unit-01 - wordpress.com · 2 lesson, we will study dual nature of radiation and de-broglie concept...
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UNIT-01
MODERN PHYSICS Introduction
The classical concept of particle, space and time stood unchallenged for
more than two hundred years and it had achieved many spectacular successes
particularly in celestial mechanics. But in the early years of the twentieth
century, the outcome of revolutionary theories like quantum theory and theory
of relativity swept away the classical concept of particle, space and time given
by Newton. A new set of laws of quantum physics and relativistic physics
replaced the laws of classical physics.
In classical physics it is assumed that light consists of minute particles
called corpuscles, which is responsible for various processes and phenomenon
associated with light; however, after the discovery of phenomenon like
interference, diffraction and polarization, it is proved beyond doubt that light is
a form of wave, more correctly electromagnetic wave and these phenomena are
successfully explained on the basis of Huygens wave theory of light. The
observed phenomena like Compton Effect and explanation of spectrum of black
body radiation required description of radiation in terms of particles of energy-
photons. Thus dual nature of light is a fact of experimental evidence.
Overview of Unit-01
This unit consists of three lessons of teaching. In the first lesson, we will study
spectrum of black body radiation, significance of Quantum theory. In the
second lesson, we will study Compton Effect and its significance; in the third
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lesson, we will study dual nature of radiation and de-Broglie concept of matter
waves and numericals.
Objectives of Unit 01
At the end of this unit we shall understand that:
The emission and absorption of energy is not continuous, but
discrete.
A particle in motion is associated with waves called matter waves.
Matter has dual characteristics i.e. it exhibits both wave and particle
properties.
Both wave properties and particle properties of moving objects cannot
appear together at the same time because there is a separable link.
De-Broglie waves are pilot waves and not electromagnetic waves.
A moving particle is described in terms of wave packet.
The dual nature of radiation has made position of a particle
uncertain.
Introduction:
In this unit we will study about the failure of classical physics to explain the spectrum of black body radiation leading to discovery of Quantum theory of radiation, which signifies the particle nature of radiation thereby opening new way of understanding physics. Hence physics developed from the year 1901 is called Modern Physics and most of the phenomena are satisfactorily explained on the basis of Quantum theory of radiation. Later it became a tool to study particles of sub atomic world. And there was a need for new mechanics to explain experimentally verified atomic phenomena.
Objectives:
At the end of lesson you shall understand that:
The classical physics cannot explain spectrum of black body radiation, which has lead to discovery of Quantum theory of radiation, hence radiation cannot be
emitted continuously as predicted in classical physics.
Introduction:
In this lesson we will study spectrum of black body radiation and various laws put forward to explain the energy distribution in the spectrum, their failure and success.
Introduction to Black Body Radiation Spectrum
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A perfect black body is the one which absorbs the entire radiations incident on it, it neither reflects nor transmits radiations, and hence it appears perfectly black. But there are no perfect black bodies. For all practical purposes we take lamp black as black body, because when a body coated with lamp black exposed to radiations, it absorbs 99 percent of it, and also when it is heated, it emits radiations containing almost all wavelengths.
The black body radiation is characteristic of its temperature; hence it is important to know how the energy is distributed among various wavelengths at different temperatures.
Number of scientists carried out experiments on this energy distribution. Among them two scientists namely Lummer and Pringsheim found that when a graph of energy density is plotted against wavelength, curves are obtained as shown in the figure. These curves are known as radiation curves or spectrum of black body radiation.
The following conclusions can be drawn from the radiation curves.
1) The energy is not uniformly distributed in the spectrum of black body radiation.
2) At a given temperature, energy density increases with wave length, becomes maximum for a particular wavelength and then decreases as wavelength increases.
3) As temperature increases, intense radiation represented by peak of the curve shifts towards shorter wavelength region.
Spectrum of Black Body Radiation or Radiation Curves
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Laws of black body radiation In order to explain the spectrum of black body radiation, number of laws have been put forward, notable among them are Stefan‘s law of Radiation, Rayleigh-Jeans Law of energy distribution, Wien‘s Law of energy distribution and Planck‘s Law of Radiation. Stefan’s law of radiation The Stefan‘s law states that energy radiated per second per unit area is directly
proportional to thefourth power of absolute temperature. E T 4, or E = T4
where is Stefan‘s constant, though this law is experimentally verified, it does not explain the energy distribution in the spectrum of black body radiation. Wien’s law of radiation
In the year 1893, Wien assumed that black body radiation in a cavity is supposed to be emitted by resonators of molecular dimensions having Maxwellian velocity distribution and applied law of kinetic theory of gases to
obtain formula for energy distribution as UdC1e–(C2/T)d, where Ud is
the energy /unit volume for wavelengths in the range, and dand C1 and C2 are constants. Drawbacks of Wien’s Law: This law explains the energy distribution only in shorter wavelengths & fails to explain the energy distribution in longer wavelength region. Also according to this law, when temperature is zero, energy density is finite. This is a contradiction to Stefan‘s law.
Lord Rayleigh –Jeans law of Radiation: Lord Rayleigh–Jeans considered the black body radiations full of electromagnetic waves of all wavelengths, between 0 and infinity, which due to reflection, form standing waves. They calculated number of possible waves
having wavelengths between and +d and by using law of equi-partition of
energy, they established distribution law as: Ud= 8kT-4dBecause of the
presence of the factor -4in the equation, the energy radiated by the black body should rapidly decrease with increasing wavelength.
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Drawbacks of Rayleigh–Jeans law: It is found that, Lord Rayleigh–Jeans law holds good only for longer wavelengths region and fails to explain energy distribution in shorter wavelength region, moreover; as per this law, as wavelength decreases, energy density increases enormously deviating from the experimental observations. The failure of the Rayleigh–Jeans law to explain the aspect of very little emission of radiation beyond the violet region towards the lower wavelength side of the spectrum is particularly referred to as Ultra-violet Catastrophe.
Planck’s Law of Radiation
In the year 1901, Max Planck of Germany put forward Quantum Theory of Radiation to explain Black Body Radiation spectrum. The following are the assumptions of Planck law of radiation.
1) The black body radiations in a cavity are composed of tiny oscillators having molecular dimensions, which can vibrate with all possible frequencies.
2) The frequency of radiations emitted by oscillators is same as the frequency of its vibrations.
3) An oscillator cannot emit energy in a continuous manner, but emission and absorption can take place only in terms of small packet of energy called Quanta, the oscillator can have only discrete energy values E given by nh ν
ν = Frequency of radiations, n = integer and ‗h‘ is Planck‘s constant, h= 6.625 x
10-34Js.
Planck using above assumptions derived a formula to explain black body radiation spectrum as,
[Since, ν=c/λ] ----------- (1)
This is called Planck’s radiation law and explains the entire spectrum of
black body radiation. From this law, we can also obtain Stefan‘s law, Wien‘s
law and Rayleigh-Jean law under suitable conditions.
1. Reduction of Planck’s radiation law to Wien’s law for shorter
wavelengths:
For shorter wavelengths, ν=c/λ is large,
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When ν is large, is very large
>>1.
( -1)≈
Making use of this in (1)
,
=C1-5 where C1=8πhc and C2=(hc/k).
This equation is Wien’s law of radiation.
2. Reduction of Planck’s radiation law to Rayleigh-Jeans law for longer
wavelengths:
For longer wavelengths, ν= c/λ is small,
When ν is small, hν/kt will be very small.
Expanding as power series, we have,
=1+ (hν/kt) + (hν/kt)2+…..
1+hν/kt [since hν/kt is very small, its higher power terms could be
neglected]
( -1) hν/kt = hc/λkt
Substituting in (1)
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This equation is Rayleigh-Jeans law of radiation.
Thus Wien‘s law and Rayleigh-Jeans law are special cases of Planck‘s law.
Rayleigh-Jeans Law
Planck’s law
Wien’s law
Ed
Energy distribution curves
Summary of Lesson –01
Here we have learnt that classical physics cannot explain black body radiation spectrum. The emission and absorption of energy takes place only in terms of quanta
and not continuously as predicted in classical physics. Quantum theory of radiation
has opened a new concept of understanding physics.
LESSON-2 Objectives:
At the end of lesson you shall understand that:
Light rays consists of invisible particles called photons.
A single electron in metal cannot absorb one photon of energy h.
Compton scattering is different from classical scattering.
Compton effect signifies particle nature of radiation.
Introduction
In this lesson, we will study Compton Effect which signifies particle nature of radiation, thereby strengthening the fact that radiation has dual characteristics.
COMPTON EFFECT In the year 1924, Compton discovered that when monochromatic beam of very high frequency radiation such as X-rays or Gamma rays is made to scatter through a substance, the scattered radiation found to contain two components; one having same frequency or wavelength as that of incident radiation, known as unmodified radiation; and the other, having lower frequency or longer wavelength than incident radiation known as modified radiation. This is called
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Compton scattering, during the process an electron recoils with certain velocity. This phenomenon is called Compton Effect. The Compton Effect is explained on the basis of Quantum theory of radiation, in which it is assumed that, radiation is composed of small packets of energy
called Quanta or photons having energy h. According to Compton, when a
photon of energy hof momentum h/ moving with velocity equal to velocity of light, obeying laws of conservation of energy and momentum, strikes an electron which is at rest, there occurs an elastic collision between two particles namely photon and electron.
When photon of energy h strikes the electron at rest, photon transfers some of its energy to electron, therefore photon loses its energy, hence, its frequency
reduces to 1 and wavelength changes to , the scattered photon makes an
angle with the incident direction, during the process an electron gains kinetic energy and recoils with certain velocity.
Compton by applying laws of conservation of energy and momentum showed
that, the change in wavelength is given by formula,
where mo = rest mass of electron.
The change in wavelength ' is called Compton shift. This shows that the
change in wavelength (Compton shift) depends neither on the incident
wavelength nor the scattering material, but depends only on the angle of
scattering.
Experimental Arrangement to Study Compton Effect
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h'
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The Experimental arrangement to study Compton Effect is, as shown in the
figure, a monochromatic beam of very high frequency such as X-rays of known
wavelength is made to fall on a scattering substance such as graphite. The
intensity of scattered X-rays for different angles of scattering is measured by
Bragg x-ray spectrometer, and then a graph of intensity versus angle of
scattering is plotted. When the angle of scattering is 90o, the Compton shift is
found to be 0.0243 Å. This value is in agreement with theoretical value obtained
from Compton formula.
Physical significance of Compton Effect
The phenomena of Compton effect is explained by Compton on the basis of
Quantum theory of radiation, in which it is assumed that radiation is composed
of small packets of energy called Quanta. The Compton Effect is an elastic
collision between two particles namely photon and electron in which exchange
of energy takes place as if it is a particle–particle collision. Also it is assumed
that photon and electron obey laws of conservation of energy and momentum.
Hence Compton Effect signifies particle nature of radiation.
Summary of Lesson -02
The Compton Effect signifies particle nature of radiation.
LESSON-3
Objectives
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At the end of this lesson we will learn that:
Any particle in motion exhibits wave like properties.
Matter waves are generated due to motion of the particle not by the charge carried by them.
Wave properties and particle properties do not appear together.
Dual nature of radiation has put the position of the particle uncertain.
Introduction:
In this lesson we shall study dual characteristics of matter waves and de-Broglie concept that all particles in motion exhibit wave properties and also de-Broglie equation.
Wave Particle Dualism and de-Broglie concept of Matter Waves
Before we discuss wave particle dualism, we must know the concept of particle and the concept of wave. The concept of particle is easy to understand, because it has mass and occupies certain fixed position in space and particle in motion has definite momentum; when slowed down, it gives out energy. Therefore particle is specified by its mass, momentum, energy and position.
The concept of wave is bit difficult to understand, because a wave is a disturbance spread over a large area. We cannot say wave is coming from here or going there. No mass is associated with wave and the wave is characterized by its wavelength, frequency, amplitude and phase.
Considering the above properties of particle and wave, it is difficult to accept the dual nature of radiation, but the acceptance is necessary because, the phenomenon like interference and diffraction has shown beyond doubt the wave nature of light radiation. And successfully explained by Huygens wave theory of light, however, experimental phenomenon like Photo electric effect, Compton Effect are successfully explained by Quantum theory of radiation, which signifies particle nature of radiation. Hence we can conclude that radiation has dual characteristics i.e. sometimes behaving like a wave and at other time as a particle, but radiation cannot exhibit both wave and particle properties simultaneously.
De-Broglie concept of matter waves
L. de-Broglie in the year 1924 put forward the concept of matter waves. According to this concept the dual characteristics of radiation is not confined only to electromagnetic waves, but also holds good for all material particles in motion i.e. all the particles like electrons, protons, neutrons, molecules, atoms etc. exhibit dual characteristics. His theory is based on the fact that nature
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loves symmetry that means when waves exhibits particle like properties then particle also should possess wave like properties. According to de-Broglie the particle in motion is associated with a group of waves and controlled by the wave. This wave is known as matter wave or de Broglie wave and wavelength associated with it is called de Broglie wavelength.
De-Broglie wavelength of a free particle
For a free particle, total energy is same as its kinetic energy given by,
E = ½ mv2
E = m2v2/2m (But p = mv)
E= p2/2m
Hence,
p = √2mE
By de Broglie hypothesis,
λ = h/p
Therefore, λ= h/√2mE = h/√2meV (since E = eV) where V is the
accelerating potential on an electron.
Substituting the constants, we get, λ = 12.27/√V Ǻ.
Characteristics of Matter Waves
• Matter waves are the waves associated with a moving particle.
• The lighter the particle larger the wavelength.
• Smaller the velocity of particle larger the wavelength.
• The amplitude of the matter wave at a given point determines the probability of finding the particle at that point at a given instant of time.
• The wavelength of a particle is given by, λ= h/p = h/mv
Summary of Lesson
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The material particle in motion exhibits wave like properties. The de - Broglie waves are pilot waves and are not electro-magnetic waves. Wave properties and particle properties cannot appear together.The dual nature of radiation has put position of particle uncertain.
Solved Examples
1. Calculate the momentum of the particle and de Broglie wave length associated with an electron with a KE of 1.5KeV.
Solution: Data p=?
λ = ?
K E =1.5x10³ eV
p² = 2mE
= 2x9.1x10‐³¹x1.5x10³x1.6x10‐¹⁹
= 2.08x10-²³ kgms-1
λ = h/p
= 6.625x10‐³⁴/2.08x10-²³
= 3.10x10‐¹¹ m.
2. Calculate the wave length of the wave associated with an electron of 1eV.
Solution:
λ= h/p
= h/2mE½
= 6.625x10-³⁴/2x9.1x10-31x1.6x10-¹⁹½
= 1.23x10-⁹ m
3. Find de Broglie wave length associated with a proton having velocity equal to 1/30th of that light. Given, mass of proton as1.67x10-²⁷kg.
Solution: v =1x3x10⁸/30 = 10⁷ m/s
λ =h/mv
= 6.625x10-³⁴/1.67x10-²⁷x10 ⁷
= 3.9x10-¹⁴ m/s
4. The velocity of an electron of a hydrogen atom in the ground state is 2.19x10⁶m/s. Calculate the wave length of the deBroglie waves associated with motion.
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Solution: λ = h/mv
= 6.625x10-³⁴ /9.11x10-31x2.19x10⁶
= 3.31x10-¹⁰ m
5. Estimate the potential difference through which a proton is needed to be accelerated so that its deBroglie wave length becomes equal to 1Å. Given that it‘s mass is 1.673x10-²⁷kg.
Solution: eV = 1/2 mv²
= p²/2m
= h²/2mλ²v² = h²/m²λ²
= h²/2meλ²
= 6.625x10-³⁴²/2x1.67x10-27x1.6x10-19x(10-¹⁰)²
= 0.082 V
6. Compare the energy of a photon with that of a neutron when both are associated with wave length of 1 Å. Given the mass of the neutron is 1.67x10-²⁷Kg.
Solution: E₁ = hν
= hc/λ₁
= 1.989x10-¹⁵ / 10-10x 1.6x10-¹⁹eV
=12411 eV
E₂ = h²/2mλ₂²
= 0.08 eV
E₁/E₂ =12411/0.08
= 1.5 x10⁵
7. Find the KE of an electron with de Broglie wave length of 0.2nm.
Solution: p = h/λ
= 6.625x10‐³⁴/0.2x10‐⁹
= 3.313x10‐²⁴ n-s
E = p²/2m
= (3.313x10‐²⁴)²/2x9.1x10‐³¹ = 37.69 eV
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QUANTUM MECHANICS
Over view This unit consists of five lessons, in first lesson we will study Heisenberg‘s uncertainty principle and its physical significance. In lesson two, we will study the applications of uncertainty principle and show that it is not possible for an electron to exist inside the nucleus. In lesson three, we will study wave function, its properties and physical significance and also we will study probability density and normalized wave functions, Eigen values, Eigen functions. In lesson four we will study Schrödinger matter wave equation and in the last lesson we will study particle in a box, energy values and wave functions.
Objectives
At the end of unit we would understand that:
In sub atomic world, it is impossible to determine precise values of two physical variables of particular pair which describes atomic system.
Both wave properties and particle properties are essential to get clear picture of atomic system.
Wave properties and particle properties are complimentary to one another.
In our daily life, we cannot realize quantum conditions. Particle in a box is a quantum mechanical problem and the probable
position of a particle can be estimated by evaluating the value of
ψ2. Quantum mechanics is an important tool to study atomic and sub
atomic state.
LESSON –1
Introduction In this lesson we will study uncertainty principle, its related equations derived from concept of wave packet and also we will study the physical significance of uncertainty principle.
Objectives
At end of lesson we understand that:
It is impossible to determine precise values of physical variables which describes atomic system. Hence we should always think of probabilities of estimating those values.
Both wave properties and particle properties of moving objects cannot appear together at the same time.
Wave properties and particle properties of moving objects are complimentary to one another.
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From uncertainty principle it is clear that, inaccuracy inherently present in its measurements.
HEISENBERG’S UNCERTAINITY PRINCIPLE
In the year 1927, Heisenberg proposed very interesting principle known as uncertainty principle, which is a direct consequence of dual nature of matter.
In the classical physics the moving particle has fixed position in space and definite momentum. If the initial values are known final values can be determined. However in Quantum Mechanics the moving particle is described by a wave packet. The particles should be inside wave packet, hence when wave packet is small; position of the particle may be fixed, but particle flies off rapidly due to very high velocity; hence, its momentum cannot be determined
accurately. When the wave packet is large, velocity or momentum may be determined but position of particle becomes uncertain.
In this way, certainty in position involves uncertainty in momentum and certainty in momentum involves uncertainty in position. Therefore, it is impossible to say where exactly the particle inside the wave packet is and what its exact momentum is.
According to uncertainty principle it is impossible to determine precisely and simultaneously, the exact values of both members of particular pair of physical variables which describes atomic system.
In any simultaneous determination of position and momentum of a particle, the product of corresponding uncertainties inherently present in the measurements is equal to or greater than h/4π
Δp.Δx ≥h/4π
These are the other uncertainty relations:
ΔE.Δt ≥ h/4π
ΔL.Δθ≥ h/4π
Δx = uncertainty in measurement of position
Δp = uncertainty in measurement of momentum
ΔE = uncertainty in measurement of energy
Δt = uncertainty in measurement of time
ΔL = uncertainty in measurement of angular momentum
Δθ = uncertainty in measurement of angular distance
Note: Heisenberg‘s uncertainty principle could also be expressed in terms of uncertainty involved in the measurements of physical variable pair like angular displacement (θ) and angular momentum (L).
