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ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 1
Engineering Physics Engineering Physics Engineering Physics Engineering Physics
STUDY MATERIAL
For B.E. I/II SemesterFor B.E. I/II SemesterFor B.E. I/II SemesterFor B.E. I/II Semester
As per JainJainJainJain University University University University Syllabus (sep 2010)
Department of Physics
Sri Bhagawan Mahaveer Jain College of Engineering
Jakkasandra Post, Kanakapura Taluk, Ramanagaram Dist.
562 112
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 2
MODULE I
QUANTUM PHYSICS
Black Body Radiation
A hot body emits thermal radiations which depend on composition and the
temperature of the body. The ability of the body to radiate is closely related to its ability to
absorb radiation. A Body which is capable of absorbing almost all the radiations incident on
it is called a black body. A perfectly black-body can absorb the entire radiations incident on
it. Platinum black and Lamp black can absorb almost all the radiations incident on them.
Emissive power of a black body:
It is defined as the total energy radiated per second from the unit surface area of a
black body maintained at certain temperature.
Absorptive power of a black body:
It is defined as the ratio of the total energy absorbed by the black body to the
amount of radiant energy incident on it in a given time interval. The absorptive power of a
perfectly black body is 1.
Spectral Distribution of energy in thermal radiation
(Black Body radiation spectrum)
A good absorber of radiation is also a good emitter. Hence when a black body is
heated it emits radiations. In practice a black body can be realized with the emission of
Ultraviolet, Visible and infrared wavelengths on heating a body. German physicists Lummer
and Pringsheim studied the energy density as a function of wavelength for different
temperatures of a black body using a spectrograph and a plot is made. This is called Black
Body radiation spectrum.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 3
The Salient features of black body radiation spectrum are as below
1) The energy density increases with wavelength then takes a maximum value Em for
a particular wavelength λm and then decrease to a value zero for longer wavelengths. Hence
the Energy distribution in the spectrum is not uniform
2) As the Temperature increases the Wavelength (λm) corresponding to the
maximum emission energy (Em) shifts towards shorter wavelength side. Thus the λm is
inversely proportional to temperature (T) and is called Wein’s Displacement Law.
Mathematically . Here b is Wein’s Constant of value 2.898x10-3 mK.
3) The total energy emitted by the black body at a given temperature is given by the
area under the curve and is proportional to the fourth power of temperature. This is called
Stefan’s law of radiation. Mathematically E = σ T4 here ‘σ’ is the Stefan’s constant of value
5.67 x 10-8 Wm2K-4.
Explanation of Black Body Radiation Spectrum
Classical Theories
Wein’s Distribution Law:
In the year 1893 Wein using thermodynamics showed that the energy emitted per
unit volume in the wavelength range λ and λ+dλ
Here C1 and C2 are empirical constants. A suitable selection for these constants helps
to explain the experimental curve in the shorter wavelength region. The drawback of this
law is it fails to explain the curve in the longer wavelength region. Also according to this
equation the energy density at high temperatures tends to zero which contradicts
experimental observations.
Rayleigh-Jeans Law:
British Physicists Lord Rayleigh and James Jeans made an attempt to explain the
Black Body radiation spectrum Based on the concepts formation of standing electromagnetic
waves and the law of equipartition of energy. According to this law the energy density of
radiation is given by
Here ‘k’ is Boltzmann constant with value 1.38 x 10-23 JK-1. This law successfully
explains the energy distribution of the black body radiation in the longer wavelength region.
According to this law black body is expected to radiate large amount of energy in the
shorter wavelength region thus leading to no energy available for emission in the longer
wavelength region. Experimental observations show that the most of the emissions of the
black body radiation occur in the visible and infrared regions. This discrepancy is called
Ultraviolet Catastrophe.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 4
Quantum theory of radiation
Planck’s law of radiation:
German physicist Max Plank successfully explained the energy distribution in the
black body radiation based on the following assumptions
1) The surface of the black body contains oscillators
2) These oscillators absorb or emit energy in terms of integral multiples of discrete packets
called quanta or photons. The energy ‘E’ of photons is proportional to the frequency ‘ν’ of
the radiation. Mathematically E=nhν here ‘h’ is a constant called Planck’s constant and its
value is 6.625 x 10-34 Js, and ‘n’ can take integer values
3) At thermal equilibrium the rate of absorption and emission of radiation are equal.
According to Planck’s law of radiation the expression for energy density of radiation
is given by
Where ‘c’ is the velocity of light, ‘k’ is Boltzmann constant and ‘h’ is Planck’s
constant. This law explains the distribution of energy in the black body radiation spectrum
completely for all wavelengths and at all temperatures. Also this law can be reduced to
Wein’s distribution law in the shorter wavelength region and to Rayleigh-Jeans law in the
longer wavelength region.
Deduction of Wein’s law, Rayleigh-Jean’s law from Planck’s law
(i) For shorter wavelengths, is very large
If ν is very large, i.e., is very large
⇒ , ≈ ≈ ……….(1)
Using Planck’s equation,
, by substituting equation (1)
Where, and
(ii) For longer wavelengths, is small,
If ν is very small, i.e., is very small
Now expand as power series
We have, ……….....
≈
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 5
(Since is very small, higher power terms could be neglected)
⇒
Again using Planck’s equation,
is reduced to
This is known as Rayleigh-Jean’s law
Photo-Electric effect
“The emission of electrons from the surface of certain materials when radiation of
suitable frequency is incident on it is called the phenomenon of Photo-Electric effect.” The
electrons emitted are called photo electrons and the material is said to be photo sensitive.
This was discovered in the 1887 by Henrich Hertz.
The experimental observations of photoelectric effect are
1) Photo electrons are emitted instantaneously as soon as the radiation is incident
2) Photo electric emission occurs only if the frequency of the incident radiation is greater
than a certain value called Threshold frequency.
3) The kinetic energy acquired by photo electrons is directly proportional to the frequency of
the incident radiation and is independent of the intensity.
4) The number of photoelectrons emitted depends on the intensity of the incident radiation
and is independent of the frequency.
Photoelectric effect signifies the particle nature of radiation.
Einstein’s explanation of the photo electric effect
When metal is illuminated with radiation of suitable frequency, the photons of the
radiation interact with electrons in the metal. When a photon interacts with an electron, the
electron absorbs it and the photon vanishes. The energy acquired by the electron from the
photon is made use in two stages. A part of the energy is used by the electron to free itself
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 6
from the metal since it is bound within metal. Thus some minimum amount of energy is
required for the electron just to escape from the metal is called Work Function (φφφφ). The
rest of the energy is carried by the electron as Kinetic Energy (KE). Since the energy of
the photon is ‘hν’ the photoelectric effect satisfies the following equation
This is called Photoelectric Equation. Here ‘ν’ is the frequency of the incident
radiation
Here ‘ ’ is the threshold frequency and ‘v’ is the velocity of electron and ‘m’ the
mass. Thus from the photoelectric equation, if the frequency of the radiation ν < no
photoelectrons are emitted.
Compton Effect
“The phenomenon of scattering of X-rays from suitable material and hence increase
in its wavelength is called Compton Effect.”
When X-rays are incident on certain materials they are scattered and the scattered
X-rays contain two components. One component has the same wavelength as the incident
X-ray and the other with wavelength greater than the incident X-rays. This is due to the
scattering of X-ray photons from the electrons present in the material. Due to the transfer
of energy from X-ray photon to electron the wavelength of X-ray increases and the electron
recoils. This can be treated as collision between two particles. Thus Compton Effect signifies
particle nature of radiation. The change in wavelength which is also called Compton Shift is
given by
Here ‘λ’ is the wavelength of incident X-rays and ‘λ’ ‘is the wavelength of scattered X-ray ‘θ’
is the scattering angle and ‘m0’ is the rest mass of electron. The quantity is called
Compton Wavelength.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 7
Experimental verification
A beam of monochromatic X-rays are allowed to fall on a graphite crystal as shown
in the figure. The intensity of the scattered X-rays is measure as a function of wavelength of
X-
rays, at different scattering angles. At each angle, two peaks appear corresponding to
scattered X-ray photons with two different wavelengths. The wavelength of one peak does
not change as the angle is varied. This is called primary or unmodified component. We
denote it by λλλλ. The wavelength of the other peak varies strongly with the angle and hence it
is called modified component. Ii is denoted by λλλλ’. This effect is called Compton effect.
The change in wavelength ∆∆∆∆λλλλ is called Compton shift.
Dual Nature of Radiation and de Broglie’s hypothesis
The phenomenon like Interference, Diffraction and Polarization are attributed to the
wave properties of radiation. The Quantum theory of radiation and experiments like
Photoelectric effect and Compton Effect describe the particle nature of radiation. Thus
radiation behaves like waves and like particles under different suitable circumstances.
Hence radiation exhibits dual nature.
In the year 1924 French physicist Louis de Broglie made a daring suggestion “If
radiant energy could behave like waves in some experiments and particles or
photons in others and since nature loves symmetry, then one can expect the
particles like protons and electrons to exhibit wave nature under suitable
circumstances.” This is well known as de Broglie’s hypothesis.
Therefore waves can be even associated with moving material particles called Matter
waves and the wavelength associated with matter waves is called de Broglie wavelength.
The de Broglie wavelength is given by where ‘m’ is the mass of the particle and ‘v’ is
its velocity.
Expression for de Broglie wavelength (Wavelength of matter wave)
According to the Einstein’s photon theory the energy of the photon is given by
Here ‘ν’ is the frequency of the incident radiation and ‘h’ is Planck’s constant. If ‘m’ is the
mass equivalent of the energy of the photon then
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 8
Since the frequency of the incident radiation could be expressed in terms of wavelength
‘λ’as we get,
⇒
Here ‘p’ is the momentum of the photon
Therefore,
Thus, according to de Broglie’s hypothesis, for a particle moving with velocity ‘v’ the above
equation can be modified by replacing the momentum of photon with the momentum of the
moving particle ‘mv’. Therefore the de Broglie wavelength associated with a moving particle
is given by
…… (1)
Here ‘m’ is the mass of the moving particle.
de Broglie wavelength of an electron accelerated with a potential difference of ‘V’
volt
Consider an electron accelerated by a potential difference of ‘V’ volts. The kinetic energy
acquired by the electron is given by
⇒
Here ‘m’ is the mass of the electron and is given by 9.1 x 10-31 kg
Substituting the value of ‘v’ in equation (1) we get
Since the electron acquires kinetic energy from the applied potential difference ‘V’ The
kinetic energy of the electron is also given by E=eV where ‘e’ is the charge on electron with
value 1.6 x 10-19C
Hence the expression for the de Broglie wavelength
Substituting the values for the constants h, m and e we get
Davisson-Germer’s experiment:
The De-Broglie’s hypothesis of possibility of wave nature of material particles under
appropriate conditions was first experimentally verified by Davisson and Germer. In order to
show that particles can also exhibit wave nature, it needs to be proved that material
particles can also produce effects such as interference, diffraction, etc which are
characteristic phenomena associated exclusively with waves.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 9
Davisson and Germer were studying the phenomenon of scattering of electrons from
material targets and they observed diffraction of electrons in a crystal of nickel, similar to X-
ray waves undergoing diffraction in crystals, thus proving the wave behavior of electrons.
FIGURE:
The experimental apparatus consists of an
electron gun to produce a beam of
electrons, a solid nickel crystal used as a target mounted on a rotatable stand and an
ionization chamber (detector) C which is connected to a galvanometer to collect and
measure the current due to the scattered electrons.
The electron gun (G) consists of a filament which upon heating by a low-tension
battery emits electrons. These emitted electrons are accelerated by applying a high
potential from a high-tension battery and using a series of metallic diaphagrams as slits, a
narrow beam of electron is obtained. This narrow beam of electrons is incident normally on
the Nickel crystal (N) mounted on a rotatable stand. The electrons incident on the Nickel
crystal undergo scattering in all directions inside the crystal, just as X-ray waves are
scattered in the phenomenon of X-ray diffraction by crystals. The scattered electrons are
collected by the ionization chamber (C) and the current due to these collected electrons is
measured by the galvanometer connected to the ionization chamber. The ionization
chamber can be rotated along a circular path S to collect the electrons at various scattering
angles φ .
In the Davisson-Germer experiment, the accelerating potential was kept constant in
G and ionization chamber C collects scattered electrons at various scattering angles φ . For
each scattering angle φ , the ionization current as measured by the galvanometer was noted
and the same procedure was repeated by applying different potentials to the electron gun
G.
Initially by applying a constant potential of 40V, the ionization current was noted as
a function of scattering angle φ and the same was repeated by applying 44,48,54,60 and 68
volts respectively.
A polar plot of ionization current and scattering angle φ , for various applied
potentials is obtained as shown in the figures. In the polar plot, for each data point, the
angle of inclination to Y-axis equals the scattering angle and the length of the arrow to the
data point gives the ionization current.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 10
At the acceleration potential of 40V, as seen from figure, the variation is found to be
smooth without any maxima or minima. However, at V=44volts a distinct maxima was
observed as seen in figure and this maxima became more and more pronounced till 54V.
With further increase in applied voltage, this maximum declines and fades away as seen in
Fig. The values of applied voltage and the scattering angle φ at which the ionization current
was maximum was found to be V=54volts and φ =50 degrees (Fig ).
If electrons were to be behaving only as particles, then from classical theories, it is
expected that with increasing applied potential, the ionization current due to scattered
electrons would increase and therefore, the nature of polar plot at higher potentials should
be similar or a laterally pulled version to that of 40volts curve. But as can be seen from the
above plots, such behavior is not observed.
To explain the observation of distinct maxima of the ionization current over a certain
scattering angle φ , Davisson & Germer proposed that the incident electron beam is
scattered from the nickel crystal as a beam of monochromatic waves associated with
electrons (i.e., De-Broglie waves or Electron waves associated with the electron) similar to
X-ray waves undergoing diffraction in crystals.
