and thereby the flux passes into the loop.

Once, the current in either of the junctions or in both exceed the critical value, the loop acts

like a gate to allow or exclude the flux.

A mathematical analysis shows that, IC that is supported in the loop is a periodic

function of the applied magnetic flux (φ ).

It can be interpreted as the consequence of interference due to the phase difference

between the reunited currents.

The phase difference is caused by the applied magnetic field. Here both the

interference effects and the quantization effects in superconductivity state are involved.

Thus the device is named as SQUID.

Applications:

When the squid is brought under the influence of the external field, the flux through

the loop changes. It causes a change in squid current Is. The variation in Is will be periodic

in nature and hence induces an emf in an adjacent coil of an electric circuit that senses

changes in the flux.

Peridoic variation of the critical current IC with

the total flux through the area of the squid.



It can be used to detect magnetic fields of heart and brain.

It can be used for logic operations in an electronic circuit as well as memory devices.

Temperature dependence of specific heat

The specific heat of the normal metal is seen to be of the form.

( ) 3TTTCn βγ ==

The first term in Eq. is the specific heat of electrons in the metal and the second

term is the contribution of lattice vibrations at low temperatures. The specific heat of the

superconductor shows a jump at Tc. Since the superconductivity affects electrons mainly, it

is natural to assume that the lattice vibration part remains unaffected, i.e., it has the same

value βT3 in the normal and superconducting states. On subtracting this, we notice that the

electronic specific heat Ces is not linear with temperature. It rather fits an exponential form.

( ) ( )TKATC Bes ∆−= exp

Temperature dependence of the electronic specific heat in the normal and

superconducting states

This exponential form is an indication of the existence of a finite gap is the energy spectrum

of electrons separating the ground state from the lowest state.



The number of electrons thermally excited across the gap varies exponentially with the

reciprocal of temperature. The energy gap is believed to be a characteristic feature of the

superconducting state which determines the thermal properties as well as high frequency

electromagnetic response of all superconductors.

(a) (b)

(a) Conduction band in the normal state.

(b) Energy gap at the Fermi level in the superconducting state E=10 eV.

Thermal Conductivity

The thermal conductivity of superconductors undergoes a continuous change between

the two phases and is usually lower in the superconducting phase suggesting that the

electronic contribution drops, the superconducting electrons possibly playing no part in heat

transfer.

The thermal conductivity of tin at 2 K is 34W cm-1 K-1 for the normal phase and 16W

cm-1 K-1 for the superconducting phase. At 4K, it is 55 W cm-1 K-1 (At 4K there is no

superconducting phase for tin as Tc=3.73K).



Thermal conductivity of a specimen of tin in the normal and superconducting

state

Phy Notes Syl 10

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