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Summary
It is impossible to determine the values of both members of a particular pair of physical variables which describes atomic system. Hence we should always think of probability to estimate those values. However precise may be the method of measurement there is no escape from these uncertainties because it is an inherent limitation of nature on the measurement.
LESSON-2 Objectives:
At the end of lesson we shall understand that the electron cannot exist inside the nucleus of an atom and we can determine frequency of radiation emitted by atom and radius of electronic orbit and binding energy of electron.
Introduction In this lesson we will study applications of uncertainty principle, mainly to show that it is not possible for an electron to stay inside the nucleus of an atom.
Applications of Heisenberg’s Uncertainty Principle
Non-existence of electrons in nucleus of atoms Calculation of frequency of radiation emitted by atom Calculation of binding energy of an electron in an atom Determination of radius of Bohr electronic orbit
Here we will discuss first two important applications.
Non-existence of electrons in nucleus of atoms
The diameter of nucleus of atom is of the order 10-14 m. If an electron exists in nucleus of atom then maximum uncertainty in determining position of electron
must be 10-14 m.
( x) max =10- 14 m
From uncertainty principle (x) max (p) min = h/4
10- 14 (p) min = h/4
If an electron exists in nucleus then it should possess minimum momentum of
0.528 x 10-20 kg-m/sec particle of having this momentum must be moving with velocity equal to velocity of light, then it must be a relativistic problem.
Hence energy of the particle is given by E = mc2 or E= (mc) (c)
E = p c = (0.528 x 10-20 kg-m/sec) (3 x 108m ) J
E = 0.990346 x 107 eV
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E = 10 MeV
If an electron exists in nucleus of atom then it should have minimum energy of 10 MeV, but beta decay experiments has shown that energy possessed by beta particle from nucleus of an atom has maximum energy of 2 to 3 MeV. Hence we can conclude that it is not possible for an electron to exist inside the nucleus of an atom.
Physical significance of Heisenberg’s uncertainty principle
Non-existence of electrons inside the nucleus of atoms. Calculation of frequency of radiation emitted by an atom. Calculation of binding energy of an electron in an atom. Determination of radius of Bohr electronic orbit.
The wave and particle properties are complimentary to one another. It is impossible to determine precisely and simultaneously values of
physical variables which describes the atomic system.
Summary
The negatively charged particle electron cannot exist inside the nucleus. The wave and particle properties are complimentary to one another rather than contradictory.
LESSON-3
Objectives At end of lesson we shall understand that:
The wave function by itself has no physical significance The wave function is a complex quantity
The value of 2 evaluated at a point gives the probability of finding the particle at that point
Introduction In this lesson we will study the wave function and its characteristics, physical significance, probability densities, and normalization of a wave function.
Wave Function, Probability Density and Normalized Functions
The concept of wave function was introduced by Schrödinger in the matter wave
equation. It is denoted by , it is a variable whose variations constitutes matter wave. Wave Function is related to position of particle. The following are some characteristics of wave function.
1) The wave function by itself has no direct physical significance.
2) The wave function cannot be interpreted by an experiment.
3) The wave function is complex quantity consisting of both real and imaginary parts.
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4) With the knowledge of the wave function we can establish angular momentum, energy and position of particle.
5) The value of 2 evaluated at point gives the probability of finding particle at that point.
Properties of Wave function:
Property 1:is single valued everywhere.
MULTIVALUED FUNCTION
A function f(x) which is not single valued over a certain interval as shown in the above figure, has 3 values f1,f2,f3 for the same value of P at x=P. Since f1≠f2≠f3, it says that the probabilities of finding the particle have 3 different values at the same location. Hence such wave functions are not acceptable.
Property 2:is finite everywhere.
FUNCTION NOT FINITE AT A POINT
A function f(x) which is not finite at x=R as shown the above figure. At x=R, f(x)=infinity. Thus if f(x) were to be a wave function, it signifies large probability of finding the particle at a single location at x=R, which violates the uncertainty principle. Hence such wave functions are not acceptable.
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Property 3:and its first derivatives with respect to its variable are continuous everywhere.
Discontinuous function
A function which is discontinuous at Q as shown in the above figure, at x=Q, f(x) is truncated at A and restarts at B. Between A and B it is not defined and f(x) at Q cannot be ascertained. Hence such wave functions are not acceptable.
Property 4: For bound states, must vanish at infinity. If is a complex
function, then must vanish at infinity.
The wave functions that possess these four properties are named in quantum mechanics as Eigen functions.
Probability Density
The wave function is a complex quantity consisting of both real and
imaginary parts. Hence it can be expressed as follows:
ψ = a + ib where a and b are real functions of (x, y, z) and ‗t‘.
Complex conjugate of ψ is,
ψ* = a – ib
The product of ψ and ψ* is ψψ* = a 2 + b2, which is called
Probability density denoted by P = ׀ψ2׀,
Where ψ and ψ* are real and positive and also if ψ ≠ 0.
Normalized functions
The value of ׀ψ2׀ evaluated at point gives the probability of finding a
particle at that point, hence the probability of finding the particle in
an element of volume δv is given by:
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2δv׀ψ׀
Since the particle must be somewhere in space, the total
probability of finding the particle should be equal to 1 i.e
2δv = 1׀ψ׀
Any function which obeys this condition is said to be normalized
Wave function. Normalized wave functions should satisfy following
conditions:
1. It should be single valued function.
2. It should be finite everywhere.
3. It should be continuous and it should have continuous first derivative
ψ tends to zero when x, y, z tends to 0.
Eigen functions and Eigen values of energy
In Quantum mechanics, the state of a system is defined by its energy, position and momentum. These quantities can be obtained with the knowledge of wave function ψ.
Hence to define the state of a system we have to solve Schrödinger wave equation, but Schrödinger equation is a second order equation. It has several solutions, and only few of them are acceptable which gives physical meaning, these acceptable solutions are called proper functions or Eigen functions.
These are single valued, finite and continuous functions.
Eigen functions are used in Schrödinger equation to solve for energy of a system, since there can only be certain restricted Eigen functions and hence only few restricted values of energy, these values of energy is called Eigen values of energy.
Summary The wave function is a variable quantity, whose variations constitute matter waves.
The wave function is related to position of particle.
With the knowledge of wave function we can establish energy, angular momentum and position of particle.
Lesson 4
Objectives
At the end of the lesson we will understand that:
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The Schrödinger Wave Equation is useful in obtaining wave function, which is related to position of a particle. We will also understand that energy of free particle is not quantized.
Introduction
In this lesson we study, which is fundamental equation of quantum mechanics and free particle.
Schrödinger Time Independent Wave Equation
According to de Broglie concept of matter waves, a particle in motion is associated with group of waves called matter waves, the wavelength is given by λ = h/mv.
If the particle behaves like a wave then there should be some sort of wave equation which describe behavior of wave, and this equation is called Schrödinger Time Independent Wave Equation.
Consider a system of stationary waves, a particle of coordinates (x,y,z,) and wave function ψ. The wave equation of wave motion in positive x – direction is given by,
Ψ = Aei(kx–ωt) = …………………. 1
The time independent part is given by,
ψ = Aeikx…………………..2
Ψ = ψe–iωt…………………3
Let us differentiate Ψ twice with respect to x,
We get,
∂2 Ψ = e –iωt ∂2 ψ ………..4 ∂x2 ∂x2
Let us differentiate Ψ twice with respect to t,
∂2Ψ = - ω2 e –iωtψ ………5 ∂t2
We have the equation for a travelling as,
d2y = 1 d2 y dx2 v2 dt2 where y is the displacement and v is the velocity of the wave.
By analogy, we can write the motion of a free particle as,
d2Ψ = 1 d2 ψ ……………6 dx2 v2 dt2
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The above equation represents waves propagating along x–axis with a velocity v and Ψ is the displacement at the instant t. Substituting 4 and 5 in 6 we get,
d2Ψ = - ω2 ψ ……………7 dx2 v2 d2Ψ = - 4π2 ψ dx2 λ2
or 1/ λ2 = - 1 d2 ψ ……………8 4π2ψ dx2
We have KE = p2/2m
Put λ = h/p, KE = h2/2m. 1/λ2 ………….9
Substituting 7 in 9 we get,
KE = -h2/8π2m. 1/ ψ .d2 ψ /dx2……………10
E = KE + PE = -h2/8π2m. 1/ ψ .d2 ψ /dx2 + V
E – V = -h2/8π2m 1/ ψ d 2 ψ /dx2
d2 ψ /dx2 + 8π2m/h2 ( E – V) ψ = 0
This is Schrödinger time independent wave equation.
Summary
The Schrödinger matter wave equation is basic equation of quantum mechanics. And it is one of the important tools to study subatomic world. The energy of free particle is not quantized.
Lesson 5
Objectives
At the end of the lesson we shall learn that:
The energy levels for a particle in a box are quantized and hence cannot
have arbitrary values. The energy corresponding to n=1 is called ground state energy or zero
point energy and all other energy states are called excited states. The energy difference between successive levels is quite large. The electron cannot jump from one level to the other level on the
strength of thermal energy, hence quantization of energy plays important role in case of electron.
Introduction
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In this lesson we will study one of the important applications of Schrödinger wave equation, that is particle in a box of infinite depth and solve for Eigen values and Eigen functions. We will also study wave functions, probability densities and energy values of a particle in a box.
Particle in a box of infinite depth
Considered a particle of mass‘ ‗m‘ moving along ‗x‘ axis between two rigid walls of infinite length at x=0 and x= a. The particle is said to be moving inside potential well of infinite depth and potential inside box is zero and rises to infinity outside box.
i.e V=0 for 0 ≤ x ≤ a, V= ∞ for 0 ≥ x x ≥ a
Schrödinger wave equation for particle in a box
δ2ψ + 8π2m (E — V) ψ = 0 δx2 h2
but V=0 for 0 ≤ x ≤ a
δ2ψ + 8π2m( E ) ψ = 0 δx2 h2
or E ψ = - h2/ 8π2m . (δ2ψ/ δx2)
put, 8π2mE = K2 h2
δ2ψ + K2ψ = 0 δx2
The general solution for above equation is of the type,
Ψ(x) = A sinKx + B cosKx
‗A‘ and ‗B‘ are constants to be determined by applying suitable boundary conditions.
The particle cannot exist outside the box and cannot penetrate through walls, hence the wave function Ψ(x) must be zero for x=o and x= a.
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i) Applying boundary conditions Ψ(x)=0 at x = 0,
We get 0 = A sin K(0) + B cosK( 0)
0 = A (0) + B (1)
ie B = 0,
Substituting in above equation, Ψ (x) = A sin Kx
ii) Applying boundary conditions Ψ(x) = 0 at x = a,
0 = A sin Ka
Since, A ≠ 0, sin Ka = 0
Hence ka = 0, π, 2π, 3π, 4π, 5π, ------nπ
Ka = nπ
or K = nπ/a.
The wave function is Ψ(x) = A sin(nπ)x for n = 1,2,3,4,5,6,7,8---- a Energy 8π2 m E = K2 h2 Hence, 8π2mE = n2π2 h2 a2
E=n2h2
8ma2
From the above equation it is clear that particle in a box cannot have arbitrary value for its energy, but it can take values corresponding to n = 1,2,3,4, --- these values are called Eigen values of energy. EIGEN VALUES FOR ENERGY
When n = 1, E1 = h2 = Eo-is called ground state energy 8ma2
Eo or zero point energy or lowest permitted energy.
When n = 2, E2= 4h2 = 4Eo - this is first excited energy state 2ma2 When n = 3, E3= 9h2 = 9Eo - this is second excited energy state 8ma2 To evaluate A in Ψ (x) = A sin(nπ/a)x, We have to perform normalization of wave function. As the particle is inside the box at any time, we can write,
a
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2δx = 1׀ψ׀ ∫0
Substituting for ψ [from Ψ(x) = A sin (nπ/a)]
We get, a ∫ A2sin2 (nπ/a)x δx = 1 0 Solving we get, A = √2/a
Substituting in the equation for Ψ(x) = A sin(nπ/a)x
We get,
Ψ(x) = √2/a sin(nπ/a)x
ENERGY EIGEN FUNCTIONS
For n=1,
Ψ1 = √2/a sin (π/a) x. Here Ψ1= 0 for x = 0 & x = a. And maximum for x = a/2.
For n=2,
Ψ2 = √2/a sin (2π/a)x. Here Ψ2= 0 for x=0, a/2 and a.
And maximum for x=a/4 and 3a/4.
For n=3,
Ψ3 = √2/a sin(3π/a)x. Here Ψ3= 0 for x=0, a/3, 2 a/3 and a.
And maximum for x=a/6, a/2 and 5a/6. The plot of ׀ψ(x)2׀ versus x is as shown
in the figure below, for n=1,2 and 3.
Summary:
The particle in a box is quantum mechanical problem. The most probable position of a particle at different energy levels can be estimated by solving for its Eigen functions. The existence of zero point energy is in conformity with Heisenberg uncertainty principle.
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Solved problems
1. An electron has a speed of 300m/s accurate to 0.01% with what fundamental accuracy can we locate the position of the electron?
Solution: Given, v= 300m/s, ∆v=0.01% of v.
∆v = (0.01/100) x 300=0.03m/s.
We know that uncertainty relation is ∆x.∆p≥ h/4π
For the given uncertainty in speed, ∆p is minimum
Uncertainty in the position is given by,
∆x= h/4π∆p= h/4πm∆v
= 6.632x10-34 / 4x 3.14x 9.11x10-31x3x10-2
=1.93x10-3m/s
2. An electron of energy 20 eV is passed through a circular hole of radius10-6 m. what is the uncertainty introduced in the angle of emergence?
Solution: Given E= 250eV= 250x1.6x10-19J
r=10-6m
∆x=2x10-6
We know that energy E=p2/2m and
p=√2mE
=√(2x9.11x10-31x40x10-18)= 8.853x10-24Kgm/s.
∆p= h/4π∆x
=6.632x10-34 / 4x 3.14x 2x10-6 = 0.263x10-28Kgm/s
Angle of emergence, =∆p/p
=0.263x10-28 /8.853x10-24
=0.0309x10-4radian
3.The average time an atom retains excess excitation energy before re-emitting it in the form of electromagnetic radiation is 10-8 sec. calculate the limit of accuracy with which the excitation energy of the emitted radiation can be determined?
Solution: Given ∆t=10-8 sec
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According to uncertainty principle, ∆E.∆t= h/4π
∆E = h/4π∆t
=6.632x10-34 / 4x 3.14x 10-8
= 0.5x10-26 J
=0.5x10-26/1.6x10-19eV
∆E=0.3x10-7eV.
4. Using Heisenberg uncertainty relation, calculate the kinetic energy of an electron in a hydrogen atom?
Solution: The uncertainty in the coordinate of an electron inside the atom is equal to the radius of the atom. The Bohr radius, r= 0.053nm,is the reasonable estimate for the uncertainty in position ∆p.
We know from Heisenberg uncertainty principle.
∆x. ∆p=h/4π
∆p= h/4π∆x
But ∆x=r (Bohr radius)
Now energy, E=p2/2m= h2/16π2r2 2m
=(6.632x10-34)2/ (4x 3.14x0.053x 10-9)2x2x9.1x10-31
=5.45x10-19 J = 3.4 eV
5. An electron is constrained in a 1-dimensional box of side 1nm. Calculate the first Eigen values in electron volt.
Solution: Given a=1nm
The Eigen values are given by
En=n2h2/8ma2
The first Eigen value given by
E1= (6.632x10-34)2 / 8x9.11x10-31x (1x 10-9)2J
=0.377eV
E1=0.377eV
Second Eigen value, E2= 22x 0.377= 1.508 eV
Third Eigen value, E2= 32x 0.377= 3.393 eV
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Fourth Eigen value, E2= 42x 0.377= 6.5032 eV
6. Is it possible to observe energy states for a ball of mass 10 grams moving in a box of length 10cm.
Solution: The Energy is given by
En=n2h2/8ma2
E1= n2(6.632x10-34)2 / 8x9.11x10-31x (0.1)2
=38x10-18 n2
When n=1, 2 , 3etc energies are 38x10-18 eV, 152.6x10-18eV,
343.35 x 10-18eV.
The energies states are so near that they appear as continuous.
7.A spectral line of wavelength 5461 Å has a width of 10-4 Å. Evaluate the minimum time spent by the electrons in the upper energy state between the excitation and de-excitation processes?
Solution: Wavelength of the spectral line, λ=5461x10-10m
Width of the spectral line ∆ λ=10-14m
Minimum time spent by the electron, ∆t=?
We have the equation, Е = hν= hc/ λ ∆Е =hc ∆(1/λ) ∆Е =hc (∆λ/λ2) ------ (1) As per uncertainty principle, ∆E ∆t ≥ h/4π
∆t ≥ h/4π∆E ------ (2)
From (1) the right hand side of (2) can be written as
h/4π∆E = h λ2/4π (hc ∆λ)= λ2/4πc∆λ
= (5461 x 10-10)2/4π x 3 x108 x10-14
=0.8X10-8s
From equation (2), we have ∆t = 0.8X10-8 second.
Therefore the minimum time spent by the electron is ≥ 0.8X10-8 second.
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UNIT-II LESSON I
Objectives At the end of this lesson we shall understand that valence electrons in metal are treated as molecules of gas. The free electrons in metal move from atom to atom among positive ion cores. The flow of current in metal is the consequence of drift velocity. Introduction In this lesson, we will study about free electron theory proposed by Drude to explain some of the outstanding properties of metals and drift velocity and its significance.
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ELECTRICAL CONDUCTIVITY
3.1 Drude – Lorentz’s Free Electron Theory of metals: [Classical Free Electron Theory] In order to explain some of the outstanding properties of metals like very high electrical, thermal conductivities in the year 1900; Drude proposed free electron theory of metals. According to this theory valence electrons in metals are free to move inside the metal among positive ionic cores, these electrons are called conduction electrons and they are responsible for most of the properties of metals. It is assumed that the metal consists of large number of atoms, which are held together, thus valence electrons move from atom to atom throughout the metal. When an electron leaves the parent atom, the atom losses its electrical neutrality, thus it is ionized. These ions are localized in the metal and the structural formation of these ions in three dimensional arrays is known as lattice. The positive ions are fixed in the lattice frame but constantly vibrating about their mean positions due to thermal agitations. The thermal agitation is responsible for random motion of electrons in the metal, hence free electrons in metal is treated as molecules of gas called electron gas.