According to de Broglie’s hypothesis for an electron accelerated by potential difference of 54
V the de Broglie wavelength is given by
mAV
o 109
1066.154
10226.126.12 −−
×=×
==λ
According to Bragg’s law the diffracted waves from a crystal undergo constructive
interference only for that angle of incidence θ which satisfies the equation
θλ sin2dn =
In the Davisson-Germer experiment the constructive interference was observed at a
glancing angle of 065=θ and it would occur only for those waves with their wavelength λ
given by θλ sin2d= (assuming the order of diffraction n to be equal to one).
We therefore have mSin 10010 1065.1)65(1091.02 −− ×=×××=λ
In the above evaluation we have taken the value of md101091.0 −×= for the lattice
spacing in a nickel crystal.
The experimentally determined value is in good agreement with the value calculated
according to de Broglie’s hypothesis. Thus Davisson and Germer experiment not only
confirms the wave associated with moving particle it also verifies the de Broglie’s
hypothesis.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 11
Phase velocity and Group velocity
Phase velocity (vp)
The velocity with which a wave travels is called phase velocity and is also called
wave velocity. If a point is marked on the wave representing the phase of the particle then
the velocity with which the phase propagates from one point to another is called phase
velocity. It is given by Where ‘ω’ is the angular frequency and ‘k’ is wave number.
Substituting for and . We get ,
Therefore Where ‘E’ is the energy and ‘p’ is momentum
Where ‘c’ is the velocity of light and ‘v’ is the velocity of the article.
From the above expression it is evident that the phase velocity is not only greater
than the particle velocity it is also greater than the velocity of light. Hence there is no
physical meaning for phase velocity of matter waves.
Properties of Matter waves
The following are the properties associated with the matter waves
1) Matter waves are associated only with particles in motion
2) They are not electromagnetic in nature
3) Group velocity is associated with matter waves
4) The phase velocity has no physical meaning for matter waves
5) The amplitude of the matter wave at a given point is associated with the probability of
finding the particle at that point.
6) The wave length of matter waves is given by
Group velocity (vg)
Since the velocity of matter waves must be equal to that of the particle velocity and
since no physical can be associated with phase velocity the concept of group velocity is
introduced.
Matter wave can be considered as a resultant wave due to the superposition of many
component waves whose velocities differ slightly. Thus a wave group or wave packet is
formed. The velocity with which the wave group travels is called group velocity which is
same as particle velocity. It is denoted by vg.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 12
Expression for Group velocity using the concept of superposition of waves
Consider a wave group formed by the superposition of a minimum of two waves
which slightly differ in their velocities with amplitudes ‘A” traveling in the same direction and
are represented mathematically as below
………….(1)
………….(2)
According to the principle of superposition the resultant wave is given by
We know that
………..(3)
But and since very small
Therefore equation (3) could be written as
……..(4)
Compare the eqn.(4) with eqn.(1)
Eqn.(4) represents the resultant wave whose amplitude varies as
Which is a constant but varies as a wave
As by definition of group velocity with which the variation in amplitude is transmitted in the
resultant wave is the group velocity and is given by,
or
Relation between group velocity and phase velocity
The phase velocity is given by
The group velocity of matter waves is given by
………….. (1)
The wave number ‘k’ is given by
Differentiate the above equation we get,
⇒ …………. (2)
Substituting eqn.(2) in eqn.(1) we get
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 13
The relation between group velocity and phase velocity is given by
Relation between group velocity and particle velocity
Consider a particle of mass ‘m’ moving with a velocity ‘v’. We know that the group
velocity is given by
But the angular velocity is given by E
Differentiating we get
The propagation constant is given by ⇒
Differentiating we get
= …………… (1)
The total energy of the particle is given by E= Kinetic energy + Potential energy
Differentiate the above equation we get
⇒ The particle velocity
Hence the group velocity and particle velocity are equal.
Heisenberg’s Uncertainty Principle
Statement: “The simultaneous determination of the exact position and momentum
of a moving particle is impossible”
Explanation: According to this principle if ∆x is the error involved in the
measurement of position and ∆p is the error involved in the measurement of momentum
during their simultaneous measurement, then the product of the corresponding
uncertainties is given by
The product of the errors is of the order of Planck’s constant. If one quantity is measured
with high accuracy then the simultaneous measurement of the other quantity becomes less
accurate.
Physical significance: According to Newtonian physics the simultaneous
measurement of position and momentum are “exact”. But the existence of matter waves
induces serious problems due to the limit to accuracy associated with the simultaneous
measurement. Hence the “Exactness” in Newtonian physics is replaced by “Probability” in
quantum mechanics.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 14
-ray microscope
Consider an imaginary experiment in which an electron is tried to be spotted using a
high resolution -ray microscope.
The limit of resolution of the microscope is given by
Here ‘λ’ is the wavelength of the scattered -ray photon, θ is the semi vertical
angle.
According to the definition of limit of resolution becomes the uncertainty in the
determination of position of the electron. In order to observe the electron, the scattered
photon from the electron must enter the microscope anywhere within angle of . The x
component of momentum ∆px may lie between and . Here p is the
momentum of the photon is given by . Since the momentum is conserved during the
collision, the uncertainty in the x component of momentum is given by
Thus the product of the uncertainties is of the order of ‘ h ’. More rigorous calculation results
in the value
Diffraction by single slit
Consider a narrow beam of electrons passes through a single narrow slit and
produces a diffraction pattern on the screen as shown in the Fig. The first minimum of the
pattern is obtained for n=1, in the equation describing the behavior of diffraction pattern
due to a single slit. Hence,
)1........(sin λθ =∆y
Where y∆ is the width of the slit and θ is the angle of diffraction corresponding to
first minimum.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 15
In producing the diffraction pattern on the screen all the electrons have passed
through the slit but we can not say definitely at what place of the slit. Hence the uncertainty
in determining the position of the electron is equal to the width y∆ of the slit. From equation
(1) we have
θ
λ
sin=∆y ………..(2)
Initially the electrons are moving along the x-axis and hence they have no component of
momentum along y-axis. After diffraction at the slit, they are deviated from their initial path
to form the pattern and have a component θsinp . As y component of momentum may lie
anywhere between θsinp and θsinp− . Uncertainty in y component of momentum is
( ) θλ
θθθ sin2sin2sinsinh
ppppy ≈≈−−≈∆ ……….. (3)
Hence from equations (5) and (6)
hh
py y 2sin2
sin≈×≈∆×∆
λ
θ
θ
λ
hpy y ≈∆×∆
Thus the product of the uncertainties is of the order of ‘ h ’. More rigorous calculation results
in the value
Wave function
A wave is constituted by a periodic oscillation of a particular physical quantity. For
ex, in case of water waves, the quantity that varies is the height surface, in sound waves it
is the pressure variation and in case of electromagnetic waves it is the variation of electric
and magnetic fields that constitutes the electromagnetic wave.
In case of waves associated with material particles (matter waves) the quantity
whose variations make up the matter waves is called the wave function and is denoted
byψ . The value of the wave function ),,,( tzyxψ of a body at the point ),,( zyx in space and
time t , determines the likelihood of finding the body at the location ),,( zyx at that instant of
time ‘ t ’
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 16
The wave function for a wave moving along x-axis in complex notation is given by
Where -angular frequency, k- wave number or propagation constant,
- Amplitude of the wave
Physical significance of wave function
The wave function just as itself has no direct physical meaning. It is more difficult
to give a physical interpretation to the amplitude of the wave. The amplitude of the wave
function is certainly not like displacement in water wave or the pressure wave nor the
waves in stretched string. It is a very different kind of wave. But the quantity, the squared
Absolute amplitude gives the probability for finding the particle at given location in space
and is referred to as probability density. It is given by
Thus, in one dimension the probability of finding a particle in the width ‘dx’ of length ‘x’
Similarly, for three dimension, the probability of finding a particle in a given small volume
dV of volume V is given by
here
Here ‘P’ Probability of finding the particle at given location per unit volume and is called
Probability Density.
According to Max Born’s interpretation
The wave function is complex the probability density is given by
Where * is the complex conjugate of and the above product results in real number.
Normalization and Normalized wave function
Since the particle exists somewhere in volume V then the probability of finding the
particle in the given volume V is equal 1.
Thus
If we are unable to locate the particle in volume V then the notion can be extended
to the whole space with
But, normally, the value of the above integral will not be unity but contains an indefinite
constant which can be determined along with sign using above considerations. The process
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 17
is called normalization and the wave function which satisfies the above condition is called
normalized wave function.
Eigen functions and Eigen values
The Schrödinger wave equation is a second order differential equation. Thus solving
the Schrodinger wave equation to a particular system we get many expressions for wave
function ( ). However, all wave functions are not acceptable. Only those wave functions
which satisfy certain conditions are acceptable. Such wave functions are called Eigen
functions for the system. The energy values corresponding to the Eigen functions are called
Eigen values. The wave functions are acceptable if they satisfy the following conditions
1) must be finite everywhere (not zero everywhere)
2) must be single valued which implies that solution is unique for a given position in
space
3) and its first derivatives with respect to its variables must be continuous everywhere.
Schrödinger Wave Equation
In quantum mechanics, the basic fundamental governing equation which describes
the state of the system is the Schrödinger Equation
We can determine the motion of an atomic particle using Schrödinger Equation just
as we determine the motion of the classical particle using Newton’s law. The solution of the
Schrödinger Equation gives the wave function of the particle that carries information about
the wave behavior associated with the particle.
The Schrödinger Equation can be set up in two different contexts. One, which is
general and takes care both position and time variations of the wave function, It is called
Time dependent Schrödinger Wave Equation.
The other one is applicable only to steady state conditions, in which case the wave
function can have variation with position not with time. It is called Time independent
Schrödinger Wave Equation.
Time dependent Schrödinger Wave Equation
Consider a particle of mass ‘m’ moving with velocity ‘ v ’ along +ve X-axis. The deBroglie wave length ‘λλλλ’ is given by
)1(..................mv
h=λ
The wave equation for one dimensional propagation of waves is given by
VelocityWaveis'v'Where
axisXvealongtv
1
x 2
2
22
2
−+∂
ψ∂=
∂
ψ∂
Here )2........(..........e )kxt(i
0
−ϖ−ψ=ψ
Where ψψψψ0 is the amplitude at the point of consideration ωωωω is angular frequency and k is Wave Number.
Differentiating equation (2) with respect to time (t), we get
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 18
)3....(..........it
ωψ−=∂
ψ∂
Differentiating equation (2) twice with respect to x, we get
)4....(..........kx
2
2
2
ψ−=∂
ψ∂
Using Einstein and deBroglie equations
h
2pkand
h
2E π=
π=ω
Substitute the above in the equations (3) and (4)
)5........(Et2i
hor
h
2iE
t)3(
ψ=∂
ψ∂
π
−
ψπ
−=∂
ψ∂⇒
)6.........(px4
hor
h
4p
x)4(
2
2
2
2
2
2
22
2
2
ψ=∂
ψ∂
π−
ψπ
−=∂
ψ∂⇒
The total energy of the moving particle is given by
)VE(m2
por
E.PE.KE
2
−=
+=
Multiply above equation throughout by Ψ, we get
ψ−ψ=ψ VEm2
p 2
Substituting, for ψψ 2pandE from equations (5) and (6) resp., we get,
2
2
2
2
2
2
2
2
2
i1)7.......(t2
ihV
xm8
hor
t2
h
i
1V
xm2
1
4
h
=−∂
ψ∂
π=ψ+
∂
ψ∂
π−
∂
ψ∂
π−=ψ+
∂
ψ∂
π−
Q
Equation (7) is the one-dimensional time-dependent Schrödinger equation.
Time-Independent Schrödinger Wave Equation
The wave equation which has variations only with respect to position and describes the steady state is called Time-Independent Schrodinger wave equation.
Consider a particle of mass ‘m’ moving with velocity ‘ v ’ along +ve X-axis. The deBroglie wave length ‘λλλλ’ is given by
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 19
)1(..................mv
h=λ
The wave equation for one dimensional propagation of waves is given by
VelocityWaveisWhere
axisXvealongtx
'v'
)2........(..........v
12
2
22
2
−+∂
∂=
∂
∂ ψψ
Here )3........(..........)(
0
kxtie
−−= ϖψψ
Where ψψψψ0 is the amplitude at the point of consideration ωωωω is angular frequency and k is Wave Number.
Differentiating ψ twice with respect to time (t), we get
)4........(..........)(
0
2
2
2kxti
et
−−−=∂
∂ ϖψωψ
Substituting equation (4) in equation (2)
Substituting for λ from equation (1) we get
)5........(..........02
18
02
24
04
2
2
2
2
2
22
2
2
2
2
2
2
2
2
=
+
∂
∂
=
+
∂
∂
=
+
∂
∂
ψπψ
ψπψ
ψπψ
mvh
m
x
vm
hx
mv
hx
The kinetic energy of the particle 2
2
1mv is given by
( )( )
( )
04
4
21
v
1
2
2
2
2
2
2
2
2
2
2
2
22
2
=
+
∂
∂⇒
−=
∂
∂∴
−=−=∂
∂
ψλ
πψ
ψλ
πψ
λψπλ
ψωψ
x
x
lengthwavetheisandwavetheoffrequencytheisfHereffx
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 20
EnergyPotentialtheisVandparticletheofEnergyTotaltheisEhereVEmv −=2
2
1
Therefore equation (5) becomes
0)(8
2
2
2
2
=−
+∂
∂ψ
πψ
h
VEm
x
Generalizing the equation for three dimensions we get
0)(8
2
2
2
2
2
2
2
2
=−
+∂
∂+
∂
∂+
∂
∂ψ
πψψψ
h
VEm
zyx
0)(8
2
22 =
−+∆ ψ
πψ
h
VEm
Here2
2
2
2
2
22
zyx ∂
∂+
∂
∂+
∂
∂=∆
Hence the Time-Independent Schrodinger Wave equation for three dimensions.