When an electric field is applied, the negatively charged electrons moves along the direction of positive field and hence produce current in the metal. In order to prevent the electrons accelerating indefinitely it is assumed that they collide elastically with metal ions, this leads to steady current, which is directly proportional to applied voltage and this explains ohm‘s law. In the year 1909 Lorentz applied Maxwell – Boltzmann distribution law to electron gas to explain free electron theory of metals & hence it is known as Drude –Lorentz theory. The assumptions of free electron theory of metals: 1. The free electrons in metal is treated as molecules of gas, hence it is assumed that they obey laws of kinetic theory of gases and in the absence of applied electric field energy associated with each electron. At temperature T is 3/2 kBT,
which is equal to kinetic energy of electron 3/2 kBT = 1/2mv2, vth= thermal velocity or r.m.s velocity 2. The field due to positive ion is constant and force of repulsion between electrons is very small hence it is neglected. 3. The flow of current in metal is due to applied field is consequence of drift velocity. Drift velocity The free electrons in metal are random in motion in the absence of electric field. However when electric field is applied, even though randomness persist but overall there is a shift in its position opposite to applied field. The electrons acquire kinetic energy and collide with ions, loses energy and this process continues until electrons acquire constant average velocity opposite to applied field. This velocity is called drift velocity. When electric field E is applied, electrons acquires constant average velocity Vd
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If ‗m‘ is mass of electron and ‗e‘ is charge carried by electron then the resistive force Fr
Acting on electron is equal to m a = dmV
τ
The driving force = - eE
Therefore dmV
τ = - e E τ = mean collision time
τ
d
eV
m
Summary:
The free electrons in metal are responsible for most of the properties exhibited
by metals
In metal the absence of electric field electrons are random in motion
The flow of current in metal is due to drift velocity
LESSON-II Objectives At the end of this lesson we shall understand that:
The current flowing through it is directly proportional to drift velocity The current density is directly proportional to electric field
Introduction: In this lesson we will study some important definitions like relaxation time, mean free path and mean collision time and expression for current in a conductor.
Relaxation time:
“The time required for the average vector velocity to decrease to 1/e
times its initial value when the field is just turned off is called the
relaxation time”.
Explanation:
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In a metal due to the randomness in the direction of motion of the conduction
electrons, the probability of finding an electron moving in any given direction is
equal to finding some other electron moving in exactly the opposite direction in
the absence of an electric field. As a result the average velocity of the electrons
in any given direction becomes zero.
i.e. vav = v'av , in the absence of the field
However, when the metal is subjected to an external electric field, there will be
a net positive value v'av for the average velocity of the conduction of the
electrons in the direction of the applied field due to the drift velocity.
i.e. vav = v'av , in the presence of the field
If the field is turned off suddenly, the average velocity vav reduces
exponentially to zero from the value v'av which is also the value of vav just when
the field is just turned off.
The decay process is represented by the equation (figure 2)
vav = v'av e-t /τr
----------------(1)
Where t: time counted from the instant the field is turned off.
τr :constant called relaxation time
Figure: Decay of average velocity
In equation (1) if t = τr , then
1' evv avav
33
avav ve
v '1
Therefore the relaxation time can be defined as:
“Consequent to the sudden disappearance of an electric field, across a
metal, the average velocity of its conduction electrons decays
exponentially to zero & the time required in this process for the average
velocity to reduce to 1/e times its value just when the field is turned off is
known as relaxation time”.
Mean free path (λ)
According to kinetic theory of gases, the mean free path is the distance traveled
by a gas molecule in between two successive collisions. In the classical theory of
free electron model, it is taken as the average distance traveled by the
conduction electrons between successive collisions, with the lattice ions.
Mean collision time (τ)
“The average time that elapses between two consecutive collisions of an
electron with the lattice points is called mean collision time”. The
averaging is taken over a large number of collisions.
If ‗v‘ is the velocity of the electrons which is the velocity due to the combined
effect of thermal & drift velocities, then the mean collision time is given by,
v
where λ is mean free path.
If vd is the drift velocity, then vd = vth
→ v ≈ vth
In the case of metals, it can be shown that the relaxation time ‗t‘ always refer to
a single total velocity called Fermi velocity.
Mean free path It is the average distance travelled by conduction electron between successive collisions.
Mean free path is about 10-7 m.
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Electric Field (E)
The electric field is defined as potential per unit length of homogeneous
conductor of uniform cross-section. i.e L
VE volts/m
If ‗L‘ is the length of a conductor of uniform cross – section & of uniform
material composition & ‗V‘ is the potential difference between its two ends, then
the electric field ‗E‘ at any point it is given by
L
VE ---------------------- (2)
Conductivity (σ):
It is the physical property that characterizes the conducting ability of a
material.
If ‗R‘ is the uniform resistance of uniform material of length ‗L‘ & area of
cross – section ‗A‘ then the electrical conductivity is given by
A
L
R
1 --------------------(3)
If we consider the product σ E, then from (2)
L
V
A
L
RE
1
RA
VE
Since I = V/R, we have
A
IE --------------------- (4)
Comparing (1) & (4) we get
EJ ----------------------- (5)
This represents the Ohm‘s law.
Resistivity (ρ):
Resistivity signifies the resistance property of the material & is given by the inverse of conductivity.
i.e
1
---------------------- (6)
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Expression for current in a conductor:
Figure: Current carrying conductor Consider a conductor of uniform area of cross-section ‗A‘, carrying a
current ‗I‘ as shown in figure (5). If ‗v‘ is the velocity of the electrons, then the
length traversed by the electron in unit time is ‗v‘.
Consider in the conductor, an imaginary plane at ‗X‘ normal to the
current‘s direction. If we consider the motion of electrons as a group starting
from ‗X‘, then they sweep a volume ‗vA‘ of the conductor in unit time as
indicated in the figure.
Let ‗n‘ be the number of electrons/unit volume. Therefore
the number of electrons in a volume )(vAnvA
In other words, the number of electrons crossing any cross-section in unit time
= )(vAn
If ‗e‘ is the charge on each electron crossing any section per second is the same
as rate of flow of charge. Therefore
the rate of flow of charge = nevA --------------------(1)
Since the velocity acquired by the electron is due to an applied electric field, it is same
as drift velocity dv i.e dvv
AvenI d -------------------- (2)
Expression for electrical conductivity:
Consider the motion of an electron, in a conductor subjected to the influence of
an electric field. If ‗e‘ is the charge on the electron and ‗E‘ is the strength of the
applied field, then the force ‗F‘ on the electron is
EeF ---------------------- (1)
36
If ‗m‘ is the mass of an electron then as per Newton‘s second law of motion force
on the electron can be written as
dt
dvmF ----------------------(2)
Equating (1) & (2)
dt
dvmEe
dtm
eEdv
Integrating both sides,
dtm
eEdv
t
0
tm
eEv ---------------------(3)
where t: time of traverse
Let the time of traverse ‗t‘ be taken equal to the collision time ‗τ‘. Since by
definition the collision time applies to an average value, the corresponding
velocity in (3) also becomes the average velocity v.
m
eEv
We have the expression for electrical conductivity ‗σ‘ as
E
J --------------------- (5)
Where ‗J‘ is the current density & J = I /A, where ‗I‘ is the in the conductor &
‗A‘ is the area of cross-section of the conductor.
AE
I --------------------- (6)
Now the distance covered by electrons in a unit time is v. They sweep a volume
equal to vA (figure) in a unit time. If ‗e‘ is the charge on the electron, ‗n‘ is the
37
number of electrons per unit volume, then the quantity of charge crossing a
given point in the conductor per unit area per unit time i.e the current ‗I‘ is
I = nevA -------------------(7)
Substituting for ‗I‘ from (7) in (6) we get
EA
Aven
E
ven -------------------- (8)
Substituting for ‗v‘ from (4) in (8) we get
m
Ee
E
en
m
en
2
----------------------(9)
Equation (9) is the expression for electrical conductivity of a conductor.
Mobility of Electrons
The mobility of electrons is defined as the magnitude of the drift velocity
acquired by the electrons in unit field. Thus if ‗E‘ is the applied electric field, in
which the electrons acquire a drift velocity vd , then the mobility of electrons
E
vd --------------------(1)
We have for Ohm‘s law,
EJ
Hence,
E
J
enEA
ven d
38
en
---------------------- (2)
We know that,
m
en
2
Therefore (2) becomes,
enm
en 12
Hence, m
e --------------------- (3)
Mobility represents the ease with which the electrons could drift in the
material, under the influence of an electric field. Different materials have
different values for mobility.
Failure of classical free electron theory
Though the classical free electron theory has been successful in accounting for
certain important experimental facts such as electrical and thermal
conductivities in metals: it failed to account for many other experimental facts
among which the notable ones are specific heat, temperature dependence of σ
and the temperature dependence of electrical conductivity on electron
concentration.
(1) Specific Heat:
The molar specific heat of a gas at constant volume is
R2
3Cv
As per the classical free electron theory, free electrons in a metal are
expected to behave just as gas molecules. Thus the above equation holds good
equally well for the free electrons also. But experimentally it was found that, the
contribution to the specific heat of a metal by its conducting electrons was
smaller than the classical value (3/2)R by a factor of about 10-2 .
(2) Temperature dependence of electrical conductivity:
39
It has been observed that for metals the electrical conductivity ‗σ‘ is
inversely proportional to the temperature ‗T‘.
i.e T
1exp -----------------(1)
But according to the main assumptions of classical free electron theory
th2vm
2
1TK
2
3
m
KT3v th
2
Tv th
Since the mean collision time τ is inversely proportional to the thermal velocity,
we can write,
thv
1
Or
T
1 ------------------- (2)
But σ is given by,
m
en
2
Therefore the proportionality constant between σ and τ can be represented as
or
T
1 -------------------- (3) [From proportionality (2)]
Now from the proportionality representations (1) & (3), it is clear that the
prediction of classical free electron theory is not agreeing with the experimental
observations.
3. Dependence of electrical conductivity on electron concentration:
40
As per the classical free electron theory, the electrical conductivity ‗σ‘ is
given by,
m
en
2
where n: concentration of the electrons, therefore
σ α n
If we consider the specific cases of Zinc and Cadmium which are divalent
metals, the electrical conductivities are respectively 1.09X 107 / Ω m & 0.15 X
107 / Ω m which are much lesser than that for Copper and Silver, the values for
which are 5.88 X 107 / Ω m & 6.3 X 107 / Ω m respectively. On the other hand,
the electron concentration for Zinc and Cadmium are13.10 X 1028 /m3 & 9.28
X 1028 /m3 which are much higher than that for Copper and Silver, the
values for which are 8.45 X 1028 /m3 and 5.85 X 1028 /m3 respectively.
LESSON-3
Objectives:
At the end of this lesson we shall understand that: All the free electrons cannot receive energy The electrons are not completely free as assumed in classical theory The free electrons obey Pauli‘s exclusion principle The energy levels of an electron in a metal are quantized
Introduction:
In this lesson, we will study Quantum free electron theory of metals, Fermi –
Dirac distribution function and Fermi energy and Fermi factor,
41
Quantum Free Electron Theory of Metals One of the main difficulties of classical free electron theory of metals is that, it allows all the free electrons to gain energy (as per Maxwell-Boltzmann statistics). Hence value obtained are much higher than experimental values After development of quantum statistics, it is realized that only one percent of free electrons can thus absorb energy . This brings importance of Pauli‘s exclusion principle. In the year 1928 Arnold Summerfeld applied quantum mechanical conditions and Pauli‘s exclusion principle to explain failures of classical free electron theory of metals. The following are some of the assumptions of quantum free electron theory: 1) The energy levels of free electrons in metals are quantized
2) The free electrons obey Pauli Exclusion Principle
3) The electrons travel in side metal with constant velocity, but they are confined within boundaries
4) The distribution of electrons are among various levels as per Fermi-Dirac statistics.
5) The force of attraction between electrons and positive ionic lattice also force of repulsion among electrons is neglected
Fermi–Dirac Statistics According to Sommerfeld the electrons are not completely free in the metal as predicted in the free electron theory, i.e they are partially free and bound to the metal as a whole, hence electrons in metal can ot be compared to gas molecules; therefore, we cannot apply Maxwell-Boltzmann statistics. Moreover electrons are assumed to obey Pauli‘s exclusion principle hence they are governed by Fermi –Dirac statistics .
The electrons obeying Pauli‘s exclusion principle are identical and indistinguishable particles called ‗Fermions. The Fermi–Dirac distribution function gives most probable distribution of electrons. Hence in equilibrium at a temperature T, the probability that an electron has an energy E is given Fermi function f (E).
E is energy level whose occupancy is being considered, EF = Fermi level, it is
constant for a particular metal.
At absolute zero f(E) = 0 for E >EF and f (E) =1 for E< E F
The Fermi level is highest state for the electrons to occupy at absolute zero, that mean at absolute zero Fermi level divides the occupied states from the unoccupied states.
42
Fermi Energy
In quantum free electron theory, the energy of electron in metal is quantized, therefore according to quantization rules, if there are N numbers of electrons, then there must be N number of allowed energy levels, since these electrons obey Pauli‘s exclusion principle.
An energy level can accommodate at most only two electrons, one with spin up and other with spin down, thus while filling energy levels, two electrons occupy the least level, two more next level and so forth, until all electrons are accommodated as shown in figure.
The energy of the highest occupied level at absolute zero temperature is called Fermi Energy and the corresponding energy level is called Fermi Level. The Fermi energy can also be defined as maximum kinetic energy possessed by free
electron at absolute zero temperature, it is denoted by EF.
At absolute zero temperature, metal does not receive energy from surroundings, therefore all the energy levels below Fermi level is completely filled up and above the Fermi level all the energy levels are empty, If there are N electrons in the metal then highest occupied level is N/2 this level is called Fermi level and corresponding energy is called Fermi energy.
Fermi Factor When the temperature is greater than the 0 K, metal receives thermal energy from the surroundings; however, at room temperature thermal energy received by metal is very small (kBT = 0.025 eV, kB = Boltzmann‘s constant), hence the electrons in the energy levels far below Fermi level cannot absorb this energy because there are no vacant energy levels above them, however the electrons just below Fermi level absorb this energy and may move to unoccupied energy levels above Fermi level, though these excitations seems to be random, the occupation of various energy levels takes place strictly as per Fermi –Dirac distribution law.
Fermi function, f (E) = 1/(1+e(E-E
F)/kT
), where f (E) is the probability of an electron
occupying energy state E.
43
(i) For T = 0K and E > EF
f (E) = 1/(1+e∞) = 1/∞ = 0. ie no electron can have energy greater than EF at
0K.
(ii) For T = 0K and E < EF
f (E) = 1/(1+e-∞
) = 1/1+0 = 1. ie all electrons occupy energy states below EF
at 0K.
(iii)For T > 0K and E = EF
f (E) = 1/(1+e0) = 1/1+1 = ½. ie 50% electrons can occupy energy states
below EF
above 0K.
Fermi level is defined as energy level at which the probability of electron
occupation is one half or 50 %.
Summary of Lesson Here we learnt that, in metal, free electrons are partially free, because they are bound to the metal as a whole, hence they cannot be compared to molecules of gas.
In metal there is extremely larger number of energy levels. The distribution of electrons among various energy levels is strictly as per Fermi -Dirac function. At 0 K all the energy levels below Fermi level is occupied and above Fermi level, energy levels are empty.
LESSON 4
Objectives At the end of this lesson we shall understand that:
The free electrons in metal can be treated as particles in a box The Fermi temperature is only theoretical concept
The total energy of free electron is 3/5 EF
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Introduction
In this lesson, we will study number of available energy states in the range E
and E + dE, Number of electrons per unit volume, Fermi energy, Fermi
temperature and Fermi velocity.
Density of states
The electron energy levels in a material are in terms of bands. The number of levels in each band is extremely large and these energy levels are not evenly distributed in the band. At the highest energy the difference between
neighboring levels is of the order of 10-6 eV. That means in a small energy interval dE there are still many discrete energy levels. Hence for easy calculations we introduce the concept of ‗Density of States‗. It is denoted by g (E). The density of state can be defined as follows:
‗It is the number of available states per unit volume per unit energy range‘.
Number of available states per unit volume between energy range,
E and E+dE= g (E) dE.
Summary Here we learnt that, in metal, free electrons are partially free, because they are bound to the metal as a whole, hence they cannot be compared to molecules of gas. In metal there is extremely larger number of energy levels. The distribution of electrons among various energy levels is strictly as per Fermi -Dirac function. At 0 K, all the energy levels below Fermi level are occupied and above Fermi level, energy levels are empty.
LESSON-5
Objectives At the end of this lesson we shall understand that:
The electron moving in metal under influence of external field possesses effective mass. The effective mass varies from solid to solid and it is a function of energy.
The drift velocity of free electrons in metal is equal to Fermi velocity.
Introduction In this lesson we study merits of Quantum free electron theory.
Concept of effective mass According to Sommerfeld, Quantum free electron theory of metals, the motion of free electrons in metals is considered not as motion of particles, but as passage of waves among periodic lattice. Hence the motion of electrons in metal can be treated as a wave packet. Hence velocity of electron is treated as group
45
velocity. When field is ,applied to an electron; the wave packet travels under combined action of applied field and potential due to periodic lattice and due to this superposition of these fields the electron responds as if it posses effective mass ; this mass is different from its true mass with which it moves under the influence of external field alone. The effective mass is interpreted in terms of true mass. The Concept of Effective Mass shows that it is possible to deal with the motion of electrons in metal as semi classical manner. In vacuum the effective mass of electron is same as true mass and the Effective Mass varies from solid to solid.
Electrical Resistivity or Conductivity According to Sommerfeld, Quantum free electron theory of metals the electrons are partially free not completely as assumed in classical free electron theory. Hence free electrons in metals obey Fermi –Dirac statistics. By applying Boltzmann transport equation and Fermi –Dirac statistics he got the equation for electrical conductivity of metals as,
In classical free electron theory it is assumed that Electrical Resistivity in metals is due to scattering of electrons and the scattering of electrons takes place due to lattice defects, dislocations, impurities etc, but according to Summerfield the motion of electrons in metals nothing but passage of waves in periodic lattice, if there is perfect periodicity and all ions are at rest, then the waves pass across the arrays without being scattering at all in such case mean free path is infinite, but no metal is free from impurities or lattice defects, that mean there is always deviations from periodicity due to this, scattering of electron waves takes place, therefore lattice defects becomes major cause of electrical resistivity in metals and scattering of electron waves becomes the deciding factor for the mean free path of electrons.
Merits of quantum free electron theory
(1) Specific Heat : According to quantum free electron theory, it is only those electrons that are occupying energy levels close to EF which are capable of absorbing the heat energy to get excited to higher energy levels. Thus only a small percentage of the conduction electrons are capable of receiving the thermal energy input and hence the specific heat value becomes very small for the metals.
Therefore on the basis of quantum free electron theory
TRE
kC
F
BV
2
Taking a typical value of EF = 5eV we get
410
2
F
B
E
k
46
CV = 10-4 RT
This agrees with the experimental values. Since CV is very small, the energy of
electrons is virtually independent of temperature.
(2) Temperature dependence of electrical conductivity:
The experimentally observed fact that electrical conductivity ‗σ‘ has a
dependence on
T
1but not on
T
1 can be explained as follows.
The expression for electrical conductivity is given by
m
en F2
As per quantum free electron theory, F
FF
v
F
F
vm
en
2
---------------- (1)
As per quantum free electron theory EF and vF are essentially independent of
temperature. But λ F is dependent on temperature, which is explained as
follows.