Applications of Schrödinger wave equation
Particle in a one dimensional box or one dimensional potential well of infinite
height
Consider a particle of mass ‘m’ bouncing back and forth between the walls of one
dimensional potential well. The particle is said to be under bound state. Let the motion of
the particle be confined along the X-axis in between two infinitely hard walls at x=0 and
x=a. Since the walls are infinitely hard, no energy is lost by the particle during the collision
with walls and the total energy remains constant.
In between walls i.e. 0 < x < a, the potential
energy V=0.
Beyond the walls i.e. x ≤ 0 and x ≥ a, the potential
energy V=∞.
Since the particle is unable to penetrate the hard
walls it exists only inside the potential well. Hence
ψ=0 and the probability of finding the particle
outside the potential well is also zero.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 21
Inside the potential well
Since the potential inside the well is V=0, the Schrodinger wave equation is given by
0)0(8
2
2
2
2
=−
+∂
∂ψ
πψ
h
Em
x
∴ 08
2
2
2
2
=+∂
∂ψ
πψ
h
Em
x
)1.......(..........02
2
2
=+∂
∂ψ
ψk
x Here )2.(..........
82
22
h
Emk
π=
For the given value of E, k is constant. The general solution for the equation (1) is given by
)3.....(..........cossin)( kxBkxAx +=ψ Where A and B are arbitrary constants. The values of
these constants can be obtained by applying the boundary conditions
I) At x=0 , ψ(x)=0 . Substituting the values in equation (3) we get
)4..(..........sin)()3(
0
0cos0sin0
kxAxbecomesequationHence
B
BA
=
=∴
+=
ψ
II) At x=a, ψ(x)=0. Substituting the values in equation (4) we get
)5.........(..........
),(0
sin0
a
nk
nkaSolutionnoOtherwiseASince
kaA
π
π
=⇒
=≠
=
Where n= 1, 2, 3,……
Thus the wave function becomes )6.....(..........sin)(a
xnAx
πψ =
Also substituting the value of ‘k’ from eq (5) into eq (2) we get
)7.....(..........8
82
22
2
22
ma
hnE
h
Em
a
n=⇒=
ππ hence the energy Eigen
values.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 22
Thus substituting n=1 in the equation (7) we get
2
2
18ma
hE = is the ground state energy of the particle and is also called zero point
energy.
Hence 12 EnEn = E2 and E3 are energies of the first and second excited states respectively
and so on. Hence for a particle in the bound state, the energy values are discrete.
Normalization of wave function
The wave function for a particle in a box is given by equation (6)
a
xnAx
πψ sin)( =
The value of the arbitrary constant ‘A’ can be determined by the process of
normalization. Since the particle has to exist somewhere inside the box we have
∫∫ ==aa
dxxdxxP0
2
0
1)()( ψ Substituting the wave function from equation (6)
∫ =
a
dxa
xnA
0
22 1sinπ
Since [ ]θθ 2cos12
1sin 2 −= we have
∫ =
−
a
dxa
xnA
0
2
12
cos12
π Integrating the equation we get
12
sin22
0
2
=
−
a
a
xn
n
ax
A π
π The second term takes the value zero for both the limits
∴ [ ]a
AaA 2
102
2
=⇒=−
Thus the Eigen function is given by
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 23
a
xn
axn
πψ sin
2)(
=
The wave functions and the probability densities for the first three values of ’n’ are as
shown in fig
Thus for ground state (n=1) The probability of finding the particle at the walls is zero and at
the centre (a/2) is maximum. The first excited state has three nodes and the second excited
state has four nodes.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 24
MODULE II
NON LINEAR OPTICS
LASERS Laser is an acronym for Light Amplification by Stimulated Emission of Radiation. Laser is
a highly “monochromatic coherent beam of light of very high intensity”. In 1960 Mainmann
built the first “LASER” using Ruby as active medium.
Interaction of Radiation with matter:
1. Stimulated Absorption: -
When an atom in the ground state say E1 absorbs a photon of energy (E2 - E1) it makes transition into exited state E2. This is called Stimulated or Induced absorption. It is represented as follows,
Photon + Atom = Atom*.
2. Spontaneous Emission: -Spontaneous emission is one in which atom in the excited state emits a photon when it returns to its lower energy state without the influence of any external energy.
Consider an atom in the excited state E2. Excited state of an atom is highly unstable. With in a short interval time, of the order of 10-8 sec,atom returns to one of its lower energy state say E1 and emits difference in energy in the form of photon of energy hv = E2 - E1 spontaneously.
If the two atoms are in the same excited state and returns to some lower energy states
two photons of having same energy are emitted. These Two photons may not travel in the
same direction. They produce in-coherent beam of light. Spontaneous emission is
represented as follows,
Atom* = Atom + Photon.
3. Stimulated Emission: -Consider an atom in the excited state E2. If a photon of energy E2 - E1 is made to incident on the atom in the excited state E2.
The incident photon forces (stimulates) the atom in the excited
state to make transition in to ground state E1 by emitting
difference in energy in the form of a photon. This type of
emission in which atom in the excited state is forced to emit a
photon by the influence of another photon of right energy is
called stimulated emission. Stimulated emission can be
represented as follows.
Photon + atom* = Atom + (photon + photon).
When stimulated emission takes place, incident photon and the emitted photon are in phase
with each other and travel along the same direction. Therefore they are coherent.
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Photon Photon Photon Photon hhhhνννν hhhhνννν
hhhhνννν
EEEE2222 EEEE1111 Photon Photon Photon Photon
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 25
Lasing Action (Laser Action) : -
Let an atom in the excited state is stimulated by a photon of right energy so that atom makes stimulated emission. Two coherent photons are obtained. These two
coherent photons if stimulate two atoms in the exited state to make emission then four
coherent photons are produced. These four coherent photons so that stimulates 4 atoms in
the excited state, 8 coherent photons are produced and so on. As the process continues number of coherent photons increases. These coherent photons constitute an intense beam
of laser. This phenomenon of building up of number of coherent photons so as to get an intense laser beam is called lasing action.
Population inversion and optical pumping: -
In an order to produce laser beam there should be more number of stimulated emissions when compared to spontaneous emission. It is possible only if number of atoms in the exited stats is grater than that is the ground state. When system is in
thermal equilibrium, then number of atoms in the higher energy level is always less than the number of atoms in the lower energy level. If by some means number of atoms in the
exited slate is made to exceed number of atoms in the ground state then population inversion is said to have established between excited state and ground state. The method of achieving the population inversion is called pumping. If light is used to pump electrons
to the higher level then the method is called Optical Pumping. If the electric field is used to pump electrons to the higher level then the method is called Electrical Pumping.
Metastable State: - Population inversion can be created with the help of three energy levels as follows.
Let E1 is the ground state of an atom. Let E2 and E3
are the two excited states. If an atom is excited into the energy state, within a short inter of time of 10-8 sec, atom makes a transition into the energy state E2. Let
lifetime of the atom in the energy level E2 is of the order of 10-2 to 10-3 sec. Then atoms stay in the excited state
E2 for sufficiently long time without making any spontaneous emission.
As more and mare atoms are excited from the ground state to E1 more and more atoms are transferred from E3 to E2. As a result, within a short interval of time population inversion
is established between energy level E2 and E1. The energy level E2 in which atoms remain for unusually longer time is called Metastabte state. When transition from E3 to E2 takes place
Excited
EEEE1111 EEEE3333 EEEE2222
Ground
Metastable
Radiation less
Laser
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 26
excited atom looses energy in the form of heat without emitting any radiation. Such transitions are called radiation less transition (Non-radiative transitions).
Requisites of a Laser System: - The Three requisites of a Laser system are
1) Energy Source or Excitation Source for Pumping action 2) Medium Supporting population inversion called Active Medium 3) The Laser Cavity
Appropriate amount of energy is to be supplied for the atoms in order excite them to
higher energy levels. If the Input energy is in the form of light energy then pumping is
called optical pumping. If it is in the form of electrical energy then pumping is called
electrical pumping.
Population inversion occurs at certain stage in the Active medium due to the absorption of energy. After this stage the Active medium is capable of Emitting laser light. The Laser Cavity consists of an active medium bound between two mirrors. The Mirrors
reflect the light two and fro through the active medium. This also helps to tap certain
permissible part of laser energy from the active medium.
The Ruby Laser
Construction
• The ruby is a crystal of Al2O3 (Corundum) with some of the Al3+ ions replaced by Cr3+ ions. The chromium ions give the characteristic colour (red) to the ruby crystal. For the purpose of laser production, the doping concentration of chromium ions is 0.05%.
• A single crystal of ruby in the form of a cylindrical rod is chosen. The length of the rods can vary from 5 to 20 cm while their diameter can vary from 0.5 to 2cm.
• The end faces of the rod are made optically flat and parallel to each other. One face of the rod is fully silvered while the other face is partially silvered
• The ruby crystal is placed along the axis of a helical Xenon flash tube. The xenon flash tube is connected to a high voltage pulse generator. For each single voltage pulse, the Xenon tube gives out flashes of powerful light which last for several milliseconds.
• Surrounding the flash tube is a cylindrical mirror whose function is to reflect light on to the ruby crystal.
• During the working of the laser a lot of heat is generated. This heat is dissipated by circulating cold water in thin tubes which surround the crystal.
• The ruby laser satisfies the four requisites needed for any laser system. The crystal rod along with the mirrored faces functions as the resonant cavity. Optical pumping is achieved using light from xenon flash tube. Chromium ions are the active medium, which support population inversion.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 27
Working
Shown here is the energy level diagram
of chromium ions.
• Light from the flash tube excites the ions from the ground state (4A1) to the two higher energy bands (4F1 and 4F2). Remember when chromium ions interacts the outermost energy levels split into many levels forming a band.
• As there are many levels present in this band, the number of photons available, for exciting the ions, are many. Therefore numerous ions are able to absorb the photons and make transition from the ground state to one of the levels of the band.
• The atoms in the 4F1 and 4F2 band reside there for a period of 10
-8s and then make a transition to the metastable levels 2Ā and Ē. The energy difference between the energy bands and the metastable levels is not released as electromagnetic energy but is taken up by the vibrating atoms of the lattice and is dissipated as heat. These kind of transitions are non- radiative in nature.
• Therefore 4F1 ⇒ 2Ā 4F1 ⇒ Ē
4F1 ⇒ 2Ā Non-radiative transitions
4F2 ⇒ Ē
• The 2Ā and Ē levels being metastable, the atoms reside in them for an unusually long period of time. In a short while the number of atoms in the two metastable levels is more than the ground level. Thus population inversion is established. Induced transitions between these metastable levels and the ground state give rise to the needed laser radiation. Transition from 2Ā to the ground state gives rise to photons of wavelength 692.8nm and the transition from Ē to the ground state gives rise to photons of wavelength 694.3nm.
• The output of the laser is taken from the partially silvered mirror. In the output the intensity of radiation of wavelength 694.3nm is more than the 692.8nm radiation. One of the probable reasons for this could be that the population of the Ē level is more than the 2Ā level. That is why more photons of wavelength 694.3nm would be released per second. The second probable reason (although not significant) is that the probability of transition from the Ē to the ground level is more than the probability of a transition from 2Ā level to the ground level. Why it is not significant is that the two levels have an energy difference of only 0.004eV. This is not too great to cause a significant difference in the probabilities.
Application of ruby laser
• It is used in holography • As drilling requires pulsed laser, ruby lasers are the most suitable The biggest disadvantage of this laser is that since the output is discontinuous, its use is
limited to only special applications. Wherever continuous laser beam is required the helium -
neon laser is more suitable.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 28
Helium-Neon Laser (Gaseous state laser): -
Construction: -It consists of quartz discharge tube of length 1m and diameter 1.5cm fitted
with Brewster’s windows on either side and filled with the mixture of He and Ne gas in the
ratio of 10:1. It is placed between two highly parallel plane mirrors one of which is
completely silvered while the other is partially silvered. The ends of the tube have two
electrodes which are connected to a high power voltage source.
Working: The energy level diagram for He and Ne atoms are as shown in the fig. When
discharge is produced in the tube large numbers of electrons are produced. These highly
energetic electrons collide with He atoms, which are abundant and excite them to energy
levels 21s or 23s of He system. This type of collision is called collision of first kind and
represented as follows,
e1+He →→→→ He + e2 Where el and e2 are energies of electrons before and after collision.
When the helium atoms collide with the neon atoms in the ground state, because of close
coincidence in the energy values SSEE 321 ≅
; SSEE 223 ≅
, resonant energy transfer takes
place from helium to neon atoms. As a result, the neon atoms get excited to 2S and 3S
levels, whereas the helium atoms return to the ground state. This is called 2nd kind of
collision and can be represented as
He* + Ne Ne* + He
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 29
Here, the states 3S and 2S are called as virtual metastable states because the energy
values of 3S and 2S of Ne are equal to the 21S and 23S metastable state of He. Thus
population inversion built up between 2S and 3S levels with the lower energy level 2p which
leads the laser transitions. [3S to 2p transition gives laser light of wavelength 6328Ao and
2S and 2p transition giving rise to 11523Ao radiation which is in the Infrared region]
Applications of Laser
Because of high intensity, high degree of monochromaticity and coherence, lasers find
remarkable applications in medicine, communication, defence, photography, material
processing etc.
Laser Welding
In performing the task of welding, laser welding is superior to other welding such as arc
welding, gas welding, electron welding, etc.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 30
• Focus the laser beam on to the spot to be welded. • Due to the excess of heat generated, only focused portion melts. • The heat produced by the beam is so intense that, impurities in the material such
as oxides float up on the surface and upon cooling the material becomes homogeneous solid structure and it makes a strong joint.
Advantages:
• Laser welding is a contact less process and thus no foreign materials can enter into the welded joint.
• In this type of welding, no destruction occurs in the shape of work piece and the heat is dissipated immediately ( since the total amount of heat supplied is very small compared to the regular welding)
• The laser beam can be controlled to a great precision, so that we can focus the laser beam precisely to the welding spot. Even we can weld difficult to reach the locations in the material.
• Since the heat affected zones are very small, laser welding is ideal at places which are surrounded by heat sensitive components.
Laser Cutting
Laser cutting of metals is generally associated by gas blowing. The oxygen gas is
passed through the nozzle and the tip of the nozzle is pointed at the spot, where the
laser beam is focused.
• The combustion of the gas burns the metal thus reducing the laser power required for cutting.
• Also the tinny splinters along with the molten part of the metal will blow away by the oxygen jet.
• The blowing action increases the depth and also the speed of cutting. • The laser, which controls the accuracy of the cutting thus, the cut edges will be high
quality. Advantages:
• The quality of cutting is very high • There will be no thermal damage and chemical change when cutting is done in inert
atmosphere. • 3-d cutting can also be done very easy.
Drilling:
• Drilling of holes is achieved by subjecting the material to 10-4 to 10-3S duration pulse.
• The intense heat generated over a short duration by the pulses evaporates the material locally, thus leaving a hole.
Advantages:
Conventional Method Laser
The tools wear out while drilling This problem doesn’t exist in laser
setup
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 31
Whereas it could be done only to a
limited extent
Drilling can be achieved at any
oblique angle
It is difficult Very fine holes 0.2 to 0.5mm
diameter can be drilled. The holes
may be even adjacent to each other
Large force has to apply to drill the
hard materials or brittle materials.
Very hard material or brittle
materials can be drilled. There is no
mechanical stress with a laser beam.
Measurement of pollutants in atmosphere:
The concentration of pollutants in the atmosphere such as carbon monoxide,
sulphur dioxide, nitrous oxide, etc, can be measured using laser the way RADAR
system is used. Hence it is called LIDAR i.e Light Detection and Ranging. The laser
technique consists of a Laser source, retro reflector, optical detector, signal processing
unit and analyzer.
Project the pulses of laser beam to the atmosphere, the area where the
pollutants are to be measured. The back scattered light by the congestion of matter is
detected by the photo detector. The reflected laser beam undergoes attenuation due to
the absorption by the pollutants in the atmosphere. Since different gases in
atmosphere absorb laser energy at different wavelength, the amount of absorbance by
each wavelength indicates the amount of pollutants in the atmosphere. The energy of
the attenuated beam received at the detector is integrated and compared with the
reference laser energy source. The difference in energy called error signal is analyzed
and convert into a readout signal by the computer. The reading indicates the
concentration and distribution of pollutants at different section in the atmosphere. But
it does not give any information about the nature of the scattered particles. However it
can be obtained by Raman back scattering experiment.
Raman Back Scattering:
Laser light is passed through the sample and the spectrum of the transmitted light is
obtained.
Since laser is a monochromatic, hence, we expect only one line in the spectrum. But
due to Raman scattering, we are observing several lines along with the expected line.
The other lines of low intensity lie symmetrically above and below to this line. These
additional lines are called side bands and their frequencies result when the oscillating
frequencies of the gas molecules are added or subtracted from the incident light’s
frequency.
Since different types of gas molecules will have different oscillating frequencies and
produce different side band.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 32
Thus by observing Raman spectra of the back scattered light in the gas sample, the
nature of the scattering particles and their compositions can be measured.
HOLOGRAPHY
HOLO - COMPLETE GRAPHY – RECORDING (WRITING)
Holography was discovered by Dennis Gabor in 1948.
Defn: Holography is a technique of capturing pictorial details of 3-d on 2-d recording
aid, by using the phenomenon of interference.
When an object is illuminated with the light source, the light gets reflected and
scattered from the various parts of the object and they carry the information of the
object in the form of intensity and phase.
** In Photography only the intensity is recorded and the phase information is lost
** In holography both intensity and phase distribution is recorded simultaneously using
interference technique. The holographic picture provides 3-d effects even though the
recording is of 2-d.
Principle of Hologram construction:
The interference pattern, which is formed due to the superposition of reference and
object beams, has the ability to produce the transmitted effect of the object beam,
without the presence of the object, by diffracting the reference beam.
** The photographic plate on which the interference fringes are recorded is the
hologram.
The holography consist of two steps process called recording and reconstructions of the
image.
Recording of the image of an object:
There are two methods of recording the image, they are:
(1) Wavefront division technique: [Wavefront: It is the locus of points where the
particles vibrate in the same phase simultaneously.]
-> The given object and a mirror are placed one below the other such that a
part of the expanded laser beam is incident on the mirror and remaining part
falls on the object.
->The part of light reflected from the mirror (plane Wavefront) called reference
beam is incident on the photographic plate.
� When the light incident on the object, every point on the object scatters the incident light.
� Hence spherical wave fronts generates from each points on the object. The reflected beam is called object beam which incident on the photographic plate.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 33
� The photosensitive surface responds to the resultant effect of interference between the spherical wavelets of the object beam, and the plane waves of the reference beam. Thus the interference effects are recorded on the plate.
� The interference pattern consists of concentric circular rings pattern that mark successive regions of constructive and distructive interference. The ring pattern is called Gabor Zone Plate. Every set of spherical wavelets that start from each point on the object generates its own zone plates. Thus recording consists of number of zone plates.
� Such a developed photographic film is called hologram.
(2) Amplitude division technique: In this method an expanded laser source is
incident on a beam splitter. The beam splitter reflects a part of the incident light and
remaining part will be transmitted.
The beam splitter is oriented such that the reflected light incident on the mirror
and transmitted light incident on the object.
The mirror in-turn reflects the beam called reference beam, directly on to a
photographic plate kept at a suitable position for recording the image of the object.
The transmitted light, which is incident on the object, gets scattered. The
spherical wave fronts associated with the scattered rays serve as object beam and
interfere with the reference beam at the photographic plate and the resultant pattern is
recorded in it.
The photographic plate after the development becomes the hologram of the object.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 34
Reconstruction Process:
It is a second step in holography. For the reconstruction of the image, the same
laser beam is directed at the hologram in the same direction as the reference beam
was incident on it at the stage of recording.
When the light is incident on the hologram, diffraction takes place and secondary
waves originates from each constituent zone plate, which interfere constructively in
certain directions and generate both a real and a virtual image of the corresponding
point of the object on the transmission side of the hologram.
A real image will be formed infront of the hologram at the same distance as that
of the virtual image behind from the hologram.
By seeing through the hologram (like seeing through a window) from the
transmission side, it appears as though the original object is lying on the other side of
it at the same place. This is virtual image due to regeneration. By switching the
direction of view, different set of points which corresponds to constructive interference
are observed, which regenerate a different prospective of the object. Therefore, it gives
three dimensional effects.
Applications of Holography:
1. Holographic interferometry:
Interferometry is This is used to study the small distortions of an object that take
place such as due to stress or vibration etc. An object beam from an object and a
reference beam are made to superpose on a photographic film forming an interference
pattern. After the film is developed, it is put back in the same place and the reference
beam is now sent again as before. Also the object now is put under stress, so that it
undergoes deformation. The object beam from the deformed object superposes on the
diffracted reference beam. The diffracted reference beam, which imitates the object
beam, forms an interference pattern with the object beam from the stressed object.
This interference pattern gives us the information about the kind and the extent of
deformation. This is very useful when the deformation is extremely small and as such
cannot be determined by other conventional techniques.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 35
2. Diffraction grating:
When two parallel beams superpose on the photographic film, the interference
pattern consists of parallel straight fringes. The film when developed appears like a
grating. The quality of such a grating is much superior when compared to the
conventional grating in the sense that the grating constant in this case is truly a
constant .
3. Acoustic grating:
In this case two coherent ultrasonic breams, one reference and the other reflected
from an object, are made incident on a medium. The resulting interference pattern
serves as a grating for laser light, which forms an optical image of the object. This is
useful for imaging the human body parts and studying physical changes.
4. Encoding : If the reference beam is sent through a mask, then the interference pattern
becomes very unique. If an attempt is made to read the hologram and if the person
reading the hologram is not aware of the masking used, then he will not be able to
decipher the image of the object. This procedure ensures further secrecy in recording
information.
OPTICAL FIBERS
Optical fiber is a device used to transmit light signals through the transparent
medium made up of dielectric materials like glass from end to other end over a long
distance.
Construction:-
1) The innermost Light guiding region called Core.
2) The middle region-covering core made of material
similar to Core is called Cladding.
The RI of Cladding is less than that of Core.
3) The outermost protecting layer for Core and Cladding
from moisture, crushing and chemical reaction etc., is
called Sheath.
The Optical Fibers are either made as a single fiber or a
flexible bundle or Cables.
A Bundle fiber is a number of fibers in single jacket.
Principles of Optical fibres:-
It is based on the principle of Total Internal reflection.
Consider a ray of light passing from denser medium to
rarer medium. As the angle of incidence increases the
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 36
angle of refraction also increases. For a particular angle of incidence called Critical
Angle the refracted ray just grazes the interface (Angle of refraction is 90o). If the
angle of incidence is greater than Critical Angle then the ray reflected back to the
denser medium. This phenomenon is called Total Internal Reflection.
TIR is not just one kind of reflections. It may be noted that, some light energy is
always lost during reflections occur at the surface of mirror, polished metallic surface.
But in case of TIR, there is no loss of light energy at the reflecting surface. The entire
incident energy is returned along the reflected light. Hence, it is called TIR. Because
of no loss of energy during reflection, the optical fibers are able to sustain the light
signal transmission over long distance inspite of infinite number of reflections that
occur within the optical fiber.
Propagation of light through fiber Optical fiber as a light guide): -
The incident light enters the core and strikes the
interface of the Core and Cladding at large angles as
shown in fig. Since the Cladding has lower RI than
Core the light suffers multiple Total Internal
Reflections. This is possible since by geometry the
angle of incidence at the interface is greater than the
Critical angle. Since the Total internal reflection is
the reflection at the rarer medium there is no energy
loss. Entire energy is transmitted through the fiber.
The propagation continues even the fiber is bent but
not too sharply. Since the fiber guides light it is
called as fiber light guide or fiber waveguide.
Numerical Aperture:
Consider an optical fiber consisting of inner cylindrical core made of glass of refractive
index n1 and is surrounded by another material called cladding of refractive index n2
such that n2 < n1.
Consider a ray of light AO incident on the core at ‘O’ at an angle θo with the fiber
axis. Then it refracts along OB at an angle of θ1 in the core.
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The refracted ray is incident on the interface between core and cladding at B an
angle of incidence (90 - θ1). Assuming this angle (90 – θ1) is equal to critical angle,
then the ray is refracted at 900 to the normal drawn to the interface. i.e. it grazes along
BC.
Now, it is clear from the figure that any ray that enters into the core at angle θi < θo will have refractive angle less than θ1 because of which its angle of incidence at
the interface (=90 - θ1) will become greater than critical angle of incidence and thus
undergoes total internal reflection.
If the angle of incidence at ‘O’ is greater than θo, then the refracted ray pass
through the cladding and it will be lost.
If AO is rotated around the fiber axis keeping θo as constant, then it forms a
conical surface called acceptance cone. One those rays which enter within this
acceptance cone will undergo total internal reflection and propagates through the fiber.
“The angle θ0 is called waveguide acceptance angle or acceptance cone half
angle and Sinθ0 is called Numerical Aperture of the fiber”. The N.A. represents the
amount of light rays that can be transmitted along the optical fiber.
Expression for N.A.:
Let n0, n1 and n2 be the refractive indices of surrounding medium, core and cladding of
the fiber respectively.
Apply Snell’s law to the surface PQ, which separates surrounding medium and
core:
n0 0θSin = n1 1θSin
0θSin = 0
1
n
n 1θSin …(1)
Apply the Snell’s law to surface RS which separates core and cladding:
1n )90( 1θ−Sin = 2n 90Sin
1n 1θCos = 2n
1θCos = 1
2
n
n …(2)
Rewrite the equation (1) => 0θSin = 0
1
n
n1
2cos1 θ−
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= 0
1
n
n
2
1
2
21n
n−
= 0
2
2
2
1
n
nn −
If the surrounding medium is air then n0 = 1, Therefore
0θSin = N.A. = 2
2
2
1 nn −
The condition for propagation is the angle of incidence θi should be less than
acceptance
angle θ0.
i.e. iθ < 0θ
i.e. iSinθ < 0θSin
iSinθ < N.A.
iSinθ < 2
2
2
1 nn −
sine of the angle of incidence must be less than numerical aperture.
Fractional RI Change (∆):-
It is the Ratio of RI difference between Core and Cladding
to the RI of core.
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Relation between N A and ∆: -
Modes of Propagation
Light propagates as an electromagnetic wave through an optical fiber. It is true that all
waves, having directions above the critical angle, will be trapped within the fiber due to
TIR. But is not true that all such waves propagate along the fiber and only certain ray
directions are allowed to propagate. The allowed directions correspond to the modes of
the fiber i.e. Mode refers to the number of paths for the light rays to propagate in the
fiber. The number of modes that a fiber will support depends on d/λ . Where d –
diameter of the core and λ is the wavelength of the wave transmitted.
[Note: As a ray gets repeatedly reflected at the walls of the fiber, phase shift occurs.
Consequently, the waves traveling along certain zigzag paths will be in phase and
intensified, however, some other paths will be out of phase and hence the signal
strength diminishes due to destructive interference. The light ray paths along which the
waves are in phase inside the fiber are known as modes.]