As the conduction electrons traverse in the metal, they are subjected to
scattering by the vibrating ions of the lattice. The vibrations occur such that the
displacement of ions takes place equally in all directions. If ‗a‘ is the amplitude
of vibrations, then the ions can be considered to present effectively a circular
cross- section of area Πa2 that blocks the path of the electrons irrespective of
the direction of approach. Since vibrations of larger area of cross-section should
scatter more efficiently, it results in a reduction in the value of mean free path
of the electrons,
2
1
aF
------------------ (2)
Considering the facts that,
(a) the energy of a vibrating body is proportional to the square of the amplitude
(b) the energy of ions is due to thermal energy (c) thermal energy is proportional to the temperature (T).
47
Therefore we can write,
Ta 2
TF
1
------------------ (3)
From (1) & (3) we get,
T
1
Thus the dependence of ‗σ‘ on ‗T‘ is correctly explained by the quantum free
electron theory.
(3) Electrical conductivity and electron concentration:
By classical free electron theory, it was not possible to understand why metals
such as Al and Ga which have 3 free electrons per atom have lower electrical
conductivity than metals such as copper and silver which possess only one free
electron per atom. But according to quantum free electron theory the same can
be explained. We have the equation for electrical conductivity as:
F
F
vm
en
2
From this equation it is clear that, the value of σ depends on both ‗n‘ and the
ratio F
F
v
. If we compare the cases of copper and aluminum, the value of ‗n‘
for Al is 2.13 times higher than that of copper. But the value of F
F
v
for copper
is about 3.73 times higher than that of Al. Thus, the conductivity of copper
exceeds that of aluminum.
Comparison between Classical free electron theory and quantum free electron
theory
Similarities:
1) The valence electrons are treated as though they constitute an ideal gas.
2) The valence electrons can move easily throughout the body of the solid.
48
3) The mutual repulsion between the electrons and the force of attraction
between electrons and ions are considered insignificant.
Differences between the two theories:
Sl.no Classical free electron theory
Quantum free electron theory
1. The free electrons which constitute the electron gas can have continuous energy values.
The energy values of the free electrons are discontinuous because of which their energy levels are discrete.
2. It is possible that many
electrons may possess same energy.
The free electrons obey the ‗Pauli‘s
exclusion principle‘. Hence no two electrons can possess same energy.
3. The patterns of distribution of energy among the free electrons obey Maxwell-Boltzmann statistics.
The distribution of energy among the free electrons is according to Fermi-Dirac statistics, which imposes a severe restriction on the possible ways in which the electrons absorbs energy from an external source.
Solved Problems
1. Calculate the drift velocity and thermal velocity of free electrons in copper at room temperature, (300 k), when a copper wire of lengths 3 m and resistance 0.022 carries of 15 A.
Given: = 4.3 × 10-3 m2 /Vs.
Solution: Given that L= 3 m, R =0.022, I= 15 A, T = 300 k, cu = 4.3 × 10-3 m2 /Vs. Vd =? and V th
= ?
Voltage drop across the copper wire is given by
V= IR = 15 × 0.022 = 0.33 V
Electric Field, E = V / L = 0.33 /3 = 0.11 V /m
Drift velocity, Vd = E × = 0.11 ×4.3 × 10-3 = 0.473 × 10-3 m/s.
Thermal Velocity, Vth = 3kT / m
= 3 × 1.387 × 10-23 ×300 / 9.11×10-31
= 1.17 ×105 m/s.
49
2. Find the relaxation time of conduction electrons in a metal of resistivity
1.54 x 10-8 ohm-m, if the metal has 5.8 x 1028 electrons /m3
Given ρ = 1.54 x 10-8 ohm-m n = 5.8 x 10-8 electrons /m3
Resistivity of metal = ρ = =
Relaxation time = ρ = 3.97 x 10-14 s
CONDUCTIVITY IN SEMICONDUCTORS
LESSON-6
Objectives:
To study the semiconductor energy level diagram
To derive an expression for hole and electron concentration.
To observe the Hall Effect in a semiconductor.
Based on the electrical conductivity of the materials they can be classified into three categories. Conductors- conductors are the materials that allow the electricity to pass through them. Eg: aluminum, copper, silver, etc. Insulators- insulators are the materials that do not allow the electricity to pass through them. Eg: paper, glass, etc.
Semiconductors- semiconductors are materials whose electrical conductivity lies between that of conductors and insulators. Eg: silicon and germanium.
Conductivity in semiconductors:
Atoms of silicon and germanium have four electrons in their outer most shell. These electrons form covalent bond with the neighbouring atom and not free at low temperature. Hence they behave like insulators. However when a small amount of thermal energy is available from the surroundings a few covalent bonds are broken and few electrons are set free to move. Even at room
temperature good number of electrons is dissociated from their atoms and this number increases with rise in temperature. This leads to conductivity. When an electron is detached from the covalent bond, it leaves a vacancy which behaves like a positive charge. An electron from a neighbouring atom can move onto this vacancy leaving a neighbor with a vacancy. Such a vacancy is called a hole. Hole acts as a positive charge.
Types of semiconductors:
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In a semiconductor there are two kinds of current carriers- Electrons and Hole. In a pure semiconductor electrons and holes are always present in equal numbers and it is called intrinsic semiconductor.
Conductivity of the semiconductors can be changed by adding small amount of impurities (other elements) to it. These impurities are called dopants. Such semiconductors are impure or extrinsic semiconductors. When a few atoms of trivalent or pentavalent element is added into pure crystals of Ge or Si an extrinsic semiconductors are produced. The process of adding impurity atoms is called Doping. When pentavalent impurity atoms like arsenic, antimony, phosphorous, etc are added to pure germanium or silicon crystal, we get an extrinsic semiconductor known as n-type semiconductor. When trivalent impurity atoms like indium, boron, gallium, aluminum, etc are added to pure germanium or silicon crystal, we get an extrinsic semiconductor known as p-type semiconductor.
Concentration of electrons and holes in intrinsic semiconductors:
With an increase in temperature covalent bonds are broken in an intrinsic semiconductor and electron- hole pairs are generated. We expect that a large number of electrons can be found in the conduction band and similarly, a large number of holes in the valance bond. As electrons and holes are charged particles, they are together called charge carriers. Charge carriers are the number of electrons in the conduction band per unit volume (n) and number of holes in the valence band per unit volume (p) of the material. Carrier concentration is known as the density of charge carriers.
Expression for concentration of electrons in intrinsic semiconductor:
Let dn be the number of electrons whose energy lies in the energy interval E
and E+dE in the conduction band. Then,
dn = g(E)f(E)dE
where g(E)dE is the density of states in the energy interval E and E+dE and f(E)
probability that a state of energy is occupied by an electron.
The electron density in the conduction band is given by integrating the
above equation between the limits EC and ∞. EC → energy corresponding to
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bottom edge of the conduction band and ∞→ energy corresponding to the top
edge of the conduction band. Since f(E) of the electrons occupying the upper
levels of conduction band rapidly approaches 0 for higher energies, the upper
limit is taken as ∞. Then,
The density of states in the conduction band is given by,
EC corresponds to the potential energy of the electron at rest. Thus (E-EC) will be
the kinetic energy of the conduction electron at higher energy levels. Thus,
The probability of electron occupying an energy level is given by,
When the number of particles is very small compared to the available
energy levels, f(E) occupied by more than one electron is very small thus it can
be written as
Using the value of g(E) and f(E) we can write
or
This integral is of the standard form which has a solution of the form,
52
where a=1/kBT and x=(E - EC)
Rearranging,
This is the expression for the electron concentration in the conduction
band of an intrinsic semiconductor. Designating to be the effective density of
states in the conduction band as,
The electron concentration in the conduction band of an intrinsic
semiconductor can be written as,
Expression for concentration of holes in intrinsic semiconductor:
Let dp be the number of holes whose energy lies in the energy interval E and
E+dE in the valence band. Then,
dn = g(E)[1- f(E)]dE
where g(E)dE is the density of states in the energy interval E and E+dE and [1-
f(E)] probability that a state of energy is occupied not by an electron. Since f(E)
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is the probability of occupancy of an electron, then the probability that the
energy state is vacant is given by [1- f(E)].
Thus,
the density of states in the valence band is given by,
The top edge of the valence band corresponds to the potential energy of a
hole at rest. Thus ( will be the kinetic energy of the hole at lower energy
levels. Thus,
Number of holes in the energy interval E and E+dE is given by,
To calculate the hole density this equation should be integrated between
limits -∞ and . Thus the hole density is given by,
This integral is of the standard form which has a solution of the form,
54
where a=1/kT and x=( - E), thus dx=-dE. The lower limit of the x
becomes ∞ when E=-∞. If we change the order of integration we introduce
another minus sign,
Rearranging,
This is the expression for the hole
concentration in the valence band of an intrinsic semiconductor. Designating
to be the effective density of states in the valence band as,
The hole concentration in the conduction band of an intrinsic
semiconductor can be written as,
Law of mass action:
Statement: for a given semiconductor, the product of the electron and hole
concentration is a constant at a given temperature and is equal to the square of
the intrinsic carrier concentration.
Considering the electron concentration (n) and hole concentration (p) in a
semiconductor,
Therefore,
for an intrinsic semiconductor n=p= ni,
55
Fermi level in intrinsic semiconductor:
In a pure semiconductor, the electrons in the conduction band clusters very
close to the bottom edge of the band, and the electrons are located at the
bottom edge of the conduction band. Similarly the holes are at the top edge of
the valence band. The electron concentration is given by,
And the hole concentration is given by,
for an intrinsic semiconductor n=p
therefore,
taking log on both sides,
Considering the value for and and rearranging,
If the effective mass of free electron is equal to the effective mass of free hole,
We get,
Figure below shows the Fermi level in an intrinsic semiconductor.
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LESSON -7
Hall Effect:
When a conductor carrying a current is placed in a transverse magnetic field, an electric field is produced inside the conductor in a direction normal to both the current and the magnetic field. This phenomenon is known as Hall Effect and the generated voltage is called as Hall voltage.
Illustration of Hall effect:
Consider a thin rectungular semiconductor wafer mounted on an insulating strip and the two parts of electrical contacts are provided on the opposite sides of the wafer. A pair of contact is connected to a constant current source and the other pair is connected to a sensitive voltmeter. The whole arrangement is mounted between two pole pieces of an electromagnet so that the magnetic field acts perpendicular to the lateral faces of the semiconductor wafer.
57
Hall coefficient:
Hall coefficient ‗RH’ is defined as Hall field per unit current density per unit
magnetic induction. When a potential difference is applied across the ends of a
p-type semiconductor, the current (I) is given by,
I=peAvd
where p- concentration of holes, e- charge of a hole, A- area of cross-section of
the end face, vd average drift velocity
Current density J is
When the force due to the magnetic field FL is in equilibrium with the force due
to the transverse electric field FE
i.e.
since , therefore,
the ratio is known as Hall co-efficient, given by,
For a p-type semiconductor the carriers are positively charged holes,
58
For an n-type semiconductor the carriers are negatively charged electrons,
Hall voltage ( ):
If a rectangular slab of semiconductor of thickness t and width w is taken. A
current I is passed through it with potential difference V. when the slab is
placed between two pole pieces of an electromagnet, the Hall voltage VH is given
by,
Hall voltage is given by,
where J=I/A, A=wt.
Two Probe method and four probe method
In four probe (four probes or wires are used) resistance or conductivity
measurements, you send current through two probes and measure the
resulting voltage with two other probes. The voltage drop in the current wires
does not contribute error to the voltage measurement. There is very little
current flow in the voltage sense wires. With two probes, the voltage drop from
the current flow will not be separable from the voltage drop in the device under
test. It may be a significant error in low resistance devices.
Summary:
By studying the energy level diagram we can derive an expression for hole and
electron concentration. With the help of Hall Effect one can find that whether
the given material is n-type or p-type.
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SUPERCONDUCTIVITY
LESSON-8
The conductors at normal conditions have some resistance in spite of their highest purity. But certain metals and alloys, the resistance drops to zero or a negligible value at very low temperature close to absolute zero (0k) and the metal is said to be superconductor. The resistance of metal drops suddenly to zero value at a particular temperature called critical temperature (Tc) Transition
temperature. This value is different for different materials. Superconductivity is an electrical phenomenon due to which the electrical resistivity of some metals falls suddenly to zero when they are cooled below their critical tempt, and will have 100% conductivity. Till now 33 elements, alloys ceramic materials and organic compounds have been identified.
Ex: Hg, Liquid He, Liquid nitrogen, lead, Thorium, Thallium, Nb, Sn, Zn, Al etc.
The phenomenon of superconductivity was first discovered by H Kamerlingh onnes in 1911.
The superconducting state of metal can be destroyed by
when the temperature of the metal is increased above critical tempt. when a strong magnetic field is applied to super conducting state when a high current density exists in the conductor
Many materials have critical temperature in the range of 1K to 20K. In certain ceramics the critical tempt. Will be closed to 100 K and is termed as high temperature Superconductors.
Temperature Dependence electrical resistivity:
60
Not all metals and good conductors are superconductors. Most superconductors are relatively poor conductors at room tempt. Even the ferromagnetic elements are not superconductor Electric current is the flow of electron amongst the positive ion in a metallic crystal. Increase in temperature will increase the thermal agitation of ions above their mean position and hence the conductivity decreases. On the other hand, as the temperature decreases the thermal agitation decreases, hence electrons are more mobile, which reduces the resistance of the conductor.In normal conductors the resistance decreases with decrease in temperature, reaches a fixed minimum value at absolute temperature. In case of super conductors basically have higher resistivity at room temperature than the good conductors, the resistance gradually decreases with decrease in temperature, at a particular temperature the value of resistance suddenly drops to zero and the metal will have 100% conductivity. The temperature at which the resistance suddenly drops at zero is called the critical temperature. The critical temperature is different for different super conductors. Pure metal will have a fixed value
EFFECT OF MAGNETIC FIELD:
MEISSNER EFFECT:
A Super conductor kept in a magnetic field expels the magnetic flux out of its body when it is cooled below critical temperature and becomes perfect diamagnetic. This is known as Meissner effect. When the magnetic field is increased or its temperature rises above TC it becomes normal conductor and allows the magnetic field to pass through it when the temperature of the specimen increases the value of the critical magnetic field decreases correspondingly.
The critical field is different for different substances
61
Demonstration of Meissner effect:
Consider two coils, a primary and a secondary are wound on a superconducting material, which is kept above its critical temperature. Connect a DC battery with a plug key. The secondary coil is connected to galvanometer BG. When the key is closed current flows in the primary coil and sets up a magnetic field in it. The magnetic flux immediately links with the secondary coil. This amounts to the change in magnetic flux in the secondary coil and therefore, momentary current flows through BG which shows deflection. When the current in the primary coil reaches a steady value, the magnetic flux also become steady. Hence there will be no change in the magnetic flux linked with secondary coil. The BG now does not show any deflection. Now the temperature of the superconductor is gradually decreased. As soon as temperature reaches below the critical temperature, the BG suddenly shows a momentary deflection indicating a change in the magnetic flux linked with the secondary coil. This change in the magnetic flux can be attributed to expulsion of the magnetic flux from the body of superconducting material. This magnetic expulsion continues for all values of T ≤Tc.
LESSON-9
B C S THEORY of Superconductivity:
62
During the flow of current in a superconductor, when an electron comes near a positive ion core of the lattice, it experiences an attractive force because of the opposite charge polarity between the electron and the core. The ion core will be displaced from its position which is called lattice distortion. The ion core while returning to original site from distorted, position overshoots and hence oscillates around its mean position before it comes to rest. While doing so, a phonon is created (quantized mechanical/ elastic oscillation in a solid, need a medium). Now an electron which comes near that place will also interact with the distorted lattice, which tends to reduce the energy of the electron. Because of the reduction of energy during the interaction the electrons are in their lower energy states get attracted and form a bound pair called cooper pair (is a bound pair of electrons formed by the interaction between electrons with opposite spin and momenta in a phonon field. This pair formation takes place through phonon interaction and it is called Electron-lattice-electron interaction. The attractive force between the electrons is maximum when the two electrons forming a pair have equal and opposite momenta and spin. The interaction of electron is attractive only when the phonon energy is more than the electronic energy. Temperature below Tc close to 0K, electron- lattice-electron interaction with the help of phonons continues and all the electrons form a cloud of cooper pair, which in turn moves through the crystal. This forms an ordered state of the conduction electrons. There is no scattering on the lattice atoms, have no resistance to flow. Therefore the conductivity becomes infinity which is named as superconductivity.
TYPES OF SUPERCONDUCTORS:
They are of two types:
Type 1 (Soft) superconductors
Type 2 (Hard) superconductors
Type 1 (soft) Superconductors:
Type1 or soft superconductors are those in which the destruction of the sc state by a magnetic field occurs sharply.
63
The magnetic field at which a superconductor transits to a normal state is called the critical magnetic field (Hc). The critical field value for type 1 sc are found to be very low The dependence of magnetic moments on H for type 1 sc is as shown in fig. If the applied field exceeds (Hc) then the material becomes normal conductor. Hence they are not used for coil in superconducting magnets. Examples: Al, Pb, tin, mercury.
Type II (Hard) superconductors:
Type II superconductors are those in which the destruction of the sc state by a mag. Field takes place gradually. The transition from the sc state to normal state occurs between lower critical field HC1 and upper critical field HC2. At HC1 the magnetic flux lines begin and HC2 the field penetrates completely.
Between Hc1& Hc2, the material exists in a mixed state.It exhibits both normal &sc
Type II superconductors have higher values of Tc &Hc and hence require a greater magnetic field. Type II superconductor has a higher resistance to magnetic field induced and therefore widely used in the coils of superconducting magnets. Examples: Niobium, Tantalum, alloys of Niobium, Zn, Chromium, niobium-tin (Nb3Sn), Nb-Al, Nb-Ge etc.
64
HIGH TEMPTURE SUPER CONDUCTOR (HTSC):
With the low temperature superconductor practical application becomes limited as they use expensive liquid helium (BP=4.2K) to reach the sc state. HTSC have a Tc value around 100K such superconductors can be achieved using liquid nitrogen (BP=77K) which is 500 times cheaper than liquid helium. High temperature superconductors are not metals & intermetallic compounds, but are oxides which fall under the category of ceramics. Ceramics are mostly anti ferromagnetic. Most of them are based on copper oxides with other metallic elements.
The modern high temperature superconductors have a complex unit cell structure with oxygen. such cells are made of 1 atom of rare earth metal, 2 atom of barium, 3 atoms of copper and 7 atoms of oxygen, referred to as 1-2-3
superconductors.
Two compounds using bismuth, strontium, calcium, copper and oxygen had Tc= 100k and using mercury, thallium barium, calcium, copper and oxygen had Tc=139K.
LESSON-10
APPLICATION OF SUPER CONDUCTORS:
MAGLEV VEHICLE:
Propulsion magnetsPropulsion magnets
Levitation Magnets
Wheels
Magnetically levitated vehicles are called Maglev vehicles. They are made use of
in transportation by being set afloat above a guideway. Here the friction is
65
eliminated. With such arrangement, great speeds could be achieved with low
energy consumption. It works based upon Meissner effect. The vehicle consists
of superconducting magnets built into vehicles base. There is an Aluminum
guideway over which the vehicle will be floating. The levitation is produced by
enormous repulsion between two powerful fields, one produced by the
superconducting magnet in the vehicle and the other one by the electric
currents in the aluminum guideway. Vehicle wheels will be retracted into the
body, when it starts levitation. While stopping, the wheels are drawn out. The
speed is around 400km/h in test drives in Japan.