V- Number:
The number of modes supported for propagation in the fiber is determined by a
parameter called V-number and is given by
V = λ
πd
on
nn 2
2
2
1− d – diameter of the core, λ - wavelength of
the light, n0 – R.I. of the surrounding medium, n1 – R.I. of the core, n2 – R.I. of the
cladding
Number of modes ≅ 2
2V
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Types of Optical Fiber:
Optical fibers are classified into 3-types based on their R.I. of core and cladding
and number of modes of propagation in the fiber.
(1) Step index single mode fiber (2) Step index multi mode fiber (3) Graded Index multi mode fiber.
(1) Step Index Single Mode optical fiber:
(2)Step index multi mode fiber
(** Construction is similar to that of a single mode fiber)
• It consists of a core of uniform refractive index n1
• The diameter of the core is about 50 - 200 mµ
• The core is surrounded by a material of uniform R.I. n2 called cladding such that n2 < n1. The external diameter of the cladding is 100 – 250 mµ
• The variation of R.I.s of core and cladding takes the shape of step as shown in fig.
• Since the core diameter is very large, therefore, it will be able to support propagation of large number of modes.
• LED or Laser can be used as a source. • Applications: It can be used in data links
which has lower band width requirements.
• It consists of a core of uniform refractive index n1
• The diameter of the core is about 10 mµ
• The core is surrounded by a material of uniform R.I. n2 called cladding such that n2 < n1. The external diameter of the cladding is 60 – 70 mµ
• The variation of R.I.s of core and cladding takes the shape of step as shown in fig.
• Since the core diameter is very small, therefore, it can guide a single mode as shown in figure.
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(3)Graded Index multi mode fiber (GRIN)
Attenuation (Fiber Loss)
The loss of light energy of the optical signal as it propagates through the fiber is called
attenuation or fiber loss.
The main reasons for the loss of light intensity over the length of the cable is
due to
(i) absorption (ii) Scattering (iii) Radiation loss
(i) Absorption Losses: In this case, the loss of signal power occurs due to absorption
of photons associated with the signal. Photons are absorbed by (a) impurities in the
silica glass (b) Intrinsic absorption by the glass material.
(a) Absorption by impurities: During the light propagation the electrons of the impurity atoms like copper, chromium, iron etc, present in the fiber glass absorb the photons and get excited to higher energy level. Later these electrons give up the absorbed energy either as heat or light energy. But the emitted light will have different wavelength with respect to the signal. Hence it is loss.
• The core material has a special feature that its R.I. value decreases in the radially outward direction from the axis and becomes equal to that of the cladding at the interface.
• It is obvious from the figer that a ray is continuously bent and travels a periodic path along the axis. The ray entering at different angles follow different paths with the same period.
• The diameter of the core is about 50 - 200 mµ
• The core is surrounded by a material of uniform R.I. n2 called cladding such that n2 < n1. The external diameter of the cladding is 100 – 250 mµ
• Since the core diameter is very large, therefore, it will be able to support propagation of large number of modes.
• LED or Laser can be used as a source.
• Applications: It can be used in
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(b) Intrinsic Absorption: some times even if the fiber material has no impurities, but the material itself may absorb the light energy of the signal. This is called intrinsic absorption.
(ii) Scattering Loss: (Rayleigh scattering) Since, the glass is heterogeneous mixture of many oxides like SiO2, P2O5, etc, the compositions of the molecular distribution varies from point to point. In addition to it, glass is a non-crystalline and molecules are distributed randomly. Hence, due to the randomness in the molecular distribution and inhomogeneties in the material, there will be sharp variation in the density (refractive index value) inside the glass over distance and it is very small compared to the wavelength of light. Therefore, when the light travels in the fiber, the photons may be scattered. (This type of scattering occurs when the dimensions of the
object are smaller than the wavelength of the light. Raleigh scattering4
1
λα ). Due to
the scattering, photons moves in random direction and fails to undergo total internal reflection and escapes from the fiber through cladding and it becomes loss.
(iii) Radiation loss: Radiation losses occur due to bending of fiber. There are two types of bends:
(a) Macroscopic bends: When optical fiber is curved extensively such that incident angle of the ray falls below the critical angle, then no total internal reflection occurs.
(b) Microscopic bends:
The microscopic bending is occur due to non-uniformities in the manufacturing of the
fiber or by non-uniform lateral pressures created during the cabling of the fiber. At
these bends some of the radiations leak through the fiber due to the absence of total
internal reflection and leads to loss in intensity.
Hence, some of the light rays escape through the
cladding and leads to loss in intensity of light
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Attenuation co-efficient (α ):
The net attenuation can be determined by a factor called attenuation co-
efficient (α ).
When light travels in a material medium, there will always be loss in its intensity
with distance traveled. The rate of decrease of intensity of light with distance traveled
in the homogeneous medium is proportional to the initial intensity called Lamberts’s
law. i.e. if P is the initial intensity and L is the distance propagated in the medium,
then,
dL
dP− α P ( negative sign indicates that it is a decrement)
Or dL
dP− = α P (1)
where α is a constant called attenuation coefficient, or simply as attenuation.
Equation (i) can be rewritten as P
dP = - α dL
By integrating on both side ∫ P
dP = α− ∫ dL (2)
If Pin is the initial intensity with which light is entering into the fiber and Pout be the
intensity of light at the end of the fiber then equation (2) becomes,
∫out
in
P
PP
dP = α− ∫
L
dL0
The unit of attenuation for light in optical fiber is Bel.
Or α = - L
1
in
out
P
P10log Bel / unitlength
In optical fiber technology, it is customary to express α in terms of decibel/kilometer.
P is in watt and 1B = 10decibel
Therefore, α = - L
10
in
out
P
P10log dB / km
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Applications of Optical Fiber:
A typical point to point communication system is shown in figure.
The analog information such as voice of telephone user is converted into electrical
signals in analog form and is coming out from the transmitter section of telephone.
The analog signal is converted into binary electrical signal using coder. The binary data
comes out as a stream of electrical pulses from the coder.
These electrical pulses are converted into pulses of optical power by modulating the
light emitted from an optical source like LED. This unit is called an Optical transmitter.
Then optical signals are fed into the optical fiber. Only those modes of light signals,
which are funneled into the core within the acceptance angle, are sustained for
propagation through the fiber by means of TIR.
The optical signals from the other end of the fiber are fed to the phtodetector, where
the signals are converted into binary electrical signals.
Which are directed to decoder to convert the stream of binary electrical signals into
analog signal which will be the same information such as voice received by another
telephone user.
Note: As the optical signals propagating in the optical fiber are subjected to two types
of degradation – attenuation and delay distortion. Attenuation is the reduction in the
strength of the signal because of loss of optical power due to absorption, scattering of
photons and leakage of light due to fiber bends. Delay distortion is the reduction in the
quality of signal because of spreading of pulses with time.
These effects cause continuous degradation of signal as light propagates and hence it
may not possible to retrieve the information from the light signal. Therefore, a
repeater is needed in the transmission path. An optical repeater consists of receiver,
amplifier and transmitter.
ADVANTAGES of Optical Fiber:
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1. Optical fibers can carry very large amounts information. 2. The materials used for making optical fibers are silicon oxide and plastic,
both are available at low cost. 3. Because of the greater information carrying capacity by the fibers, the
cost, length, channel for the fiber would be lesser than that for the metallic cable.
4. Because of their compactness, and light weight, fibers are much easier to transport.
5. There is a possibility of interference between one communication channel and the other in case of metallic cables. However, the optical fiber are totally protected from interference between different communication signals, since, no light can enter a fiber from its sides. Because of which no cross talk takes place.
6. The radiation from lightning or sparking causes the disturbance in the signals which are transmitting in the metallic cable but cannot do for the fiber cable.
7. The information cannot be tapped from the optical fiber. 8. Since signal is optical, no sparks are generated as it could in case of
electrical signal. 9. Because of it superior attenuation characteristics, optical fibers support
signal transmission over long distances.’
Limitations of Optical fiber communications system:
1. Splicing is skilful task, which if not done precisely, the signal loss will be so much. The optic connectors, which are used to connect (splicing) two fibers are highly expensive.
2. While system modifications or because of accidents, a fiber may suffer line break. To establish the connections, it requires highly skilful and time consuming. Hence, maintenance cost is high.
3. Though fibers could be bent to circles of few centimeters radius, they may break when bent to still smaller curvatures. Also for small curvature bends, the loss becomes considerable.
4. Fibers undergo expansion and contraction with temperature that upset some critical alignments which lead to loss in signal power.
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MODULE III SOLID STATE PHYSICS Crystal Structure
A Crystal is a solid composed of atoms or other microscopic particles arranged in an
orderly repetitive array. The study of Crystal physics aims to interpret the macroscopic properties in terms of properties of the microscopic particles of which the solid is composed. The arrangement of atoms in a Crystal is called Crystal Structure.
Lattice points and Space Lattice: -
Points can be imagined in space about which atoms or molecules are located. Such points are called Lattice Points. The totality of such points is called Space Lattice or Crystal Lattice. A Three-Dimensional space lattice (3-D space lattice) may be defined as a finite array of lattice points in three-dimension such that each and every lattice point has identical surrounding in the array.
(2-D Space Lattice)
(Lattice point)
Lattice + Basis = Crystal Structure
Basis and Crystal structure: -Every lattice point can be associated with one or unit assembly of atoms or molecules identical in composition called Basis or Pattern. The regular periodic three-dimensional arrangement of Basis is called Crystal Structure. Space lattice is imaginary. Crystal structure is real.
Bravais and Non-Bravais lattice: - A Bravais lattice is one in which all lattice points are identical in composition. If the lattice points are not identical then lattice is called Non -Bravais lattice.
The set of lattice points ‘s together constitutes a Bravais lattice. Similarly the set of lattice points ‘s together constitutes a Bravais lattice. But set of all lattice points ‘s and ‘s together constitute a Non – Bravais lattice. Hence a Non-Bravais lattice could be considered as the superposed pattern of two or more interpenetrating Bravais lattices.
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Unit cell and Lattice parameters: - In every crystal some fundamental grouping of particles is repeated. Such fundamental grouping particles is called unit cell. A unit cell
is chosen to represent the symmetry of the crystal. Hence the unit cell with maximum
symmetry is chosen. They are the basic building blocks of the crystal. When these unit
cells are translated in three dimensions that will generate the crystal.
Each crystal lattice is described by type of unit cell. But each unit cell is described three vectors a, b and c when the length of the vectors and the angles (α,β,γ) between them are specified. They are nothing but the intercepts of the faces and the interfacial angles. All together they constitute lattice parameters.
Primitive Cell :-Some times reference is made to a primitive cell. Primitive cell
may be defined as a geometrical shape which, when repeated indefinitely in three dimensions, will fill all space and it consists of lattice points only at corners.
It consists of only one atom per cell. There fore unit cells may be primitive
(simple) or Non-primitive.
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Crystal systems and Bravais space lattices:- Based on lattice parameters crystals are classified into seven basic systems. If atoms are placed only at corners seven crystal systems yield seven lattices. But few more lattices could be constructed by placing atoms at face center, body center etc., Bravais showed that there are 14 such lattices exist in nature. Hence the name Bravais space lattices. Each crystal system differs from the other in lattice parameters.
1) Cubic Crystal system (Isometric) a = b = c and α = β = γ = 90°
(Simple or primitive, Face centered (FCC) and Body centered (BCC)),
2) Tetragonal Crystal system a = b ≠ c and α = β = γ = 90°
(Simple and Body centered)
3) Orthorhombic Crystal system a ≠ b ≠ c and α = β = γ = 90°
(Simple, Face centered (FCC), Body centered (BCC) and Base centered),
4) Monoclinic Crystal system a ≠ b ≠ c and α = β = 90° ≠ γ
(Simple and Base centered).
5) Triclinic Crystal system a ≠ b ≠ c and α ≠β ≠ γ ≠ 90°
(Simple).
6) Trigonal Crystal system (Rhombohedral) a = b = c and α = β = γ ≠ 90°
(Simple).
7) Hexagonal crystal system a = b ≠ c and α = β = 90°, γ =120°
(Simple).
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Directions and Planes: -
Directions: -In crystals there exists directions and planes in which contain concentration
of atoms. It is necessary to locate these directions and planes for crystal analysis. Arrows in two dimensions show directions. “The directions are described by giving
the coordinates of the first whole numbered point ((x, y) in two dimension,(x,y,z) in
three dimension) through which each of the direction passes”. Directions are enclosed within square brackets.
Planes: - The crystal may be regarded as made up of an aggregate of a set of parallel
equidistant planes, passing through the lattice points, which are known as lattice planes. These lattice planes can be chosen in different ways in a crystal. The problem in
designating these planes was solved by Miller who evolved a method to designate a set
of parallel planes in a crystal by three numbers (h k l) called Miller Indeces.
Different lattice planes
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Steps to determine miller Indeces of given set of parallel planes: - “Miller indices
may be defined as the reciprocals of the intercepts made by the plane on the crystallog4raphic axes when reduced to smallest numbers.
Consider a plane ABC which is one of the planes belonging to the set of parallel planes with miller indices (h k l). Let x, y and z be the intercepts made by the plane along the Three crystallographic axes X, Yand Z respectively.
1) Determine the coordinates of the intercepts made by the plane along the three crystallographic axes.
2) Express the intercepts as multiples of the unit cell dimensions, or lattice parameters along the axes
3) Determine the reciprocals of these numbers
4) Reduce them into the smallest set of integral numbers and enclose them in simple brackets. (No commas to be placed between indeces)
Eg,
1) The intercepts x=2a, y=2b & z=5c Generally x=pa, y=qb, z=rc.
2) The multiples of lattice parameters are
x 2a y
= = 2, = 2, &
z
= 5
a a b c
3) Taking the reciprocals
a 1
= ,
b 1
= ,
c 1
& =
x 2 y 2 z 5
4) Reducing the reciprocals to smallest set of integral numbers by taking LCM.
10 × 1 , 10 ×
1 , 10 ×
1
2 2 5
5 5 2
Miller indices of plane ABC = (h k l) =(5 5 2).
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Note: - a) All parallel equidistant planes have the same Miller indices.
b) If the Miller indices have the same ratio, then the planes are parallel.
c) If the plane is parallel to any of the axes, then the corresponding
Intercept is taken to be ∞.
Expression for Interplanar spacing in terms of Miller Indeces:
Consider a Lattice plane ABC, which is one of the planes belonging to the set of
planes with Miller indeces )( lkh . Let x, y and z be the intercepts made by the plane along the Three crystallographic axes X, Y and Z respectively.
Let OP be the perpendicular drawn form the origin to the plane. Let α|,β| and γ| be the angles made by OP with the crystallographic axes X, Y and Z respectively. Let another consecutive plane parallel to ABC pass through the origin. Let a, b and c be the lattice parameters. OP is called interplanar spacing and is denoted by dhkl.
From right angled triangle OCP
cos α|
cos β|
cos γ |
= OP
=
OC
= OP
=
OB
= OP
=
OA
8dhkl
x
8dhkl
y
8dhkl
z
but we know that
h = a , k = b & l =
c O
x y z
⇒ x = a , y =
b & z =
c
h k z
There fore
cos α| = h d
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hkl hkl
d
hkl
a hkl
cos β| = k d
b
cos γ| = h d
hkl
c hkl
for the rectangular Cartesian coordinate system we have (cos α|)2 + ( cos β|)2 + ( cos γ|)2 = 1
h2
d2 + d2
k 2
+ d2
l2
= 1
a2 b
2 c
2
⇒ 2
hkl
=
h2
1
k 2
l2
+ +
a2
d =
b2
1
c2
hkl h2 k 2 l2
+ +
a2
b2
c2
is the expression for Interplanar spacing. For a cubic lattice a=b=c, There fore
dhkil =
a
h2 + k 2 + l 2
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Definitions
1) Coordination number: - It is the number of equidistant nearest neighbors that an
atom has in a crystal structure.
2) Nearest neighbor distance: - It is the distance between two nearest neighbors in a crystal structure.
3) Atomic packing factor (APF) or Packing fraction: - It is the fraction of space occupied by atoms in a unit cell. It is defined as the ratio of volume occupied by atoms in unit cell to the volume of the unit cell. If the number of atoms per unit cell are ‘n’ and if Va is the volume of atoms in the unit cell and V is the volume of the unit cell then,
n × V
APF = a
V
4) Lattice Constant: - In a cubic lattice the distance between atoms remains constant
along crystallographic axes and is called Lattice Constant.
Simple Cubic Structure:
In simple cubic structure each atom consists of 6 equidistant nearest neighbors. Hence
its co-ordination number is 6.
Eight unit cells share each atom at the corner. Hence only 1/8th of the volume of the atom lies in each cell. Since the atoms are present only at corners, the No. of atoms per unit cell is given by
n = 1
× 8 = 1 atom
8
We know that the APF is given by
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n × V
APF = a
V
4πR 3
1 ×
APF = 3
a3
In this structure the atoms touch each other along the sides of the cube. There fore a = 2
R, Where R is the radius of each atom.
4πR 3
APF =
3(2R)3
APF =
4πR 3
3 (8 R 3 )
APF = 0.5235 Hence atoms occupy 52.35% off the volume of the unit cell.
Body Centered Cubic (BCC) Structure:
Each atom has 8 equidistant nearest neighbors. Hence the co-ordination number is 8.
Since there are eight atoms at corners and 1 atom at the body center, the no. of
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3
atoms per unit cell is given by
n = 1 + 1
× 8 = 2 atoms
8
Also in this structure the atoms touch each other along the body diagonal. There fore
(4R)2 = ( 2 a)2 + a2
Where R is the radius of the atom
16R 2 = 2 a2 + a2 = 3a2
2
a2 = 16R
3
a = 4R
3
Now the APF is given by
n × V
APF = a
V
4πR 3
1 ×
APF = 3
a3
4πR 3
1 ×
APF = 3
4R
3
APF = 3π
8
APF = 0.6802 Hence atoms occupy 68.02% of the volume of the unit
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Face Centered Cubic (FCC) Structure:
In FCC structure in addition to atoms at corners, atoms are present at face centers.
Each atom consists of 12 equidistant nearest neighbors. Hence the coordination number
is 12.
The number of atoms per unit cell is
n = 1
× 6 + 1
× 8 = 4 atoms
2 8
In this structure atoms touch each other along the face diagonal. There fore
(4R)2
16R 2
= a2 + a2
= a2 + a2
2
= 2a2
Where R is the atomic radius.
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a =
3
a2 = 16R
2
4R
= 2 2R
2
The APF is given by
APF = n×Va
V
4πR 3
1 ×
APF = 3
a3
1 × 4πR
APF = 3
(2 2R )3
APF = 2π
6
APF = 0.7405 Hence atoms occupy 74.05% of the volume of the unit cell.
X-Ray Diffraction
The wavelength of x-ray is of the order of Angstrom (Å). Hence optical grating
cannot be used to diffract X-rays. But the dimension of atoms is of the order of few angstroms and also atoms are arranged perfectly and regularly in the crystal. Hence
crystals provide an excellent facility to diffract x-rays.
Bragg’s X-Ray Diffraction and Bragg’s Law: -
Bragg considered crystal in terms of equidistant parallel planes in which there is regularity in arrangement of atoms. These are called as Bragg planes. There are different families of such planes exist in the crystal and are inclined to each other with certain angle.
In Bragg’s Diffraction the crystal is mounted such that an X-ray beam is inclined
on to the crystal at an angle θ. A detector scans through various angles for the diffracted
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X-rays. It shows peaks for (maximum current) for those angles at which constructive interference takes place. Bragg’s law gives the condition for constructive interference.
Derivation of Bragg’s Law:
Consider Monochromatic beam of X-Rays. It is incident on the crystal with glancing
angle ‘θ’. Ray AB, which is a part of the incident beam, is scattered by an atom at ‘B’ along BC. Similarly the ray DE is scattered by an atom in the next plane at ‘E’ along EF. The two scattered rays undergo constructive interference if path difference between the rays is equal to integral multiple of wavelength.
Construction: -Bp and BQ are the perpendiculars drawn as shown in the fig. The path difference δ = PE + EQ = nλ ----------- (1)
From Right angled triangle PBE
Sinθ = PE
BE
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Where BE = d (Interplanar spacing)=dhkl
Therefore PE= BE Sinθ = d Sinθ
Similarly From Right angled triangle QBE
QE = BE Sinθ = d Sinθ
Substituting in (1) δ = d Sinθ +d Sinθ = nλ δ = 2 d Sinθ = nλ
Therefore the condition for constructive interference is integral multiple of Wavelength of
X- Rays is
2 d Sinθ = n λ Hence Bragg’s Law.
Since Bragg diffraction satisfies the laws of reflection it is also called Bragg reflection.
Bragg’s X-ray Spectrometer(Determination of wavelength and Interplanar
spacing): -
It is an instrument devised by Bragg to study the diffraction of X-Rays using a crystal as Grating. It is based on the principle of Bragg Reflection.
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Construction: - Monochromatic X-Ray Beam from an X-Ray tube is collimated by slits
s1 and s2 and is incident on the crystal mounted on the turntable at a glancing angle θ. The crystal can be rotated using the turntable. The reflected X-Ray beam is again collimated by slits s3 and s4 and allowed to pass through ionization chamber fixed on the Mechanical Arm. Due to ionization in the medium current flows through the external circuit, which is recorded by the Quadrant Electro Meter (E). In order to satisfy the laws
of reflection the coupling between the turntable and the mechanical arm is so made that,
if the turntable is rotated through an angle θ then mechanical arm rotates through an angle 2θ.
Experiment: Rotating the turntable increases glancing angle. Ionization current is measured as a function of glancing angle. The Ionization current is plotted versus glancing angle. It is as shown below.
The angles corresponding to intensity maximum are noted . The lowest
angle θ, corresponding to maximum intensity corresponds to the path difference λ .
∴ 2d sinθ1=n1λ=λ
Similarly for next higher angles
2d sinθ2=n2λ=2λ
2d sinθ3=n3λ=3λ and so on…
⇒ sinθ1: sinθ2:sinθ3 = 1: 2 : 3 ………(1).
If equation (1) is satisfied for θ1,θ2, and θ3 etc. then the Bragg’s law is verified.
By determining θ using Bragg’s Spectrometer and by knowing the
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Value of Interplanar separation (d), Wavelength (λ) of X-Ray beam can be calculated.
By determining θ using Bragg’s Spectrometer and by knowing the value of
Wavelength (λ) of X-Ray beam, Interplanar separation (d) can be calculated.
Crystal structure of Sodium Chloride (NaCl): - NaCl is an ionic compound. Hence both Na and Cl are in ionic state. The molecule is under equilibrium because; the attractive force due to ions is balanced by repulsive force due to electron clouds.
The Bravais lattice of NaCl is FCC with the basis containing one Na ion and one Cl ion.
The bond length is 2.813Å. For each atom there are 6 equidistant nearest neighbors of opposite kind. Hence the co ordination number is 6. There are 12 next nearest neighbors
of the same kind. The conventional cell which consists of four molecules of NaCl is as shown in the figure. The coordinates of the ions in the conventional cell is as given below. (taking Na ion as origin)
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This structure could be considered as the superposed pattern of two interpenetrating
Bravais lattices each made of one type of ion.
Crystal structure of Diamond (Allotropic form of Carbon): - Diamond is an allotropic form of carbon. The Bravais lattice is an FCC similar to ZnS. There are 18 carbon atoms in the unit cell. 8 at corners, 6 at face centers and 4 at intermediate tetrahedral positions. The unit cell is as shown in the fig.
In the unit cell, each carbon atom bonds to four other carbon atoms in the form of a
tetrahedron. Since each atom has four equidistant nearest neighbors the coordination number is 4.This structure could be considered as the superposed pattern of two
interpenetrating Bravais FCC lattices each made of Carbon with one displaced from the other along 1/4th of the body diagonal. The interatomic distance is 1.54Å and the lattice constant 3.56Å.
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MODULE IV DIELECTRIC PROPERTIES OF SOLIDS DielectricsDielectricsDielectricsDielectrics are electrically non-conducting materials such as glass, porcelain etc, which exhibit remarkable behaviour because of the ability of the electric field to polarize the material creating electric dipoles. Dielectric ConstantDielectric ConstantDielectric ConstantDielectric Constant Faraday discovered that the capacitance of the condenser increases when the region between the plates is filled with dielectric. If C0 is the capacitance of the capacitor without dielectric and C is the capacitance of the capacitor with dielectric then the ratio C / C0 gives εεεεr called relative permittivity or Dielectric constant. Also for a given isotropic material the electric flux density is related to the applied field strength by the equation D D D D = ε EEEE, Where εεεε is Absolute permittivity. In SI system of units the relative permittivity is given by the ratio of absolute permittivity to permittivity of free space. ε = ε0 εr ε0 is permittivity of free space. εr is relative permittivity or dielectric constant. For an isotropic material, under static field conditions, the relative permittivity is called static dielectric constant. It depends on the structure of the atom of which the material is composed. Polarization of dielectricsPolarization of dielectricsPolarization of dielectricsPolarization of dielectrics: - “The displacement of charged particles in atoms or molecules of dielectric material so th“The displacement of charged particles in atoms or molecules of dielectric material so th“The displacement of charged particles in atoms or molecules of dielectric material so th“The displacement of charged particles in atoms or molecules of dielectric material so that at at at net dipole moment is developed net dipole moment is developed net dipole moment is developed net dipole moment is developed inininin the material along the applied field the material along the applied field the material along the applied field the material along the applied field direction direction direction direction is called polarization is called polarization is called polarization is called polarization of dielectric.” of dielectric.” of dielectric.” of dielectric.” Polarization is measured as dipole moment per unit volume and is a vector quantity. µ
rrNP= Where µ
r is average dipole moment per molecule and N is number of molecules per unit volume. Also Err
αµ = where α is a constant of proportionality called polarizability.
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In Polar dielectric materials, When the external electric field is applied all dipoles tend to align in the field direction and hence net dipole moment develops cross dielectric material. This is the polarization of polar dielectric materialspolarization of polar dielectric materialspolarization of polar dielectric materialspolarization of polar dielectric materials. In non polar dielectric materials dipoles are induced due to the applied electric field which results in the net dipole moment in the dielectric material in the direction of the applied field. This is the polarization of nonpolarization of nonpolarization of nonpolarization of non----polar dielectric materialspolar dielectric materialspolar dielectric materialspolar dielectric materials. As the polarization measures the additional flux density arising from the presence of the material as compared to free space it has the same unit as D and is related to it as litysusceptibiElectricisWhere
E
p
Also
EP
PEE
EDSince
PED
r
r
r
r
χ
χεε
εε
εεε
εε
ε
=−=
−=∴
+=
=
+=
)1(
)1(
0
0
00
0
0 Electrical Polarization mechanismsElectrical Polarization mechanismsElectrical Polarization mechanismsElectrical Polarization mechanisms The electrical polarization takes place through four different mechanisms. They are 1. Electronic polarization. 2. Ionic polarization. 3. Orientation polarization. 4. Space charge polarization. The net polarization of the material is due to the contribution of all four polarization mechanisms. PPPP = P = P = P = Peeee + P + P + P + Piiii + P + P + P + Poooo + P + P + P + Pssss 1) 1) 1) 1) Electronic polarizationElectronic polarizationElectronic polarizationElectronic polarization: - This occurs through out the dielectric material and is due to the separation of effective centers of positive charges from the effective center of negative charges in atoms or molecules of dielectric material due to applied electric field. Hence dipoles are induced within the material. This leads to the development of net dipole moment in the material and is the vector sum of dipole moments of individual dipoles.