UNIT-III LASERS & OPTICAL FIBERS
Overview
Unit III consists of 5 lessons in LASERS. In lesson 1 you will be
introduced to properties, principle of Laser & important requisites for Laser
action. The lesson 2 will give you the idea of conditions of a laser system.
The lesson 3, gives the idea of thermal equilibrium, Einstein‘s constants and
based on these knowledge there is a derivation of energy density equation. In
lesson 4, you will be introduced to principle, construction and working of
semiconductor laser and Co2 laser system. In lesson 5, you will learn
important applications of lasers.
Objectives:
At the end of this unit-III, you should be able to know:
What is laser? Understand the properties of laser. Explain the requisites of laser emission Conditions for laser actions
66
Principle of laser action Understand the dependency of energy density for laser system Production of laser beam by different methods Important applications of laser
Introduction
The term LASER is the acronym for Light Amplification by Stimulated
Emission of Radiation. It is basically opto-electronic device. In the year 1916,
the theory behind laser was established by Albert Einstein and he got the
Nobel Prize in the year 1921.
Albert Einstein framed a theory that if an electron in an exited state is
collided by a photon of the light energy, then the exited electron would drop
down to a lower energy state and emit a photon of the same energy and
wavelength that would move in phase and direction as that of the colliding
photon. Such coherent sources are known as lasers. The light beam from laser
source with coherence lengths up to 1014 Hz will make many applications and
experiments possible. Prior to Einstein‘s theory, the light sources of coherence
length available was up to 107 only, which would not possible to discover the
new era applications.
Lesson-1 Properties & Principle of Laser
Objectives:
At the end of this lesson you will be able to:
Understand what is laser
Know how laser beam is different from other light sources
Understand the concept of absorption & emission of energy
Understand atomic excitation
3.1 Introduction:
Interaction of radiation with matter requires certain conditions. When
radiation interacts with matter, the energy state or atomic state of the matter
gets disturbed. It leads to the change in its energy state. If the transition is from
lower state to higher state, it absorbs the incident energy of the transition. In
reverse direction, such as higher state to lower state, it emits its energy by the
way of photon.
67
3.1.1Properties of Laser
Laser is a light beam which is
Monochromatic
Coherent in nature
Highly directional
Low convergence
Travel as a narrow beam
Spreads very little
Does not fade even after long distance
3.1.2 Requisites of a laser system
There are three requisites for a laser system:
A source of pumping energy in order to establish a population inversion. An active medium with a suitable set of energy levels to support laser
action. A medium in which light gets amplified is called an active medium. The medium may be solid liquid or gas. Out of the different atoms in the medium only small fraction of the atoms are
responsible for stimulated emission and consequent light amplification.
An optical cavity or resonator to introduce optical feedback and so
maintain the gain of the system overcoming all losses.
That means: an excitation source for pumping action, an active medium
which supports population inversion, a laser cavity.
The process of achieving a large number of atoms in excited state than ground state is called ―Population Inversion‖.
The number of atoms in the higher (excited) energy state must be greater than in lower (ground) state (i.e., N2 > N1) and the process by which population inversion is achieved is called ―Pumping‖. Excitation of
atoms from lower energy state to a higher energy state by supplying
68
energy from an external source is called pumping. There are various types of excitation or pumping mechanisms available, the most
commonly used ones are optical, electrical, thermal or chemical techniques. For example, Solid state lasers usually employ optical pumping from high energy xenon flash lamps (e.g.Ruby, Nd:YAG). Gas
lasers use an AC or DC electrical discharge through the gas medium, electron beam bombardment or a chemical reaction. The DC electrical discharge is most common for 'small' gas lasers (e.g., Helium-neon,
Argon ion).
The process which leads to emission of stimulated photons after
establishing the population inversion is referred to as lasing. The transition from the metastable state back to the ground state is the ‗lasing‘ transition, induced by a passing photon. In order to sustain laser
action, one has to confine the laser medium and the pumping mechanism in a special way that should promote stimulated emission rather than spontaneous emission. This is achieved by bounding the laser medium
between two mirrors as shown in figure below. On one end of the active medium is the high reflectance mirror (100% reflecting) or the rear mirror and on the other end is the partially reflecting or transmissive mirror or
the output coupler. The laser emanates from the output coupler, as it is partially transmissive. Stimulated photons can bounce back and forward along the cavity, creating more stimulated emission as they go. In the
process, any photons which are either not of the correct frequency or do not travel along the optical axis are lost.
69
3.1.3 Basic principles
The interaction of radiation with atoms leads to the following three distinct processes in the medium.
1. Absorption 2. Spontaneous Emission 3. Stimulated Emission
Absorption: When an atom in the state E1 absorbs an incident photon of energy hν (= E2- E1) and makes a transition to higher energy state E2, the process is known as induced absorption or simply absorption.
70
A + hν = A*
Where A is an atom in the lower state and A* is the excited state of the atom.
Spontaneous Emission: The excited state is highly unstable and atoms always seek out the least available energy state. When atom in the excited state E2 comes down to a lower energy state E1 by emitting a photon of energy hν (= E2-
E1) without the aid of any external agency, the process is termed as
spontaneous emission.
A* A + hν
The direction of the emitted photons is random and radiation is incoherent.
Stimulated Emission: When a photon of right energy is incident on an atom in the excited state, the incident photon stimulates the atom to make downward transition. The photon thus emitted will have same phase and energy as that of the incident photon. This process is known as stimulated emission.
71
A* + hν A + 2hν
The importance of this interaction is that the two photons emerging out will travel in the same direction, with exactly same energy and perfectly in phase. This is the interaction responsible for the generation of laser beam. This is the basic principle of laser system.
Self test questions 1. Explain the properties of laser.
2. Explain the principle of laser system.
3. Define the term induced absorption.
4. Define the term spontaneous emission.
5. Define the term stimulated emission. Lesson 2 Objectives: At the end of this lesson, you will be able to:
Understand two important conditions for laser
Understand the term population and population inversion
Familiar the term metastable state
Understand the term 3 level & 4 level atom interaction
3.2 Introduction
In this lesson, first we will go through conditions for laser emission. That is population inversion and metastable state. These two conditions are unusual for a atom in the thermal equilibrium conditions.
72
3.2.1 Conditions for laser action
The lasing action does not work easily for two reasons.
1. It is difficult to maintain the atoms in their excited states until they
are stimulated to emit the photon. The excited atoms have a natural
tendency to drop back to their ground state due to the spontaneous
emission.
2. Atoms in their ground state undergo absorption by using photons
from the beam that is being built, there by hindering the process of
continuous stimulation and emission.
For continuous laser beam emission these two problems are to be solved.
The conditions for continuous laser action are the population inversion and
metastable state.
3.2.2 Population inversion
The term population means the number of atoms available at a given energy
state. For good lasing action to take place, the population of atoms must be
more in an excited state than in the ground state. Under normal thermal
equilibrium conditions, the population is more in the lower energy state than in
the excited state. This condition is to be reversed and it is known as population
inversion. Population inversion is an unusual situation but very much essential
for lasing action. This can be achieved by some artificial means known as
pumping [the process of raising the energy level using an external source of
energy] energy in to the active medium.
3.2.3 Metastable state
There are two types of metastable state. They are
1. Three energy level atom
2. Four energy level atom
3.2.3.1 Three energy level atom transition
The second important condition for lasing action is to retain the excited
atom in a metastable state. Metastable state is a lower excited energy state in
which atom stays more time than the excited state.
E3
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Spontaneous
emission
E2
Pumping of energy lasing action (stimulated emission)
E1
E3------- short lived state E2-----------------Metastable state E1------ground
state
In a lasing material, the atoms originally in the ground state are pumped
in to the excited state using an external source of energy. However, this is a
short lived state [about10-8 sec] and due to spontaneous emission, the excited
state decays rapidly to the lower excited state. This lower excited state is
referred as metastable state. In metastable state, the atom stays longer time
[about 10-3 sec] than the excited state. Stimulated emission can occur by
passing a photon of right energy during the transition of atoms from metastable
state to the ground state. This is basically a three-level atom lasing action and
used in a ruby laser system. The laser produced from this lasing action is a
pulsed laser.
5.2.3.2 Four energy level atom transition
Short lived state
E4
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Meatstable state E3
pumping of energy lasing action
Intermediate short lived state E2
Ground state
E1
In this lasing action, the atoms from ground state are pumped to an
excited state. As described in the three level atoms, here also it decays rapidly
to the metastable state. Now the lasing transition proceeds from the metastable
state to an intermediate short-lived excited state, from where it decays rapidly
to the ground state. Since the intermediate short-lived state decays rapidly, its
population is much less than that of the metastable state, thus maintaining the
population inversion which is essential for lasing action. This type of lasing
action is used in He-Ne gas laser system. The output of this type of laser is
continuous laser.
Self test questions
1. What are two conditions for laser action?
2. What is meant by population?
3. What do understand from the term population inversion?
4. Explain metastable state of energy.
5. What is active medium?
6. What is meant by pumping?
Lesson 3 Energy Density
Objective:
At the end of this lesson you will be familiar with what is energy density,
thermal equilibrium, Einstein‘s coefficients for energy absorption, emission. There is an energy density derivation based on Einstein‘s coefficients at thermal equilibrium. Based on the final equation you can solve few
problems.
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3.3 Introduction
Consider two energy levels E1 and E2 of a system having population
densities N1 and N2 respectively. A radiation of energy density Eν is
incident on the system.
Rate of absorption R12 ∝ EνN1
R12 = B12EνN1 ---------------------- (1)
B12 is proportionality constant known as Einstein‘s coefficient of absorption of radiation.
Rate of spontaneous emission R21 ∝ N2
R21 = A21N2 ---------------------- (2)
Where A21 is called Einstein coefficient of spontaneous emission radiation.
Rate of stimulated emission R*21 = B21EνN2 ------------------ (3)
where B21 is called Einstein‘s coefficient of stimulated emission of radiation.
In thermal equilibrium state, the populations of different energy states E1,
E2 etc. are fixed by the Boltzman factor.
The population ratio is given by the equation
where k is the Boltzmann constant. The negative exponent in the above equation indicates N2 << N1 at equilibrium. This means more number of atoms is in the lower energy level than excited energy level. This state is
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known as normal state of energy. Atoms in the lower energy level E1 occasionally absorb radiation and make a transition to upper energy
level E2. Similarly atoms in the upper energy level will occasionally emit radiation and make a transition to the lower level. In order to maintain N1 & N2 constant, the number of upward transition must be equal to the number
of downward transitions. Thus total absorption is equal to total emission.
3.3.1 Expression for energy density at thermal equilibrium using Einstein’s coefficients
Let us consider an assembly of atoms at temperature T in thermal equilibrium with radiation frequency ν and energy density E (ν). Let N1 and
N2 be the number of atoms in lower energy state E1 and higher energy state E2 respectively at any given instant.
In thermal equilibrium,
Rate of upward transition = rate of downward transition.
R12 = R21 +R*21
Ie B12 N1E (ν) = A21 N2 + B21N2 E(ν) ---------------- (4)
where E(ν) is the energy density.
This relation was predicted by Einstein and the coefficients B12, B21, and A21 are known as Einstein‘s coefficients of induced absorption, spontaneous emission and stimulated emission respectively.
N1 B12 E(ν) = N2 [A21 + B21E(ν)] ------------- (5)
E (ν) [N1 B12 - N2 B21] = N2 A21 ------------ (6)
E (ν) = -------------------- (7)
Multiplying equation (6) by 1/N2 B21 we get,
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E (ν) = ---------------------- (8)
Thermodynamically it was proved by Einstein that the probability of stimulated
absorption must be equal to the probability of stimulated emission. Thus,
B12 = B21 ------------------------- (9)
E (ν) = ------------------- (10)
Boltzmann has shown that the atomic population at different energy levels at a
given temperature T is given by the equation,
------------------- (11)
The negative sign in the exponent indicates that N1 << N2 under equilibrium
condition.
------------------- (12)
------------------- (13)
where h = (E2 - E1) and ‗k‘ is Boltzmann constant, h is the Planck‘s constant
and ‗ ‘ is the frequency.
Therefore equation (10) becomes,
E ( ) = ------------------- (14)
This is the formula for energy density of photons in equilibrium with atoms in
energy states E1 and E2 at temperature T and frequency .
We know from Planck‘s radiation formula,
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E(v) = --------------------(15)
While comparing equation 14 & 15 we get,
------------------------------- (16)
The equation (16) is the ratio between spontaneous emission & stimulated
emission coefficients. The ratio is proportional to 3. This shows that the
probability of spontaneous emission increases rapidly with the energy difference
between the two states.
Solved Examples
1. For the given laser system, calculate the difference in energy between
metastable state and ground state electron. The wavelength of the ruby
laser beam is 6493 Å.
Given data λ = 6493 Å
Constant h = 6.63 x 10-34 Js
Constant c = 3 x 108 m/s
Solution:
The difference in energy between any two states is given by the relation
ΔE = hc
ΔE =
ΔE = 3.06 x 10-19 J
To convert into electron volt divide the value by charge of the electron,
ΔE = 3.06 x10-19/1.6 x10-19
ΔE = 1.91 eV
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- - -
2. A He – Ne laser is emitting a laser beam with an average of 4.5mW. Find the
number of photons emitted per sec by the laser. The wavelength of emitted
radiation is 6328 A˚.
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Wavelength of the emitted light, = 6328 A˚ = 6328 x 10-10 m
Power output = 4.5mW
No of photons emitted/sec = ?
We know that the energy difference,
E = h = hc/ Joule
= 6.63x10-34 x 3 x 108/6328x10-10
This energy difference becomes the energy of each of the emitted photon. If N is
the number of photons emitted per sec to give a power of 4.5mW, then
N x E = 4.5mW = 4.5 x10-3 J/s
Hence N = 4.5x10-3/3.143x10-19
= 1.43x1016.
-- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Self test question
2. What are Einstein‘s coefficients?
3. Derive an expression for energy density at thermal equilibrium using
Einstein‘s coefficient.
4. Write plank‘s radiation formula.
5. Which type of emission is independent of energy radiation?
Lesson 4
Objectives
After end of this lesson you will able to:
Know the working of semiconductor laser with energy level diagram.
Know the working of Co2 laser with energy level diagram.
3.4 Introduction
Lasers are classified into 4 types. They are 1.Solid state lasers, 2. Gas lasers 3.
Liquid dye lasers and 4. Solid state diode laser (semiconductor laser). In this
lesson you will understand the construction and working of two types of laser
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system, namely Co2 laser and semiconductor laser. First one is based on gas
and second one is based on solid state diode laser.
3.4.1 CO2 Laser:
Introduction: It is a molecular gas laser which operates in the middle IR
region involving a set of rotational and vibrational transitions. It is a four level
laser producing continuous wave (cw) or pulsated output. It was discovered by
CKN Patel in 1964.
Modes of vibration of a CO2 molecule:
1. In the symmetric stretching modes, both the oxygen atoms oscillate along the axis of the molecule simultaneously approaching and departing the carbon atom which is stationary. Fundamental frequency ν1= 1337 cm-1.
2. In the bending mode, atoms move perpendicular to the molecular axis. The bending vibration is doubly degenerate; it can occur in the plane of the figure and the plane perpendicular to it.
Fundamental frequency ν2= 667 cm-1
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3. In the asymmetric stretching mode, all the three atoms oscillate along the molecular axis; but while both oxygen atoms move in one direction, carbon atom moves in the opposite direction.
Fundamental frequency ν3= 2349 cm-1
Construction of CO2 Laser:
Brewster window
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Quartz tube
It consists of a long tube of length 5 cm and diameter 2.5 cm. The ends of
the tube are closed with alkali halide (NaCl) Brewster windows (for plane
polarized waves). Outside the ends of the tube, silicon mirrors coated with
aluminum are arranged. This forms the resonant cavity. Active medium
consists of a mixture of CO2, N2 and He gases. The pressure of gases in the
mixture is PHe = 7 Torr, PN2 = 1.2 Torr and PCO2 = 0.33 Torr (1m bar = 0.76
Torr). When a high DC voltage is applied to the mixture, pumping
mechanism based on electric discharge is used to create population
inversion. The electric discharge breaks down CO2 into O2 and CO. If water
vapor is present in the mixture, then CO2 is regenerated from CO. The rear
mirrors act as optical feedback resonators providing the necessary feedback
for the emitted photons. The Brewster angle windows are provided to give
polarized output.
Working:
POWER SUPPLY
Fully reflecting mirror
Outlet gases Inlet gases
Semi transparent mirror
Brewster window
LASER
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CO2: ENERGY LEVEL DIAGRAM
When electric discharge takes place in the gas mixture both N2 and CO2
atoms absorb energy and are excited to higher energy level. This energy level
matches with one of the vibrational-rotational level of CO2, shown by C5
(001) in the figure. More CO2 atoms are raised to level C5 by colliding with N2
molecules. There is an efficient transfer of energy between a N2 excited level
and CO2 excited level. This is called resonant energy transfer creating a
population inversion between the levels C5 and C4 and also between C5 and
C3. The transition from C5 to C4 (100) produces 10.6 m and C5 (001) to C3
(020) produces 9.6 m both lying in the IR region. Other transitions from C3
and C2 (010) to C1 (000) are accomplished through inelastic collision with
Helium atoms. Helium atoms are used to deplete the lower energy levels.
Also due to high thermal conductivity of He, the heat is conducted away
from the laser cavity.
Transitions:
N1 – N2 excitation; and C1 – C5 excitation
C5 – C4 laser 10.6 m and C5 – C3 laser 9.6 m
C3 – C2 and C2 – C1 inelastic collision.
The laser output is 100 kW in CW (continuous wave) mode.
CO2 lasers are more efficient compared to other lasers, efficiency is also high.
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3.4.1 Semiconductor laser
Introduction:
The first semiconductor laser was designed by U.S scientists in the year
1962. Semiconductor lasers are very popular because of their compactness, size
and very high efficiency. The very first p-n junction laser was built using GaAs
gallium- arsenide (infrared type) and GaAsP gallium- arsenide-
phosphide (visible) semi conducting materials. Semiconductor laser are again
continuous wave laser and are widely used in CD players, optical
communication etc.
A semiconductor is a material whose conductivity lies between those of
conductor and insulator. Semiconductors are of two types:
a) Intrinsic semiconductors or pure semiconductors
b) Extrinsic semiconductors or doped semiconductors
Extrinsic semiconductors are further classified into two types depending
upon the type of majority carriers:
i) n-type semiconductors where electrons are majority carriers.
ii) p- type semiconductors where holes are majority carriers.
When a p-type semiconductor and a n- type semiconductor is joined by
special techniques, there will be flow of electrons from n side to p side and flow
of holes from p side to n side. After some time, an electric field will be created
which will oppose this flow and flow stops. Thus, there will be formation of
depletion region. This region is called so because it is depleted from charge
carriers.
Construction
One of the example of semiconductor lasers is with the semiconductor
Gallium arsenide (GaAs). It is used as a heavily doped semiconductor. Its n-
region is formed by heavily doping with tellurium in a concentration of 3 x
1018 to 5 x 1018 atoms/cm3 while its p-region is formed by doping with zinc in a
concentration of around 1019atoms/cm3.
85
Active medium: Active medium is GaAs. But it is also commonly said
that depletion region is the active medium in semiconductor lasers. The
thickness of the depletion layer is usually very small (0.1 μm).
Pumping Source: Forward biasing is used as the pumping source. The
p-n junction is made forward biased that is p side is connected to positive
terminal of the battery and n side to negative. Under the influence of forward
biased electric field, conduction electrons will be injected from n side into
junction area, while holes will enter the junction from the p side. Thus, there
will again be recombination of holes and electrons in depletion region and thus
depletion region becomes thinner.
Optical resonator system: The two faces of semiconductor which are
perpendicular to junction plane make a resonant cavity. The top and bottom
faces of diode, which are parallel to junction plane, are metalized so as to make
external connections. The front and back faces are roughened to suppress the
oscillations in unwanted direction.
Working of semiconductor laser
Achievement of population inversion:
When p-n junction diode is forward biased, then there will be injection of
electrons into the conduction band along n-side and production of more holes
in the valence band along p-side of the junction. Thus, there will be more
number of electrons in conduction band comparable to valence band, so
population inversion is achieved. When the electrons and holes are injected into
the junction region from opposite sides with forward biasing, then population
inversion is achieved between levels near the bottom of the conduction band
and empty levels near the top of the valence band.
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Energy band diagram for semiconductor laser
Achievement of laser
When electrons recombine with the holes in junction region, then there will be
release of energy in the form of photons. This release of energy in the form of
photons happens only in special types of semiconductors like Galium Arsenide
(GaAs with a band gap Eg = 1.4 eV). Otherwise in semiconductors like silicon
and germanium, whenever holes and electrons recombine, energy is released in
the form of heat, thus Si and Ge cannot be used for the production of laser.
The spontaneously emitted photon during recombination in the junction
region of GaAs will trigger laser action near the junction diode. The photons
emitted have a wavelength from 8200 Å to 9000 Å in the infrared region.
Since Eg = 1.4eV, the wavelength of emitted light is
= hc/Eg = 8400 Å
Laser ablation process
Laser ablation is the process of removing material from a solid (or occasionally
liquid) surface by irradiating it with a laser beam. At low laser flux, the material
is heated by the absorbed laser energy and evaporates or sublimates. At high
laser flux, the material is typically converted to a plasma. Usually,
laser ablation refers to removing material with a pulsed laser, but it is possible
to ablate material with a continuous wave laser beam if the laser intensity is
high enough.
The depth over which the laser energy is absorbed and thus the amount of
material removed by a single laser pulse depends on the material's optical
properties and the laser wavelength and pulse length. The simplest application
87
of laser ablation is to remove material from a solid surface in a controlled
fashion. Laser machining and particularly laser drilling are examples; pulsed
lasers can drill extremely small, deep holes through very hard materials. Very
short laser pulses remove material so quickly that the surrounding material
absorbs very little heat, so laser drilling can be done on delicate or heat-
sensitive materials, including tooth enamel (laser dentistry). Several workers
have employed laser ablation and gas condensation to produce nano particles of
metal, metal oxides and metal carbides.
Also, laser energy can be selectively absorbed by coatings, particularly on metal, so CO2 or Nd:YAG pulsed lasers can be used to clean surfaces, remove paint or coating, or prepare surfaces for painting without damaging the underlying surface. High power lasers clean a large spot with a single pulse.
Lower power lasers use many small pulses which may be scanned across an area.
Self test questions
1. Explain the construction and working of Co2 laser with energy level
diagram.
2. Explain construction and working of Semiconductor laser with energy
level diagram.
3. Explain the term recombination.
4. What is the function of Brewster window in Co2 gas laser?
5. Why one end of semi-conductor is partially polished?
Lesson 5 Applications of Laser
Objective
At the end of this lesson we will be able to know about the important
applications of laser in general.
3.5 Introduction
Laser finds applications in almost all fields of life like engineering,
entertainment, defense, medical, communication etc. The other important
applications are laser welding, laser drilling, laser cutting of metals and alloys,
fiber optic communication, creation of holography, laser scanning, laser
printing, read & write on CD, DVD etc., accurate measurement of distances,
laser shows, navigation of aircrafts, eye-retina surgery etc. Out of these
applications we will see some of these in detail.
Applications:
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In industry: Cutting
Welding
Drilling
3.4.1 Laser welding
Lasers are widely used in metal welding operations. Carbon dioxide (CO2)
and YAG lasers are used for laser welding. CO2 lasers generate 10.6 μm
wavelength laser beams with continuous wave with high power densities. The
intense beam from the source melts the metal at the surface and enters deep
into the metal which causes welding action. Laser welding can be used to weld
dissimilar metals with widely varying physical properties. Metals with different
sizes and masses at high electrical resistance can be successfully welded.
Temperatures as high as 10,000 ˚C can be attained easily with laser welding.
Advantages:
1. Very small parts say few micron thicknesses can be welded.
2. No contamination on base metal.
3. Welding can be done in side glass chamber
4. Localized heating which will not spread the heat.
3.5.2 Laser drilling
The principle of laser drilling is to heat the metal to its boiling point and
vaporize it or remove it by high pressure vapor. Drilling is done by high power
pulsed laser of the order of 10-4 s to 10-3 s duration. The spot to be drilled is
focused by a laser beam. The metal vapor interacts with laser beam and
electrons get accelerated by electromagnetic radiation. The molten metal get
ejected from the hole there by forming drilling operation. Nd-YAG laser is used
for metals while CO2 laser is used for both metallic and non-metallic materials.
Advantages:
89
1. Holes can be drilled in close vicinity.
2. Drills with high precision.
3. Drill in any direction.
4. No wear of laser tool.
5. Ceramic materials can also be drilled without breaking.
3.5.3Laser cutting
Laser cutting can be done by melting and blowing out molten metal. For
blowing out of metal a high velocity gas jet of inert gas is used. Such a process
of cutting is termed as gas assisted laser beam machining.
Advantages
1. No need of coolant for cutting.
2. Cutting operation is clean and fast.
3. No mechanical stresses are induced.
4. No problem of wear and tear tools.
5. The heat affected zone is limited.
Self test questions
1. What are the industrial applications of Laser?
2. Give few advantages of laser being used for welding, cutting and drilling?
3. Write a note on Laser ablation?
Optical Fiber
Objective
Know total internal reflection.
Know principle of Optical Fiber.
Know construction of optical fiber.
Understand acceptance angle.
Understand numerical aperture.
Know fractional refractive index change.
Describe types of optical fibers and its mode of propagation.
Know what attenuation is.
Describe point-to-point communication.
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Lesson 5 Optical fibre-Principle and Mechanism Objective At the end of this lesson we will be able to:
Understand how total internal reflection takes place in optical fibre.
Know the parts of optical fibre and its functions.
Know acceptance angle and numerical aperture. 6.5 Introduction
Optical fibers are very thin about 5-50 μm thickness, flexible and are made
up of pure silicon glass. They act as light guide or optical wave guide in the
medium of communication using light rays. Optical fibers are dominating
the communication world with their capacity to carry a large volume of data,
with negligible loss, over the long distances. Optical communication makes
use of visible and infrared portion of the electromagnetic spectrum to
transmit digital data. It operates with high frequency/short wavelength
electro-magnetic transmission. Frequency of operation is 1012 to 1015 Hz;
hence a large volume of data/information can be transmitted through a
single channel.
6.5.1 Construction of optical fibre.
91
Optical fibers are made up of highly pure silica glass thin cylinder called
core at the centre, surrounded by a hollow cylinder called cladding, which is
also made up of glass but with a lower refractive index than the core
material. The cladding is protected by a poly urethane jacket. The core is the
transfer medium, while the cladding reflects the light medium inward, while
the jacket protects the fiber from the environment.
6.5.2. Total internal reflection
The basic principle of optical fibre communication is based on total internal reflection. The core has a slightly higher refractive index (n1) than the cladding (n2). Consider a set of two mediums, having refractive indices n1 and n2 whose interface is a plane surface. Let n1 > n2. When a ray of light R1 is incident on the interface through the first medium at an angle θ1, a part of the beam R‘1 gets reflected at the same angle θ1, while the other part of the beam is transmitted through the second medium at an angle θ2. Now by snell‘s law,
= ---------------- (1)
According to snell‘s law, θ2 is greater than θ1, when n1 is greater than n2. The
values of θ2 will reach its physical limit of 90˚ for some incident angle θ1 < 90˚.
This value of incident angle at which θ2 = 90˚ is called critical angle θc. For this
condition, equation 1 becomes,
= 90˚
= /
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= -1( / )
For incidence angles greater than the critical angle (θ1 ≥ θc), there is no
transmitted beam, and the energy of the beam or rays R3 is completely reflected
back into the first medium at an angle θ1. This is termed as total internal
reflection. For absorption losses to be zero, the purity of the medium has to be
very high. Hence the core used in optical fibers is of the highest purity.
6.5.3 Acceptance angle
Let n0 be the refractive index of the medium (air) from where light is launched
into the fiber. The light ray refracts at an angle θr and strikes the core-cladding
interface at an angle θ1. If the angle θ1 is greater than critical angle θc, the light
ray undergoes total internal reflection at the interface since n1 > n2. According
to snell‘s law, we have
=
= ( / ) -------------- (1)
For the reflected ray at the core/cladding interface, we know that from Snell‘s
law,
=
( ) = [ = for total internal reflection]
=
93
= / -------------------- (2)
By rewriting equation (1) we get
Substituting the value of c , we get
----------------(3)
Assuming the surrounding medium is air then, = 1
Hence equation (3) becomes,
The maximum angle is called the acceptance angle or acceptance cone half
angle of the fiber. Acceptance angle may be defined as the maximum angle that
a light ray can have relative to the axis of the fiber and propagate through the
fiber. The light rays contained within the cone having a full 2 are accepted
94
and transmitted to the far end. Therefore, the cone is called the acceptance
cone. Larger acceptance angles make easier launching.
6.5.4 Fractional refractive index change
It is defined as the ratio of difference between the refractive indices of the core,
cladding to the refractive index of the core in an optical fiber. The fractional
difference ‗Δ‘ between the refractive index of the core and the cladding is known
as fractional refractive index change Δ.
It is given by Δ = ( – )/
Δ is always positive since is always greater than for total internal
reflection. In order to guide light rays effectively through a fiber, Δ <<1. Typical
value of Δ is 0.01. Larger value of Δ will not be useful for optical communication
since multi-path dispersion will take place.
6.5.5 Numerical Aperture
Numerical Aperture is defined as the sine of the maximum acceptance angle. It
is a measure of its light gathering power.
Numerical Aperture (NA) =
Therefore,
NA =
NA =
95
NA =
NA =
Since ~ , ( + ) = 2
NA =
NA =
Numerical aperture is a measure of the amount of light that can be accepted by
a fibre. Numerical aperture is dependent only on and . Its value ranges
from 0.13 to 0.50 for good optical communication. Self test questions
1. What is total internal reflection?
2. What is Snell‘s law equation for different medium?
3. Describe the principle of optical fibre.
4. Explain the construction optical fibre.
5. What is acceptance angle?
6. Explain fractional refractive index change and its typical value for good propagation?
7. Explain numerical aperture and its typical value for good communication?
8. Derive an expression for acceptance angle for the optical fibre.
9. Derive an expression for numerical aperture in terms of fractional index change.
Lesson 6 Types of Optical Fibers and modes of propagation
Objective
96
At the end of this lesson we will able to:
Know different types of optical fibres.
Know different types of propagation modes.
Important application of optical fibre. 6.6 Introduction All the types of optical fibres and the modes of propagations depend upon the refractive indices of the core and the cladding materials. The refractive index of the cladding is generally kept constant, while that of core may be fixed or varied radically to achieve a suitable propagation mode. Based on the above factors, they are classified into three categories.
1. Single Mode Step-Index fibre. 2. Multi Mode Step-Index fibre.
3. Multi Mode Graded-Index fibre.
6.6.1 Single Mode Step-Index Fiber
The construction geometry, the refractive index profile, the propagation mode
and the waveform for a step-index single mode fiber are illustrated in the above
diagram. This fiber is made up of a small core say about 5-10 μm diameter with
a thick cladding say about 40-100 μm and a suitable protective sheathing. Both
the core and the cladding have uniform, but different refractive indices. Since
the profile forms a step due to sharp change in the refractive index between the
core and the cladding, it is termed as step-index fiber. This design can transmit
only one mode of wave propagation. Most single mode telecommunication fibers
are manufactured with a diameter ~4 μm. Since there is only one mode of wave
propagation, it eliminates the effect of intermodal dispersion and hence there is
no pulse broadening effect. Hence the output pulse closely resembles the input
pulse without any change in its shape or intensity (no distortion). Such a fiber
with large and fully definable bandwidth is most suitable for long distance, high
data rate communication. Also due to small core diameter, only lasers are
suitable to effectively couple the light signals into the core of such fibers.
97
6.6.2 Multi-mode Step-index Fiber
This type of fiber is made up of a thick core about 50-100 µm with a
thin cladding about 20-40 µm and a suitable protective plastic sheathing.
Here both core and cladding have uniform but different refractive indices.
Since it has a large core size, it can transmit a number of modes of wave
propagation. The rays travel in a zigzag manner, in which the high angle
modes travel a longer distance as compared to the low angle modes, causing
intermodal dispersion. Due to this reason a sharp input pulse broadens as it
travels long distances in the fiber and the output pulse will be widened pulse
resulting with a waveform of distortion. In such fiber, the scattering and
absorption losses are more; it is suitable for low bandwidth, short distance
communications only. Both Lasers and LEDs can be used as source to input
the optical signals into such fibers.
6.6.3 Multi-mode graded-index fiber
98
The construction of this fibre is similar to that of the multimode step-index fibre, except the refractive index of the core. The refractive index of the core varies across the core diameter (radially graded), while the refractive index of the cladding is fixed. In this type, a number of modes can be transmitted. The rays move in a sinusoidal path through the core. Light travels at a lower velocity in the high index region of the core than that of lower index region. Since the fastest components of the rays take the longer path, and the slower components take the shorter path in the core, the travel time of the different modes will be almost same. This reduces the effect of inter-mode dispersion. Due to this, the losses are minimum, with little pulse broadening. These fibres are most suitable for medium distance communication with large bandwidth. Either Laser or LED‘s can be used as the source.
Self test questions
1. What is a step-index profile?
2. What is a graded –index profile?
3. What is single mode propagation?
4. What is multi mode propagation?
5. Explain different types of optical fibres and its mode of propagation with neat diagrams.
Lesson 7 Attenuation and point-to-point communication
Objective
At the end of this lesson we will be able to: Explain what attenuation is.
Know how effectively the light rays are propagated in point to point communication
6.7.1 Introduction to attenuation
Attenuation is also known as fiber loss is an important parameter in the design
of optical fibers for communication systems. Attenuation is the loss of intensity
as it travels in the fiber due to the increasing distances, which reduce the
average power reaching at the receiving end. A minimum amount of light
intensity is required at the receiving end to recover the signal accurately. In fact
the transmission distance is desired by the attenuation property of the fiber.
The fiber losses are measured in unit of dB/Km which will vary with the
wavelength of the optical signal transmitted. The typical losses in the 0.8 to
1.8μm wavelength range, is of the order of 0.2 to 5 dB/Km. The fiber loss is
more for shorter wavelengths which is about 5 dB/Km in the visible region of
the light spectrum. The fiber losses may be due to various reasons like
99
scattering, absorption, dispersion and extensive fiber bends. The net
attenuation is calculated by the equation,
Where ‗ ‘ is the attenuation coefficient,
‗L‘ is the length of the fiber,
‗Pin‘ is the power input of signal sent into the fiber,
And ‗Pout‘ is the power output at the receiving end.
Solved Examples
1. An optical fiber has a clad of R.I 1.498 and numerical aperture is 0.446.
Find its R.I of the core and the acceptance angle.
Solution:
Given = 1.498, N.A = 0.446
We know NA =
The R.I of the core is 1.562.
100
The angle of acceptance
26.48o
2. Calculate the numerical aperture of an optical fiber. Given the R.I of
the core is 1.623 and cladding is 1.522 and also find out the angle of acceptance.
Solution:
The numerical Aperture
NA =
= 0.0563
The angle of acceptance,
34.30o
101
6.7.2 Applications of optical Fibers
Optical fibers have many applications in the field of data transfer,
medical,
engineering, entertainment, audio-video transfer, communications etc. One of
the most important applications is point to point communication which will be
discussed below.
Point-to-point communication
An optical fiber acts as the channel of communication and
transmits the information/data in the form of optical waves. A simple
point to point communication system has few components like an input
information coder, laser transmitter, fiber cable, repeater, receiver and
decoder. The information/data will be in the form of electrical signal
which will be converted into optical signal by the coder. This optical
signal (pulses) will be sent into the fiber cable with required incident
angle (within the limit of acceptance cone angle) by modulating the light
source (laser/LED/diode). After certain distances, depending upon the
type of fiber, repeaters will be kept to boost the signal. Finally, the
optical signal arrives at the receiver‘s end. There, first it will be converted
into the electrical signal by the decoder and then it will be amplified
up to the requirement. This is how information is transmitted from one
point to the other point by the optical fiber.
Advantages
102
1. The materials used for making optical fibers are dielectric in nature. So it does not produce or receive any electromagnetic and RF interferences.
2. Not affected by corrosion and moisture. 3. It does not get affected by nuclear radiation. 4. It is easily compatible with electronic devices. 5. No sparks are generated because the signal is optical. 6. It carries very large amounts of information in either digital or
analog form due to its large band width.
Self test questions
1. What is attenuation?
2. Explain the point-to-point communication using optical fibers.
103
UNIT-4
Objectives
At the end of lesson we shall understand about,
Static dielectric constant.
Types of Polarizations.
Internal or local fields in solids and liquids.
Lorentz field in cubic materials.
Clausius Mossotti equation.
Frequency dependence of dielectric constant.
Ferroelectric materials and applications.
Introduction
A dielectric is a substance that is highly resistant to the flow of an electric
current. In other words a dielectric is electrically non conducting material that
provides electrical insulation between two media (conductors) which are at
different potentials. Eg - Glass, Wax paper, Ceramics, Porcelain. When a
dielectric medium interacts with an applied electric field, charges are
redistributed within its atoms or molecules. This redistribution alters the shape
of an applied electrical field both inside the dielectric medium and in the region
nearby. When two electric charges move through a dielectric medium, the
interaction energies and forces between them are reduced.
Dielectric 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 εr 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 = ε E, Where E ε is Absolute
104
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.
Dipole: A dipole is an entity consisting equal number of positive and negative
charges separated by a small distance. A dipole moment is a vector directed
from positive field.
l
-q q
Polarization: The displacement of charges in the atoms or molecules of a
dielectric under the action of applied field leading to the development of dipole
moment is called polarization.
Electrical polarization
The polarization of the dielectric is the process of formation of dipoles or
alignment of already existing dipoles by the application of an electric field on
the dielectric material. The ratio of induced dipole moment to the effective
applied electric field is called polarizability.