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2)2)2)2) Ionic polari Ionic polari Ionic polari Ionic polarizationzationzationzation: - This occurs in ionic solids such as sodium chloride etc. Ionic solids possess net dipole moment even in the absence of external electric field. But when the external electric field is applied the separation between the ions further increases. Hence the net dipole moment of the material also increases. It is found that the ionic dipole moment also proportional to the applied field strength. .litypolarisabiioniciswhere
Ehence
i
ii
α
α=µ Ionic Polarization is given by ionic dipole moment per unit volume. .volumeunitperatomsof.noisNwhere
Nphence ii µ= 3)3)3)3) Orientation Polarization Orientation Polarization Orientation Polarization Orientation Polarization: - This occurs in polar dielectric material, which possesses permanent electric dipoles. In polar dielectrics the dipoles are randomly oriented due thermal agitation.
-
+
-
-
-
+
+
+
+
+
+ +
-
- -
-
+ +
E = 0. E > 0.
( )N
bygivenislitypolarizabielectronicThe
mperatomsofnumberNiswhereENNP
litypolarizabielectronicwhere
EE
ERmomentdipoleelectronic
re
eee
e
eee
e
1
.
4
0
3
3
0
−=
==
=⇒∝⇒
==⇒
εεα
αµ
α
αµµ
πεµ
E<0 E>0
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Therefore net dipole moment of the material is zero. But when the external electric field is applied all dipoles tend to align in the field direction. There fore dipole moment develops across the material. This is referred to as orientation polarization (PPPPoooo). Orientation polarization depends on temperature. Higher the temperature more the randomness in dipole orientation smaller will be the dipole moment. The orientation polarizability is given by Tk3
2
0
µα =
4)4)4)4) Space charge polarization Space charge polarization Space charge polarization Space charge polarization: - This occurs in materials in which only a few charge carriers are capable of moving through small distances. When the external electric field is applied these charge carriers move. During their motion they get trapped or pile up against lattice defects. These are called localized charges. These localized charges induce their image charge on the surface of the dielectric material. This leads to the development of net dipole moment across the material. Since this is very small it can be neglected. It is denoted by PPPPssss. Internal Field or Local FieldInternal Field or Local FieldInternal Field or Local FieldInternal Field or Local Field:- When dielectric material is placed in the external electric field, it is polarized creating electric dipoles. Each dipole sets electric field in the vicinity. Hence the net electric field at any point within the dielectric material is given by “The sum of external field and the “The sum of external field and the “The sum of external field and the “The sum of external field and the field due to all dipoles surrounding that point”field due to all dipoles surrounding that point”field due to all dipoles surrounding that point”field due to all dipoles surrounding that point”. This net field is called internal fieldinternal fieldinternal fieldinternal field or Local fieldLocal fieldLocal fieldLocal field. Expression for Internal field in case of Solids and LiquidsExpression for Internal field in case of Solids and LiquidsExpression for Internal field in case of Solids and LiquidsExpression for Internal field in case of Solids and Liquids ( ( ( (One dimensional One dimensional One dimensional One dimensional )))): Consider a dipole with charges ‘+q ‘and ‘–q’ separated by a small distance ‘dx’ as shown in fig. The dipole moment is given by µ = q dx. Consider a point ‘P’ at a distance ‘r’ from the center of dipole.
E = 0. E > 0.
Where ‘kkkk’ is Boltzman constant, T T T T is absolute temperature and µµµµ is permanent dipole moment.
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The electric field ‘E’ at ‘P’ can be resolved into two components. 1) The Radial Component along the line joining the dipole and the point. It is given by 3
0
r r4
Cos2E
επ
θµ=
r 2) The Tangential component or Transverse component perpendicular to the Radial component is given by 3
0 r4
SinE
επ
θµ=
θ
r Where ‘θ’ is the angle between the dipole and the line joining the dipole with the point ’P’, ‘ε0’ is permitivity of free space and ‘r’ is the distance between the point and dipole. Consider a dielectric material placed in external electric field of strength ‘E’. Consider an array of equidistant dipoles within the dielectric material, which are aligned in the field direction as shown in the figure. E C A X B D F
a a a a a
2a 2a
3a 3a
θ
r
P
Err
Eθ
E
dx +q -q
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Let us find the local field at ‘X’ due all dipoles in the Array. The field at ‘X’ due to dipole ‘A’ is given by θ
+= EEE rXA
3
0
XA
3
0
r
a2E
.0E,a4
0Cos2EHence
0andarHere
επ
µ=⇒
=επ
µ=
=θ=
θ
o
o The field at ‘X’ due to dipole ‘B’ is given by
θEEErXB
+= ( )
30
XB
3
0
r
a2E
.0E,a4
180Cos2EHence
180andarHere
επ
µ=⇒
=−επ
µ=
=θ−=
θ
o
o Hence the Total field at ‘X’ due to equidistant dipoles ‘A’ and ‘B’ is given by
XBXA1 EEE += 3
0
1
3
0
3
0
1
aE
a2a2E
επ
µ=⇒
επ
µ+
επ
µ=
Similarly, the total field at ‘X’ due to equidistant dipoles ‘C’ and ‘E’ is given by XDXC2 EEE += ( )a2r
)a2(E
)a2(2)a2(2E
3
0
2
3
0
3
0
2
=επ
µ=⇒
επ
µ+
επ
µ=
Q
Similarly, the total field at ‘X’ due to equidistant dipoles ‘D’ and ‘F’ is given by XFXE3 EEE +=
( )a3r)a3(
E
)a3(2)a3(2E
3
0
3
3
0
3
0
3
=επ
µ=⇒
επ
µ+
επ
µ=
Q
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The net field at ‘X’ due to all dipoles in the array is given by
3
0
|
1n3
1n33
0
333
0
3
0
3
0
3
0
|
4321
|
a
2.1E
2.1n
1but
n
1
a
3
1
2
11
a
)a3()a2(aE
EEEEE
επ
µ=∴
≅
επ
µ=
+++
επ
µ=
+επ
µ+
επ
µ+
επ
µ=
++++=
∑
∑
∞
=
∞
=
LLL
LLL
LLL
3
0
i
3
0
|
i
a
E2.1EE
.EWkt
a
2.1EEEE
bygivenis'X'atFieldLocalThe
επ
α+=∴
α=µ
επ
µ+=+=
0
0
ε
γα
ε
αγ
PEEENPonpolarizatibut
ENEE
i
i
+=∴=
+=∴
For Three-Dimension the above equation can be generalized by replacing 1/a3 by ‘N’ (where ‘N’ the number of atoms per unit volume), and 1.2/π by γ called Internal Field Constant.
Since γ, Ρ and ε0 are positive quantities Ei > E. For a Cubic Lattice γ = 1/3 and the Local field is called Lorentz field. It is given by 03ε
PEE L +=
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ClausiusClausiusClausiusClausius----Mosotti Relation: Mosotti Relation: Mosotti Relation: Mosotti Relation: Consider an Elemental solid dielectric material. Since they don’t posses permanent, dipoles, for such materials, the ionic and orientation polarizabilities are zero. Hence the polarization P is given by )1(...............
31
31
3
3
0
0
0
0
eqnN
ENP
ENN
P
PNENP
PEN
FieldLorentzisEWhereENP
e
e
e
e
ee
e
LLe
−
=∴
=
−
+=
+=
=
ε
α
α
αε
α
εαα
εα
α Where ‘N’ is the no. of dipoles per unit volume, eα is electronic polarizability 0ε is permittivity of free space, and E is the Electric field strength. The polarization is related to the applied field strength as given below
)2(...................)1(0
00
0
0
eqnEP
PEE
EDSince
PED
r
r
r
−=∴
+=
=
+=
εε
εεε
εε
ε Where ‘D’ is Electric Flux Density and rε is Dielectric Constant. Equating equations (1) and (2) E
N
ENP r
e
e )1(
31
0
0
−=
−
=∴ εε
ε
α
α 1
3)1(
31
)1(
00
00
=+−
−=
−
ε
α
εε
α
ε
α
εε
α
e
r
e
e
r
e
N
E
EN
N
E
EN
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+
−=∴
=
+
−
2
1
3
11)1(
3
3
0
0
r
re
r
e
N
N
ε
ε
ε
α
εε
α
The above equation is called Clausius – Mosotti relation. Using the above relation if the value of dielectric constant of the material is known then the electronic polarizability can be determined. Properties of Dielectrics under alternating field conditions( Frequency dependence of Dielectric Properties of Dielectrics under alternating field conditions( Frequency dependence of Dielectric Properties of Dielectrics under alternating field conditions( Frequency dependence of Dielectric Properties of Dielectrics under alternating field conditions( Frequency dependence of Dielectric constant, Dielectric loss & comconstant, Dielectric loss & comconstant, Dielectric loss & comconstant, Dielectric loss & complex dielectric constant)plex dielectric constant)plex dielectric constant)plex dielectric constant)::::- It is found under alternating field conditions of high frequency, the dielectric constant is a complex quantity. When dielectric materials are placed in alternating field the polarization tend to reverse as the polarity changes. If the frequency of the field is low (less than 1M Hz), then the polarization can follow the alternations of the field and hence the dielectric constant remains static. Under alternating field conditions of high frequency (Greater than 1MHz) the oscillations of dipoles lag behind those of field. If the frequency is increased further they are completely unable to follow the alternations in the field and hence the molecular process Orientation polarization ceases due to dielectric relaxation. This occurs in the frequency range 106 Hz to 1011Hz.As the frequency is increased further other polarizing mechanisms start to cease one after another. The ionic polarization ceases in the frequency range 1011 Hz to 1014Hz. Finally only electronic polarization Frequency
Polariza
tion
Space Charge
Orientation
Ionic
Electronic
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remains. Hence as the frequency of the field increases the polarization decreases and hence the dielectric constant decreases. This is known as Anomalous Dielectric DispersionAnomalous Dielectric DispersionAnomalous Dielectric DispersionAnomalous Dielectric Dispersion. Dielectric LossDielectric LossDielectric LossDielectric Loss: - In the alternating field conditions during the rotation of dipoles they have to overcome some sort of internal friction, which is dissipated as heat by the material. This is called as dielectric loss. Complex Dielectric ConstantComplex Dielectric ConstantComplex Dielectric ConstantComplex Dielectric Constant: - The complex dielectric constant is given by εr* =εr’ - εr’’. Where εr’’ determines Dielectric Loss. εr’ determines the component of current out of phase by 90° with the field. Important applications of Dielectric Materials: Important applications of Dielectric Materials: Important applications of Dielectric Materials: Important applications of Dielectric Materials: Dielectric materials find a wide range of applications as insulating materials. 1) Plastic and Rubber dielectric are used for the insulations of electrical conductors 2) Ceramic beads are used for the prevention of electric short circuiting and also for the purpose of insulation. 3) Mica and Asbestos insulation is provided in electric Iron in order to prevent the flow of electric current to outer body. 4) Varnished cotton is used insulators in transformers. 5) Dielectric materials are used in the energy storage capacitors. 6) Piezoelectric crystals are used in oscillators and vibrators.
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MODULE V ELECTRICAL CONDUCTIVITY IN METALS
Classical free electron theory of metals: In order to explain electrical conductivity in metals, Lorentz and Drude put forward a theory called free electron theory of metals. It is based on the following assumptions. 1. Free electrons in a metal resemble molecules of a gas. Therefore, Laws of kinetic theory of gasses are applicable to free electrons also. Thus free electrons can be assigned with “average velocity c”, ‘Mean free path λ” and “mean collision time τ”. 2. The motion of an electron is completely random. In the absence of electric field, number of electrons crossing any cross section of a conductor in one direction is equal to number of electrons crossing the same cross section in opposite direction. Therefore net electric current is Zero. 3. The random motion of the electron is due to thermal energy. Average kinetic energy of the electron is given by
Where c=average velocity or thermal velocity
4. Electric current in the conductor is due to the drift velocity acquired by the electrons in the presence of the applied electric field. 5. Electric field produced by lattice ions is assumed to be uniform throughout the solid. 6. The force of repulsion between the electrons and force of attraction between electrons and lattice ions is neglected. The Drift Velocity :
In the absence of the applied electric field, motion of free electron is completely random. During their motion electrons undergo collisions with the residual ions and during each collision direction and magnitude of their velocity changes in general. When electric field is applied, electrons experiences force in the direction opposite to the applied field. Therefore in addition to their random velocity, electron acquires velocity in the direction of the force. Since electrons continue to move in their random direction, with only a drift motion due to applied field, velocity acquired by the electrons in the direction opposite to the applied field is called Drift velocity and is denoted by vd. Note the vd is very small compared to c, the average thermal velocity. Electric current in a conductor is primarily due to the drift velocity of the electrons. Electronic Conduction in Solids
Relaxation Time, Mean collision time and Mean free path:
Mean free path (λ): The average distance traveled by electrons between two successive collisions during their random motion is called mean free path, it denoted by λ. Mean collision Time(ττττ): The average time taken by an electrons between two successive collisions during their random motion is called mean collision time, it is denoted by ττττ. The relationship between λ and ττττ is given by λ=cττττ.
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Relaxation Time(ττττr): In the presence of an applied electric field, electrons acquire drift velocity vd in addition to the thermal velocity c. if electric field is switched off, vd reduces and becomes zero after some time. Let electric field is switched off at the instant t = 0, when drift velocity vd=v0. The drift velocity of the electron after the lapse of ‘t’ seconds is given by
Where ττττr is called relaxation time. Suppose t=ττττr , then vd=v0 , e
-1=1/ev0
Thus the relaxation time is defined as the time during which drift velocity
reduces to 1/e times its maximum value after the electric filed is switched off.