Polar and non-Polar dielectrics
In dielectrics there are no free electrons, the center of positive charges are
centered or concentrated at the center of atom and center of negative charges
105
are concentrated in the electron cloud. With the center of gravity positive
charges coincide with center gravity of negative charges, then it neutralizes
each other effects; hence their dipole moment is zero. Such dielectrics are called
non-polar dielectrics.
In some other dielectrics like water, center of gravity of positive charges never
coincides with center of gravity of negative charges even in the presence of
applied field. In such dielectrics each molecule behaves as if it contains a pair
of positive and negative charges separated by a distance (10-30m). Hence they
have permanent dipole moment. They are known as polar dielectrics.
Consider a dielectric material placed between two plates of a parallel plate
capacitor as shown in figure 2.
Let DC potential be applied between the plates; the atomic dipoles in the
material align in the electric field. The mean position of electrons will align
towards the positive plate of capacitor and mean position of positively charged
nucleus will align towards negative plate of capacitor. Inside the material the
dipoles formed, align such that positively charged particles are attracted
towards the negatively charged particles. In fact, at the surface of dielectric
layer, negative charge is formed near positively charged plate of capacitor; a
layer of positive charge is formed adjacent to the negatively charged plate of
capacitor, these charges on the surface of dielectric material is called polarized
charges. +
-
+
Conducting plate
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
E0
106
- Different Types of Polarization Mechanisms
The polarization is alignment of permanent or induced atomic or molecular dipoles under action of applied field; hence depending on dielectric material and
manner of applied electric field, there are four types of polarization mechanisms:
1. Electronic polarization.
2. Ionic polarization.
3. Orientation or molecular polarization.
4. Space change polarization.
Electronic Polarization
This is the most common type of polarization, which occurs in most of the
dielectrics. The electronic polarization is due to displacement of center of gravity
of negatively charged particles relative to center of gravity of positive charges.
This is called electronic because the dipole moment results due to shift of the
electron cloud relative to the nucleus as shown in Fig 3. This type of
polarization is due to induced dipole moments. The electrons has very high
natural frequencies of order 1015Hz, hence light of frequency 1015Hz can cause
electronic polarization. The electronic polarization is temperature independent.
The electronic polarization occurs in over short interval of time 10-15sec.
Conducting plate
Fig 2: Polarization
107
Since the induced dipole moment is directly proportional to applied field
strength E
μe E or μe = eE or e=μe / E
e is called electronic polarizability. e = μe
E
Electronic Polarization Pe = N μe N= number of atoms per m3
Pe = N eE e = 4 П Єr R3
Macroscopic equation is P = Єr ( Єr — 1 ) E
N eE = Єr ( Єr — 1 ) E
Electronic polarizability. e = Єr ( Єr — 1 )
N
2. Ionic Polarization.
The ionic polarization occurs only in ionic materials like Nacl etc. In this type of
materials under equilibrium conditions, the cations and anions remain at their
mean equilibrium conditions. When the field is applied the cations and anions
get displaced from their mean positions in opposite directions and give rise to a
net dipole moment as shown in Fig 4.As the dipole moment occurs only under
Fig 3: Atom without
Electric field
Atom with Electric field
108
an applied electric field, ionic polarization is due to induced dipoles; also ions
are heavier than electrons. This type of polarization is slow process and ionic
polarization is limited to frequencies up to 1013 hertz and hence light
frequencies of 1015 cannot cause ionic polarization.
3. Orientation Polarizations or Molecular Polarization
The orientation polarization occurs in polar dielectrics in which there are
molecules with permanent dipole moment. The orientation of these molecules
are random due to thermal agitation, because of randomness in orientation, the
material has net zero dipole moment in the absence of electric field. When
electric field is applied each dipole undergo rotation so as to orient along the
direction of the field, which exert a torque in them, thus material itself develops
the dielectric polarization as shown in Fig 5. In the orientation polarization
restoring forces do not exists, however dipole alignment is balanced by thermal
agitation and this type of polarization is strongly temperature dependent.
-
+
+
+ - -
+ - + - +
Fig 4: Ionic polarization
109
Fig 5: Randomly oriented permanent dipoles
The orientation polarizability is given by α0 = µ2/3kT
The orientation polarization Po is given by Po = N µ2 E/3kT
4 Space Charge Polarizations
The space charge polarization occurs in multiphase dielectric substances in
which there is a change of resistivity between different phases when electric
field is applied at high temperature. The electric charges get accumulated at the
interface due to sudden change in conductivity. This accumulation of charges
with opposite polarities at opposite parts in low resistivity phase leads to
development of dipole moment (Fig 6).
+ -
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Aligned dipoles in electric field
+ - + - + -
+ - + - + -
+ - + - + -
+ _
Fig 6: Space charge polarization
110
The space charge polarization is not an important factor in most common
dielectrics. The total polarization α of a material is thus given by the sum of
electronic, ionic and orientation polarizations,
i.e. α = αe+ αi+ αo
Internal Field When the electric field is applied to dielectric material either liquid or solids, each atom in the material develops dipole moment and acts like electric dipole, since atoms either in liquids or solids are surrounded on all sides by polarized atoms, the internal field at given point inside the material is equal to the electric field created by the neighboring atoms and the applied field.
―The internal field is defined as the electric field that acts at the site of any given atoms of a solid or liquid. Dielectric field subjected to an external field and is resultant of the applied field and the field due to all the effects of the surrounding atoms.‖
Expression for Internal or local fields in solids and liquids.
Consider a dielectric material solid or liquid the under action of electric field of intensity ‗E‘. In dielectric imagine an infinite string of similar equidistant atomic dipoles parallel to field,
The components of the electric field at ‗P‘ due to an atomic dipole in polar
form are given by
3
0
rr2π
cosθμ E
3
0
θr2π
sinθμ E
Fig 7: Internal field
111
(1)
Dipole at A1:
The distance of X from A1 is d. i.e., r = d and = 0
(2)
Field at X due to A1: Er+ Eθ =
Dipole at A2:
Since it is situated symmetrically on the other side of X its field at X will
also be
Field at X due to A2: (3)
Therefore field at X due to both dipoles A1 and A2
i.e
Field at X due to A1 & A2:
Field at X due to B1 & B2, located at a distance of 2d: (4)
The Total Field E' at X due to all dipoles:
3
0
rd2π
μ E
0Eθ
3
0 d2π
μ
3
0 d2π
μ
3
0
1dπ
μE
3
0
2(2d)π
μE
1n33
0
333
0
3
0
3
0
3
0
321
n
1
dπ
μ
....]3
1
2
1[1
dπ
μ
....(3d)π
μ
(2d)π
μ
(d)π
μ
....EEEE'
3
0 d2π
μ
112
where n = 1, 2, 3,….∞
But we know that by summation of infinite series
1n3n
1 =1.2
E' = 1.23
0 dπ
μ
(5)
The total field at X which is the internal field Ei, is the sum of the applied field E
and the field due to all the dipoles, i.e. E' .
Ei = E + E' (6)
If αe is the electronic polarizability for the dipoles, then
µ = αe Ei (7)
Ei = E + 3 d
1.2
0
ie E
By rearranging the terms in the above equation we have
Ei = e2.11
E
∏ ε0 d 3 (8)
This is the expression for internal field in case of one–dimensional array of
atoms in dielectric solids or liquids.
Lorentz Field for a cubic lattice:
In 3D the general equation for internal field is expressed as
113
E i = E + (γ/ε0) P, where P is the polarization and γ is a proportionality constant
called internal field constant.
In the 3D if it is a cubic lattice then, γ =1/3 and the internal field is named as
Lorentz Field given by
E Lorentz = E + P/3ε0
The above equation is known as Lorentz relation. One of the important results
that follow from this relation is Clausius-Mossotti relation.
CLAUSIUS-MOSSOTTI RELATION
Consider an element solid dielectric of a dielectric constant εr .If N is the number
of atoms/unit volume of the material, is the atomic dipole moment, then we
have ,
Dipole moment,
i (1)
Polarization of the medium is
i P (2)
114
Therefore
N
Pi
(3)
For a medium with dielectric isotropy
P=0 (r-1) E (4)
Therefore
)1(
P
0
r
i
(5)
In 3D
0
PEΕ
i
(6)
Using (3), (5) and (6)
(7)
00
P
)1(
P
N
P
r
115
(8)
This is Clausius-Mosotti equation.
Dielectric losses:
It is the loss of energy in the form of heat by a dielectric medium due to the
internal friction that is developed as consequences of switching action of
molecular dipoles under certain ac conditions.
Dipolar Relaxation:
Relaxation time is the time required for the dipole to reach the equilibrium
orientation from the disturbed position in alternating field conditions.
Frequency dependence of dielectric constant:
The dielectric constant εr for a material remains unchanged when subjected to a
d.c voltage. But it is subjected to a.c voltage the value of εr undergoes changes
depending on the frequency of the applied voltage. In addition, εr becomes a
complex quantity. It is then represented as εr* and is expressed as,
εr* = εr
' - εr" ------------(1)
3
1
)1(
11
N
1
0 r
)1(3
2
N
0
r
r
03
N
2
1
r
r
116
where εr' and εr
" are the real and the imaginary parts of the dielectric constant.
εr' actually represents the part of dielectric constant that is responsible for
increase of capacitance and on the other hand εr" represents the loss. The
variation of the two with frequency is represented in Fig 8 .Since the
dependence is logarithmic; the frequency is plotted in logarithmic scale. As
there is a direct relation between various polarizations and the dielectric
constant, the variations shown in the Fig could be understood easily once we
know the dependence of various mechanisms on the frequency of the applied
field.
All the four different polarization mechanisms that occur in a dielectric material
will be effective in static field conditions. But, each of them responds differently
at different frequencies under alternating field conditions since the relaxation
frequencies of different polarization processes are different. If τe, τi, τo are the
relaxation times for the electronic, ionic and orientation polarizations, then in
general,
τe <τi <τo
When the frequency of the applied field matches with the relaxation frequency
of a given polarization mechanism the absorption of energy from the field
becomes maximum. But, when the frequency of the applied field becomes
greater than the relaxation frequency for a particular polarization mechanism,
the switching action of the dipole cannot keep in step with that of the changing
field, and the corresponding polarization mechanism is halted. Thus the
frequency of the applied a.c. is increased, different polarization mechanisms
disappear in the order, - orientation, ionic and electronic.
117
In the figure, it can be observed that in the beginning (i.e., when the frequency
variation starts from a value 10-2 Hz), all the four polarization mechanisms will
be responding and hence the total polarization will be maximum. Accordingly εr'
value starts increases. But as the frequency increases from low frequency to
higher frequencies in the radio frequency range, the space charge polarization
mechanism and the orientation polarization mechanism come to a halt in order,
through the other two mechanisms remain active throughout. Accordingly εr'
experiences decrement at the corresponding stages. When the frequency
crosses the infrared range, εr' will be next step down in its value as the ionic
polarization steps. Finally the electronic polarization fades away while the
frequency crosses the ultraviolet limit and εr' steps down for the last time to
stabilize at its least value. The peaks in the variation of εr" over frequency
regions corresponding to the decrements in εr' indicates the losses that the
material suffer over those frequencies.
The loss that occurs in a dielectric material is essentially due to the phase lag of
voltage behind the current in the capacitor between the plates of which the
dielectric material lies. Such a loss in a capacitor is expressed by a factor called
118
tan δ. A large value for tan δ signifies higher dielectric loss. It is also referred to
as tangent loss.
It can be shown that for a capacitor with a dielectric material,
tan δ = εr"/ εr
'
Ferro Electricity
The term ferro electricity is applied to certain dielectric materials that exhibit
electric polarization analogous to the magnetization exhibited by ferromagnetic
materials. The groups of materials, which exhibit polarization even in the
absence of external electric field are called ferroelectrics. Polarization in the
absence of an electric field is called spontaneous polarization. This spontaneous
polarization results from the displacement of ions due to internal fields. When
an electric field is applied, it produces displacement in the crystal structure.
The polarization associated with this displacement creates an internal field,
which further increases and stabilizes the polarization. As a result, a portion of
the polarization remains even when the applied field is removed. Rochelle salt,
barium titanate, potassium dihydrogen phosphate and potassium niobate are
some of the examples of ferroelectric materials. Ferroelectric materials have
extremely high dielectric constants at relatively low applied field frequencies.
Ferroelectric Hysteresis
Magnetic materials exhibit hysteresis when subjected to a magnetic field. A
similar behavior of the ferroelectric materials is seen for the variation of
polarization with the applied field. The hysterisis curve is as shown in the
figure. When an electric field of strength E is applied to a ferroelectric material,
119
the polarization first rises rapidly with the applied field along OABC to a value
above which the dependence is linear. This indicates the saturation at C. The
linear portion CD is extrapolated to meet the polarization axis and the intercept
gives the saturation polarization. When the field is reduced to zero (E=0), it is
observed that there is a certain amount of polarization in the materials called
remanent or residual polarization. This corresponds to spontaneous
polarization. At the point F, if the field is reversed, the path FG is traced and
the polarization reduces to zero at E=Ec. The negative field required to reduce
the polarization to zero is called coercive field. At H, where the specimen gets
saturated, the field is decreased to zero. The field is again increased in the
opposite direction and the path HIJC is traced, thus completing the cycle. The
hysteresis loop of a ferroelectric material changes its shape as the temperature
is increased. At a certain temperature known as Curie temperature Tc, a
ferroelectric material loses its ferroelectric property. The spontaneous
polarization vanishes above the Curie temperature.
Applications:
1. The nonlinear nature of ferroelectric materials can be used to make
capacitors with tunable capacitance.
Fig: Ferroelectric hysteresis curve
120
2. The spontaneous polarization of ferroelectric materials implies
a hysteresis effect which can be used as a memory function, and ferroelectric
capacitors are indeed used to make ferroelectric RAM for computers
and RFID cards.
3. Ferroelectric capacitors are used in medical ultrasound machines (the
capacitors generate and then listen for the ultrasound ping used to image the
internal organs of a body), high quality infrared cameras (the infrared image is
projected onto a two dimensional array of ferroelectric capacitors capable of
detecting temperature differences as small as millionths of a degree Celsius),
fire sensors, sonar, vibration sensors, and even fuel injectors on diesel engines.
Summary:
1. Dielectrics are insulators and posses high electrical resistivity. Dielectric
constant is characteristic of materials and it measures polarization ability of
dielectric subjected to electric field
2. Dielectrics are broadly divided into polar and non-polar dielectrics.
3. The polarization phenomenon accounts for the ability of materials to
increase storage capability of capacitors.
4. The total polarization of materials is sum of electronic, ionic and orientation
polarizations.
5. The Clausius-Mossotti equation holds good for crystals of high degree of
symmetry and non polar dielectric materials.
6. The term Ferro electricity is applied to certain dielectric materials that
exhibit electric polarization analogous to the magnetization exhibited by
ferromagnetic materials.
7. Ferroelectric materials are used make capacitors with tunable capacitance,
memory devices, Medical instruments, fire sensors, sonar, vibration sensors,
and as fuel injectors on diesel engines.
121
Solved examples:
1. Find the polarization produced in a dielectric medium of relative permittivity
15 in the presence of an electric field of 500V/m.
Solution:
Given: εr = 15, we know that, ε0 = 8.854 * 10-12 F/m
Ε =500 V/m
P=?
P = ε0 (εr -1) E
=8.854 * 10-12 (15-1) 500
= 6.195 * 10-8 C/m2
2. A parallel plate capacitor of area 650 mm2 and a plate separation of 4mm has
a charge of 2 * 10-10 C on it. What should be the resultant voltage across the
capacitor when a material of dielectric constant 3.5 is introduced between the
plates?
Solution:
Given
Area of the capacitor s = 650 mm2 = 650 * 10-6 m2
Distance of separation between the plates, d = 4 mm = 4 *10-3 m
Charge on the capacitor, Q = 2*10-10 C
Dielectric constant εr = 3.5
We know that,
122
C = ε0 εr s /d
Also, C = Q/V
Equating the above relations,
Q/V = = ε0 εr /d
Or V= Q d/ ε0 εr s
= 2*10 -10 *4*10-3/8.85 *10-12*3.5 *650*10-6
= 39.73V
3. The dielectric constant of sulphur is 3.4. Assuming a cubic lattice for
its structure calculate the electronic polarizability of sulphur.
Solution:
Since the crystal structure of sulphur is cubic we can apply Clausius –
Mossotti equation,
Hence αe =
Now, N the number of atoms/unit volume can be written as,
N = 3.89*1028 /m3
Hence αe =
03
N
2
1
r
r
Nr
r 03
2
1
eratomicnumb
D*10*NN
3
A
07.32
10*07.2*10*025.6N
326
Nr
r 03
2
1
28
12
10*89.3
10*854.8*3
24.3
14.3
123
αe =
αe = 3.035 *10-40 Fm2
5. A elemental solid dielectric material has polarizability 7*10-40 Fm2.
Assuming that internal field to be Lorentz, calculate the dielectric
constant for the material if the material has 3*1028 atoms/m3.
Solution:
αe = 7* 10-40 Fm2
No of atoms/m3 =3*1028
The internal field is Lorentz field
Since the internal field is Lorentz field, we can apply Clausius –Mossotti
equation,
= 0.7906
( r -1) = ( r +2) *0.7906
r -1 = 0.7906 r + 1.5812
r (1-0.7906) = 2.5812
r =2.5812/0.2094
r = 12.33
UNIT-5
Nano science
03
N
2
1
r
r
2
1
r
r
12
4028
10*854.8*3
10*7*10*3
124
Objectives
At the end of lesson we shall understand about,
nano science.
Density of states (1D, 2D, 3D)
Synthesis (Top-down and bottom-up)
Ball milling method.
Sol-Gel methods.
Carbon nano tube (CNT)- Properties, synthesis)
Applications of carbon nano tubes.
Introduction:
Nanotechnology is the science and technology of small things that are less than
100nm in size. One nanometer is 10-9 meters or about 3 atoms long. For
comparison, a human hair is about 60-80,000 nanometers wide. When the
dimensions of the material such as its, length, breadth, thickness fall in the
range of 1-100 nm, the resulting structures exhibit characteristics that are
specific to their sizes and dimensions. Such materials are called as
nanomaterials. Further Nanoparticles are particles between 1 and 100
nanometers in size. Thus confining the dimensions of a bulk can show different
physical and chemical properties. The confinement of dimensions of a bulk can
result in 2D,1D, 0D.
Density of states (DOS) in 0D, 1D, 2D and 3D:
In case solid-state and condensed matter physics, the density of states (DOS) of
a system gives the number of states per interval of energy at each energy level
that is available to be occupied.
Basically the nanoparticles are classified on the basis of dimensions as zero
dimensional (0D), one dimensional (1D), two dimensional (2D) and three
dimensional (3D) nano particles.
Zero dimensional (0D): A zero dimensional structures the simplest building
block that may be used for nonmaterial design.
Eg: Quantum dots
125
Density of states for 0D is as shown below,
One dimensional (1D): In one dimensional structure only one dimension of the
material is reduced to nanometer range and the other dimensions remain large.
Eg: nanorod, nanowire.
Density of states for 1D is as shown below,
Two dimensional (2D): In 2D structures two of dimensions of the material are
reduced to nanometer range and one dimension remains large called quantum
wire. Eg: Quantum wire, fibers, plate lets etc. Density of states for 2D is as
shown below:
126
Three dimensional (3D): If bulk material there is no confinement in any of its
material dimensions.
Eg: Any bulk structure
Density of states for 3D is as shown below:
Synthesis of materials:
Nanomaterials can be natural or can be synthesized by various processes that
can be categorized into two approaches namely,
1. Top-down
2. Bottom-up
In case of Top-Down approach the synthesis is initiated with the bulk material
and undergoing size reduction becomes powder and then a nano particle and
cluster finally yielding atoms. One of the examples of top-down approach is Ball
milling method.
127
Where as in Bottom-up approach the synthesis is initiated with atoms as the
starting element and undergoes polymerization giving nano particles as the end
product. One of the examples of approach is Ball milling method.
The steps of synthesis of nanoparticles are represented in the diagram given
below:
Example of Top-Down method: Ball milling method.
Example for Bottom-up method: sol-gel method.
Ball milling method: (Top-Down method)
128
The ball mill consists of a hollow cylindrical chamber that can rotate about
axis. There are hard and heavy balls made of tungsten/steel inside the
chamber. Larger balls are used for milling to produce smaller particle size. The
chamber is mounted such that, its axis is slightly inclined to the horizontal to
enable the material inside to slide and accumulate around in one region. The
given material is crushed into small grain size and fed into the chamber. As the
cylindrical chamber is rotated around its own axis, the balls get carried
upwards. But under gravity they drop down and hit the sample with high
speed. This happens repeatedly and the material will be pounded to get reduce
to nano size particles. However the speed of the rotation must be less than a
critical speed beyond which the balls instead of falling down will be carried
along the periphery of the chamber all along. Then the material size misses the
hit and reduction in size stops before attaining the nano particle size.
Advantages:
This method is suitable for large scale production at low cost.
It can be used to grind material irrespective of hardness
Disadvantages:
Because of the nature of use, the purity of the material is affected
Sol-gel method: Bottom-up method
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Sol-gel is a process in which precipitated tiny solid particles agglomerate to
form long networks which spared continuously throughout a liquid in the form
of a gel. Sols are solid particles suspended in liquid medium. Gels comprise of
long networks of particles like polymers in which the interspaces form pores
that contain liquid.
In sol-gel method, precursors which have a tendency to form gel are selected. A
solution of the precursor is obtained by dissolving it in a suitable solvent. The
precursors are generally inorganic metal salts or alkoxides which undergo
hydrolysis. By poly condensation process, nucleation of solid particles starts
and sols are formed.
The sols undergo polymerization which turns the solution into a gel. The sol-gel
is then centrifuged from which a form of gel called Xerogel which has no traces
of the dispersion medium is obtained. The xerogel is then dried by heating it up
to a temperature of 800°C during which time, the pores of the gel network
collapse. This is called densification after which we obtain the desired
nanomaterials. The steps are represented in the diagram above.
Advantages:
Highly pure and uniform nanostructures can be obtained in sol-gel
processing
It is an inexpensive technique with fine control of the product‘s chemical
composition
With this method powder/fiber/thin film coating can be made.
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Disadvantages:
Precursors having a tendency of forming gel can only be selected.
Carbon Nano Tubes (CNTs):
A carbon nanotube is a sheet of carbon (graphite) atoms joined in a pattern of
hexagons rolled into a cylinder.
Structure of carbon nano tube:
Carbon nano tubes are appearing in 3 different structures namely
Armchair Zigzag Chiral
CNTs can be single or multi walled depending on the number of graphene layers
present in the formation of tube structure.
Properties of CNT‘s:
Mechanical properties of CNT’S:
The tensile strength of CNT‘S is 100 times as that of steel. The weight of the
CNT‘S is about 1/10 as that of steel. They have the highest tensile strength of
all known materials so far. They are also highly elastic.
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Thermal properties of CNT’S:
They possess thermal conductivity which is 2 times that of diamonds. They
retain their physical structure in vacuum even up to 2800° C. They are
thermally stable.
Electrical properties of CNT’S:
They can be metallic or semiconducting depending on their structure and size.
Their current carrying is 1000 times that of copper.
Synthesis of carbon nano tubes:
Arc discharge method(bottom up approach)
Pyrolysis method
Arc discharge method:
The set-up used in the arc discharge method consists of a vacuum chamber in
which two graphite rods are mounted on two supports as shown in the
diagram. There is a gap of 1-2 mm between the two tips. The chamber is
evacuated by using a vacuum pump and the methane gas at a certain pressure
is introduced into the chamber. The two rods are maintained at a suitable dc
potential difference (70 A at 18 V) to enable the discharge. On application of the
voltage, the arc discharge starts. Carbon evaporates from the anode. Some part
of the evaporated carbon, deposits on the cathode tip layer by layer. This is
called hard deposit and the rest condenses on the other parts of cathode (called
cathode soot) and on the walls of the chamber (called the chamber soot).Both
the cathode soot and the chamber soot yield either single or multi walled
carbon nanotubes, the hard deposit doesnt yield any product of interest.
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Though this method enables production of large quantities of nanotubes, it
involves purification of the soot by oxidation, centrifugation, filtration and acid
treatment as the products will be highly impure.
Pyrolysis method: (bottom up approach)
Pyrolysis is decomposition of a chemical compound of higher molecular weight
into simpler compounds by heating in absence of oxygen so that, no oxidation
occurs. It takes place usually at a temperature in the range of 400 °C to 800 °C.
The experimental set up is as shown below,
In this method, a hydrocarbon gas such as methane is passed through a heated
quartz tube in which a catalyst is present. Due to Pyrolysis, the gas
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decomposes. Carbon atoms are freed, which bind with the catalyst such as Ni,
Fe, or Co. CNT‘S grow on the catalyst and the same is collected after cooling.
Graphene
Graphene is an allotrope of carbon in the form of a two-dimensional, atomic-
scale hexagonal lattice. It is the basic structural element of other allotropes,
including graphite, charcoal, carbon nanotubes and fullerenes. It can also be
considered as an indefinitely large aromatic molecule, the limiting case of the
family of flat polycyclic aromatic hydrocarbons. Graphene has many
extraordinary properties. It is about 200 times stronger than steel by weight,
conducts heat and electricity with great efficiency and is nearly transparent.
Researchers have identified the bipolar transistor effect, ballistic transport of
charges and large quantum oscillations in the material.
Scientists have theorized about graphene for decades. It is quite likely that
graphene was unwittingly produced in small quantities for centuries through
the use of pencils and other similar applications of graphite, but it was first
measurably produced and isolated in the lab in 2003.Andre
Geim and Konstantin Novoselov at the University of Manchester won the Nobel
Prize in Physics in 2010 "for groundbreaking experiments regarding the two-
dimensional material graphene."
Applications of carbon nanotubes:
Using semiconducting nano tubes it is possible to make electronic
components such as transistors logic gates and nano capacitors, energy
storage, super capacitors, field emission transistors, high-performance
catalysis, photovoltaic, and biomedical devices and implants.
Carbon nano tubes are used in medical field for delivering complex
natured drugs.
As these materials are strong as well as light weight, they are being used
in building aircrafts and making of Micro Electro Mechanical Systems
(MEMS).
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Thin films Introduction
Films can be divided into two categories depending on the thickness of the film and the
process of deposition. They are thick films and thin films as explained in the following sections.
Thick films
A thick film is the one whose film thickness falls in the range of micrometers (µm) to few
millimeters (mm). Major applications of thick films are in the field of electronics as surface
mount devices, sensors and hybrid integrated circuits. Some of the methods used to coat thick
films are thermal spray technique, screen printing and so on.
Thin films
Basically a thin film is considered to be a layer deposited of any material whose thickness ranges
from a few nanometers (nm) to some micrometers (µm).
Applications: Thin film solar cells
Major applications of thin films are in the field of electronics as semiconductor devices and
sensors. In polycrystalline solar cells, layers of n and p type semiconductors are deposited as
thin films.
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Vacuum ranges
Various degrees of vacuum are mainly classified as follows:
1. Low vacuum - 760 - 25 Torr
2. Medium vacuum - 25 - 10-3 Torr
3. High vacuum - 10-3 - 10-6 Torr
Stages of thin film growth
There are several stages in the growth process of thin film, from the initial nucleation stage to
final continuous three dimensional film formation state. These stages of growth were observed
Application Fields Examples
Electronics Semiconductor devices, solar cells and sensors
Optics Antireflection coating; on lenses or solar cells,
Reflection coatings for mirrors. Coatings to
produce decorations (color, luster), Interference
filters, CD's, DVD's and Waveguides.
Chemistry Diffusion barriers, Protection against corrosion /
oxidation. Sensors for liquid / gaseous chemicals.
Mechanical Engineering "Hard" layers (e.g. on drill bits).
Adhesion providers, Friction reduction.
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by the scientists using microscopic studies. Nucleation includes condensation of vapours,
adsorption of atoms, migration of atoms, formation of critical nuclei and and stable clusters.
Nucleation is the first step in the formation of a new structure. Nucleation is often found to be
very sensitive to impurities in the system. Because of this, it is often important to distinguish
between heterogeneous nucleation and homogeneous nucleation.
a b c d
a: nucleation on glass, b: growth, c: coalescence or agglomeration with
island formation d: continuous film.
Coalescence and agglomeration is the action or process of collecting in mass or cluster of
materials. Larger islands grow together, leaving channels and holes of uncovered substrate.
These islands consist of comparatively larger nuclei with size more than 10 Å generally with
three dimensional nature. The process of formation of these islands takes place either by the
addition of atoms from the vapor phase or by diffusion of atoms on the substrate. A nucleus of
5 Å size is made up of 20 atoms or so and the will be made up of around 100 to 150 atoms.
During coalescence many small islands disappear rapidly leading to some definite shapes of
larger islands. The time of coalescence is less say about 0.6 sec. As the deposition continues, the
gaps disappear to form a continuous film by filling up of the channels and holes in the aggregate
mass.
Block diagram of a thin film unit
A standard vacuum coating unit consists of a pumping system which comprises of a rotary
vacuum pump, a diffusion pump and additional fully integrated and wired parts like valves,
baffles, and gauges.
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Photograph of a vacuum coating unit.
Photograph of the vacuum coating unit is shown in the figure above. A schematic diagram of a
vacuum coating unit is shown in the figure below. A rotary pump (Pirani gauge to measure
vacuum till 10-3 Torr) along with a diffusion pump (a penning gauge to measure vacuum above
10-3 Torr) is used to ensure clean and better vacuum of the range of 10-6 Torr. Thus coating unit
is fitted with a Pirani gauge used to measure the roughing vacuum and a Penning gauge to
measure very high vacuum.
Schematic diagram of vacuum coating unit.
1. Bell jar 2. Crystal thickness monitor 3. Substrate holder 4. Filament with source material
5. Penning gauge 6. High vacuum valve 7. Diffusion pump 8. Backing valve 9. Pirani gauge
Besides, inside the coating unit, substrates can be cleaned by an ion bombardment facility in
partial vacuum. A low tension transformer of 10V-100A is used to heat the filament for resistive
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evaporation. Most of the evaporations are processed in a vacuum of about 1-8 x10-6 Torr. Once
the high vacuum is reached, resistive heating is done by heating the source (semiconductor or
metallic) material with a resistively heated filament or boat (fill the boat or filament with the
material to be coated), generally made of refractory metals such as W (tungsten) or Mo
(molybdenum). Vapor sources of various types, geometrics and sizes can be easily constructed
or obtained commercially. Some forms of these sources are basket, spiral, crucible heater
dimple boat, etc. if a material has sufficiently high vapor pressure before melting occurs, it will
sublime and the condensed vapors form a film.
a. Boat b. coils or filament
On heating the source material in vacuum, it gets deposited onto the substrate (usually glass)
for the required thickness, say 1000 Å and the film is kept in vacuum for few hours for settling.
Later the bell jar can be opened and the films can be used for various studies and
characterizations.
Pirani gauge
A heated metal wire (also called a filament) suspended in a gas will lose heat to the gas as its molecules collide with the wire and remove heat. If the gas pressure is reduced the number of molecules present will fall proportionately and the wire will lose heat more slowly. Measuring the heat loss is an indirect indication of pressure. The electrical resistance of a wire varies with its temperature, so the resistance indicates the temperature of wire. In many systems, the wire is maintained at a constant resistance R by controlling the current I through the wire. The resistance can be set using a bridge circuit. The power delivered to the wire is I2R, and the same power is transferred to the gas. The current required to achieve this balance is therefore a measure of the vacuum. The gauge may be used for pressures between 0.5 Torr to 10−4 Torr.
Thus, pressure change in the vacuum unit results in an increase or decrease in the quantity of
gas molecules that is there in the chamber. This in turn increases or decreases the thermal
conductivity of the given gas. Therefore the loss of heat of the filament that is electrically
heated by the constant voltage in the system fluctuates with any variation in pressure. The head
filament of the pirani gauge has a temperature coefficient of resistance that is relatively high. So
even a small change in pressure of the system can result in a considerable change in the
resistance of the filament, as a result of which the current in a Wheatstone’s bridge becomes
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out-of balance that can be measured as the vacuum on a meter. A schematic diagram of a Pirani
gauge is shown in the figure.
Schematic diagram of a Pirani gauge.
H- High vacuum, V- Vacuum system, R- Reference wire, S- Sensing wire,
R1 and R2- Resistances
Penning gauge
It is a cold cathode ionization gauge head and has two electrodes. Through a current limiting
resistor, a 2.3 kV potential difference is given between the electrodes. A magnetic field of nearly
about 800 gauss is introduced by a permanent magnet which acts at perpendicular to the plane
which consists of the electrodes in order to increase the ionization current. Electrons coming out
from the cathode (negative electrode) are there by deflected by the applied magnetic field.
These electrons are forced to have a course which is helical in shape before they reach the
anode loop. There by going through a very long path, even at considerably low pressures the
chances of collision with gas molecule are high. Then there is increase in the rate of ionization
and also the secondary electrons that are produced by ionization. Eventually, the anode
captures the electrons and the equilibrium is said to be attained. This happens when the
number of electrons produced per second is the sum of the positive ion current to the cathode
and the electron to the anode thus measuring the vacuum. The schematic representation of a
penning gauge is shown in figure.
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Schematic diagram of a penning ionization gauge
C- Cathode (-), A- Ring anode (+), M- Magnetic field, V- 2kV,
B- Ballast resistor (1MΩ)
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Rotary pump
This is considered as the commonly used mechanical pump so as to attain the necessary initial
vacuum for high vacuum pumps. This pump comprises of a rotor that is eccentrically mounted within
a stator. Pressure valve leading into an oil reservoir closes the exhaust end. At the time of operation
the air from the inlet side is drawn into the pump from the rotor when the vanes slide in and out.
The volume of air having a crescent-shape is then condensed up to a pressure of about one
atmosphere so as to open the outlet valve and to discharge the air out into the atmosphere from the
oil seal. An outlet valve which is vented admits a small airflow to the compression section decreases
the condensation of water vapor by reducing the value of compression ratio, which is called gas
ballasting.
The fluids that are usually preferred to be used in rotary pump are mineral oils or diphenyl ethers
because the high vapor is solved by installing fore-line traps.
Schematic representation of an oil sealed rotary pump
Diffusion pump
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Schematic representation of a diffusion pump
Gaede was the first person to describe the thought of draining a vessel. This process is performed by
means of molecular momentum transfer. Oils are used by the diffusion pumps for the purpose of
pushing out the gas molecules. Its performance can be improvised by making use of traps and baffles
along with the pumps to nullify the effect of streaming back of vapor. The work fluid is vaporized
with the help of a heater and the hot vapor rises through a chimney. The jet cap reverses the
direction of the flow of vapor and the vapor leaves via a nozzle at a very high speed which is because
of the difference in pressure that is present between the inside and the outside of the chimney.
Momentum is imparted to the moving incoming gas molecules by this high speed jet of molecules.
Thus the gas molecules start moving towards the outlet where they are taken out by means of a
backing pump. These vapors will then condense on the pre-cooled walls of the pump eventually and
returning to the boiler. This working fluid or diffusion oil is supposed to have a high molecular
weight, a relatively low vapor pressure, essential thermal stability and has to be less reactive. The
range of vacuum for diffusion pump is about a value of 10 -2 to 10 -8 Torr. Usually, the diffusion pump
used in the coating unit has a withdrawal speed of about 500 liters/s.
Quartz crystal thickness monitor
A commonly used thickness monitor is the quartz crystal thickness monitor which is employed to
measure the thickness of the deposited thin films. It is a digital thickness monitor with a crystal
holder cum feed-through and an oscillation box. It has a 3 digit LED auto-ranging display for rate of
deposition and 4 digit LED auto-ranging display for thickness deposition.
Block diagram of the quartz crystal thickness monitor.
The concept of this type of measurement and control is that an oscillator crystal can be suitably
mounted inside the vacuum chamber to receive deposition in real time and be affected by it in a
measurable way. Specifically the oscillation frequency will drop as the crystal's mass is increased by
the material being deposited on it. To complete the measurement system, an electronic instrument
continuously reads the frequency and performs appropriate mathematical functions to convert that
frequency data to thickness data, both instantaneous rate and cumulated thickness.
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Process of thermal evaporation
Thin film deposition techniques can be broadly classified into two categories which depend on the
process being mainly physical or chemical.
In physical deposition technique, methods include mainly resistive evaporation and electron beam
evaporation techniques. In this method mechanical, electromechanical or thermodynamic
techniques are used to deposit a thin film of any solid. Starting solid substance that is deposited has
to be taken in a vacuum chamber and the source material is heated in vacuum chamber using
tungsten coils or baskets so that the particles of the solid substance escape from the surface. Thus
the complete unit is placed in the vacuum deposition chamber, so that the particles can move freely.
These moving particles when reaches a surface (usually on glass substrate) which is cooler takes
away the energy from these particles thereby letting them to shape into a solid thin layer. Examples
of physical deposition include vacuum (thermal) evaporation (or resistive heating) and electron
beam evaporation.
A thermal evaporation technique uses an electric resistance (tungsten/ molybdenum) coil or
a basket so as to liquefy the starting matter and to increase its vapor pressure up to a required level.
The process happens in a considerably high vacuum so that the vapor comes in contact with the
substrate and does not interact or does not spread with other gas molecules present inside the
vacuum chamber. This also reduces the merging of impurities from the remaining gas. Hence, only
the materials which are having a vapor pressure much higher than the heating element gets
deposited with minimum contamination of the film.
An electron beam evaporation technique also called as EBG evaporation is as same as the
thermal evaporation but uses an electron beam to melt and vaporize the source material. When a
beam of electron falls on the source material, the atoms of the source material moves to the
substrate and settles. Here the temperature of the source material is made above its
boiling/sublimation temperature so as to deposit a film on the required surfaces. E-beam
evaporation has a major advantage over thermal evaporation that this method yields a higher
density film with a better adhesion to the substrate. As the electron beam heats only the source
material and does not heat the entire crucible, the level of contamination due to the crucible is
relatively less as in the case of thermal evaporation.