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HALL EFFECT
Suppose a material carrying an electric current is placed in a magnetic field. Then an electric field is produced inside the material in a direction which is at right angles to both the current and magnetic field. This is called Hall effect discovered by Edwin H. Hall in 1879.
Let us consider a metal. Let the current J be in the X direction as shown in figure. The current carriers which are the free electrons in the metal have a drift velocity v in the negative X direction. Let a magnetic field B be applied along the Z direction. Then the electrons will experience a force of magnitude Bev. The direction of the force is by Fleming’s left hand rule in the negative Y direction. This force is called Lorentz force. Under the action of this force the free electrons will move in the negative Y direction and collect on the bottom surface of the material. There will be a deficit of electrons on the top surface. This is equivalent to a collection of positive charges on the top surface. The collection of positive
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and negative charges on the top and bottom faces of the material results in an electric field in the negative Y direction. This electric field is called Hall field. If its magnitude is EH the force due to it on an electron is eEH in the positive Y direction. This force opposes the Lorentz force. Soon an equilibrium condition will set up when
BeveEH = …….(1)
If the current density in the X direction is J then,
nevJ = ……..(2)
Where n is the concentration of the free electrons. Then by combining (1) & (2) we get
Or JBRE HH =
Where ne
R H
1= is called the Hall coefficient.
ne
BJEH =
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MODULE VI
MAGNETIC PROPERTIES OF SOLIDS
Classification of Diamagnetic, Paramagnetic and ferromagnetic materials
Diamagnetic Materials
The substances which tend to move from stronger to weaker region of the magnetic field
are called diamagnetic substances. These are weakly repelled by a strong magnet. In these
substances, there are equal numbers of electrons spinning and orbiting in opposite
directions so that the electrons of an atom remain paired in such away that their net
magnetic moment is zero in the absence of any external magnetic field.
Examples: Antimony, bismuth, copper, gold, lead, mercury, silver etc…
Properties
1. The relative permeability is less than unity
2. The susceptibility is negative
3. The susceptibility does not vary with temperature.
Paramagnetic Materials
The substances which tend to move from weaker to stronger regions of magnetic field are
called paramagnetic substances. These are feebly attracted by a strong magnet. In these
substances, the atoms or molecules with one or more unpaired electrons possess a
permanent magnetic moment. But they are oriented randomly in the absence of an external
magnetic field.
Examples: Aluminium, chromium, manganese, platinum, sodium etc…
Properties
1. Low magnetization, low susceptibility
2. Susceptibility is small, positive and varies inversely with temperature
3. Permeability is greater than one
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Paramagnetic susceptibility χ = T
C Where C is curie constant
This result namely the magnetic susceptibility of atoms varies as T
1 is known as Curies law.
Ferromagnetic materials
The substances which are strongly attracted by a magnet are called ferromagnetic substances. These
are the permanent magnets which exhibit hysteresis. It arises due to the self alignment of groups of
atoms carrying permanent magnetic moment in the same direction. The magnetic moment is an
account of spin of the electrons. Ferromagnetic materials are characterized by curie temperature
above which they become paramagnetic materials.
Examples: iron, cobalt, nickel and their alloys.
Properties
1. The relative permeability is very high
2. The susceptibility is very high and it depends on the temperature.
3. They exhibit magnetostriction and hysteresis
Ferromagnetic susceptibility χ = ( )θ−T
C Where C is curie constant
B H Curve ( Hysterisis)
Bsat Br
-Hc
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1. A ferromagnetic solid can be assumed to be comprised of small number of small
regions called domains each of which is spontaneously magnetized. The magnetic
moments of all the exams are all aligned in a particular direction. However the
different domains are so oriented as to make the net magnetization zero.
2. The process of magnetization consists in rotating the different domains in the
direction of applied field so that the specimen exhibits net magnetization.
3. When a magnetic field is applied on a ferromagnetic material, the domains nearly
parallel to H can grow in size at the expense of antiparallel domains and gradually all
the domains align along the applied field at which the material is said to be
saturated.
The hysteresis can be explained as follows.
1. As H is further increased, the rate of increase of B falls and ultimately becomes zero
and the flux density B reaches a saturation value indicated as Bsat in the figure.
2. As the applied field H is reduced from the saturation value to zero, the reduction of
flux density does not follow the same path.
3. When H becomes zero, their remains certain amount of flux in the material called the
remnant flux density Br . The material remains magnetized even in the absence of an
external field.
4. To reduce the remnant flux Br to zero, it is necessary to apply H in the reverse
direction and the amount Hc required to make Br zero is called the coercive force.
5. As the field is increased beyond Hc , the flux density reaches saturation.
6. When a ferromagnetic material is taken over one cycle of magnetic field, it exhibits
hystresis loop.
7. The area of the hystresis loop signifies the amount of energy required for
magnetization.
Soft magnetic materials:
Magnetic materials that have high permeability and low coercive force are called soft
magnetic materials.
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These are temporary magnets which can retain magnetism for a short interval of time. They
possess small hysteresis loss.
Properties:
1. Low remnant magnetism
2. High permeability
3. High susceptibility
4. Low coercivity
5. Low hysteresis energy loss
6. Thin hysteresis loop
7. Low eddy current loss
Ex: Soft iron, mild Steel, Sendust, perm alloy, pure nickel
Uses:
1. They are used in the construction of cores of transformers.
2. Relatively pure iron is frequently used as the magnetic core for direct current
applications.
3. The magnetic mild steel is used for relays, reed switches and pole pieces for
electromagnets.
4. Iron-nickel alloys are used for audio frequency applications.
Hard magnetic materials :
Magnetic materials that have large energy product (BH) and high coercivity are called hard
magnetic materials.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 94
Properties:
1. Large hysteresis loop
2. High coercivity
3. High remanent magnetization
4. High permeability
5. High hysteresis energy loss
6. High saturation flux density
Ex: Carbon, tungsten-steel alloy, alloy steel, alloys of aluminium
Uses:
1. The permanent magnets are used in instruments like galvanometers, ammeters,
voltmeters, speedometers and recorders.
2. Tungsten steel are used in chucks and latches, tool holders, magnetic bearings and
mixers.
3. They are used in electronic devices such as telephones and tape recorders, loud speakers
and hearing aids.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 95
Superconductivity:
Temperature dependence of resistivity of metal
Thus net Resistivity of a metal can be written as
ρ = ρ 0+ ρ (T)
Thus net Resistivity of conductor is equal to sum temperature independent part and
temperature dependent part as shown in the graph
Superconductivity
Kamerlingh Onnes discovered the phenomenon of superconductivity in the year 1911. When
he was studying the temperature dependence of resistance of Mercury at very low
temperature he found that resistance of Mercury decreases with the decrease in
temperature up to a particular temperature Tc= 4.15K . Below this temperature the
resistance of mercury abruptly drops to zero. Between 4.15K and Zero degree Kelvin
Mercury offered no resistance for the flow of electric current. The phenomenon is reversible
and material becomes normal once again when temperature was increased above 4.15K. He
called this phenomenon as superconductivity and material which exhibited this property as
superconductors.
The variation of resistivity with temperature for a metal is as shown in the fig. Resistivity in the case of pure metal decreases with the decrease in temperature and becomes zero at absolute zero temperature. While in the case of impure metals the resistivity of metal will have some residual value even at absolute zero temperature. This residual resistance depends only on the amount of impurity present in the metal and is independent of the temperature.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 96
Thus the phenomenon of super conductivity is defined as:
“The phenomenon in which resistance of certain metals, alloys and compounds
drops to zero abruptly, below certain temperature is called superconductivity
The temperature, below which materials exhibit superconducting property is called
critical temperature, denoted by Tc. Critical temperature Tc is different for different
substances. The materials, which exhibit superconducting property, are called
Superconductors.
Above critical temperature material is said to be in normal state and offers resistance
for the flow of electric current. Below critical temperature material is said to be in
superconducting state. Thus Tc is also called as transition temperature.
Meissner Effect
In 1933, Meissner and Ochsenfeld showed that when a superconducting material is
placed in a magnetic field, it allows magnetic lines of force to pass through, if it’s
temperature is above Tc and if temperature is reduced below the critical temperature Tc, it
expels all the lines of force completely out of the specimen to become a perfect diamagnetic
material. This is known as Meissner effect.
Since superconductor exhibits perfect diamagnetism below the critical temperature Tc,
magnetic flux density inside the material is zero.
Therefore B=0, for T< Tc
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 97
Relationship between flux density and the strength of the magnetizing field is given by
µµµµ0 = Absolute permeability of free space
M = Intensity of magnetization of the material &
H = Strength of the magnetizing field
Thus superconductor possesses negative magnetic moment when it is superconducting
state.
Consider a superconducting material above its critical temperature. A primary coil and
secondary coil are wound on the material. The primary coil is connected to battery and the
secondary coil is connected to a Ballistic Galvanometer. When the primary circuit is closed,
current flows through it, which sets up a magnetic field in it.
The magnetic flux immediately links with the secondary coil. This change in flux across
the secondary coil produces the momentary current and hence the galvanometer shows the
deflection.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 98
After certain time, the primary current becomes steady; flux linkage with the secondary
coil becomes unchanging. As a result, no change in the flux linkage in the secondary coil
and hence no more current driven in the secondary circuit.
Now, decreases the temperature of the super conductor gradually. As soon as the
temperature crosses below the critical temperature, the B.G. suddenly shows a deflection,
indicating that the flux linkage with the secondary coil has changed.
The change in flux linkage is attributed to the expulsion of the magnetic flux
from the body of the superconducting material as shown in figure.
Critical field
We know that when superconductor is placed in a magnetic field it expels magnetic lines
of force completely out of the body and becomes a perfect diamagnet. But if the strength of
the magnetic field is further increased, it was found that for a particular value of the
magnetic field, material looses its superconducting property and becomes a normal
conductor. The value of the magnetic field at which superconductivity is destroyed is called
the Critical magnetic field, denoted by Hc. It was found that by reducing the temperature
of the material further superconducting property of the material could be restored. Thus,
critical field doesn’t destroy the superconducting property of the material completely but
only reduces the critical temperature of the material
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 99
Critical magnetic field Hc depends on the temperature of the material. The relationship
between the two is given by
Isotopic effect
The variation of critical temperature of an element with isotopic mass is called isotopic
effect.
The transition temperature is inversely proportional to the square root of the atomic mass of
the isotope of a single superconductor.
aC
MT
1α a= constant equal to ½, M= atomic weight
Critical current density
The application of a large value of electric current to superconducting material destroys the
superconducting property. Consider a coil of wire wound on superconductor .Let I be the
current Flowing through the wire. The application of the current induces a magnetic field.
Thus, induced Magnetic field in the conductor destroys the superconducting property.
The induced critical current is given by
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 100
Ic = 2πrHc Ic=critical current density
Hc= critical field
BCS theory of Superconductivity
Bardeen, Cooper and Schrieffer explained the phenomenon of superconductivity in the
year 1957. The essence of the BCS theory is as follows.
We know that resistance of the conductor is due to the scattering of electrons from the
lattice ions.
Consider an electron moving very close to a lattice ion. Due to coulomb interaction
between electron and ion, the ion core gets distorted from its mean position. It is called
lattice distortion. Now another electron moving close to this lattice ion interacts with it.
This results in the reduction in the energy of the electron. This interaction can be looked
upon as equivalent to the interaction between two electrons via lattice. During the
interaction exchange of phonon takes place between electron and the lattice. This
interaction is called electron-lattice –electron interaction via the phonon field.
Because of the reduction in energy between the two electrons, an attractive force comes
into effect between two electrons. It was shown by Cooper that, this attractive force
becomes maximum if two electrons have opposite spins and momentum. The attractive
force may exceed coulombs repulsive force between the two electrons below the critical
temperature, which results in the formation of bound pair of electrons called cooper pairs.
At temperatures below the critical temperature large number of electron lattice- electron
interaction takes place and all electrons form a cloud of cooper pairs.
Cooper pairs in turn move in a cohesive manner through the crystal, which results in an
ordered state of the conduction electrons without any scattering on the lattice ions. This
results in a state of zero resistance in the material.
Persistent current
Once a current is started in a closed loop of superconducting material, it will continue to
keep flowing of its own accord, around the loop as long as the loop is held below the critical
temperature. Such a steady current Which flows with undiminishing strength is called a
persistent current. The persistent current does not need external power to maintain it
because there does not exist I2R losses. In one instance, a current is maintained in a
superconducting loop for more than two years. Persistent current is one of the most
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 101
important properties of a superconductor. Superconductor coils with persistent currents
produce magnetic fields and can therefore serve as magnets. Such a superconducting
magnet does not require a power supply to maintain its magnetic field.
Types of Superconductors
Type I or Soft Superconductors:
Superconducting materials, which exhibit, complete Meissner effect are called Soft
superconductors.
We know that below critical temperature, superconductors exhibit perfect diamagnetism.
Therefore they possess negative magnetic moment.
The graph of magnetic moment Vs magnetic field is as shown in the Fig. As field strength
increases material becomes more and more diamagnetic until H becomes equal to Hc. At
Hc, material losses both diamagnetic and Superconducting properties to become normal
conductor. It allows magnetic flux to penetrate through its body. The value of Hc is very
small for soft superconductors. Therefore soft superconductors cannot withstand high
magnetic fields. Therefore they cannot be used for making superconducting magnets
Type II or Hard Superconductors
Superconducting materials, which can withstand high value of critical magnetic fields, are
called Hard Superconductors.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 102
The graph of magnetic moment Vs magnetic field is as shown in the Fig. Hard
superconductors are characterized by two critical fields Hc1 and Hc2. When applied
magnetic field is less than Hc1 material exhibits perfect diamagnetism. Beyond Hc1 flux
penetrates and fills the body partially. As the strength of the field increases further, more
and more flux fills the body and thereby decreasing the diamagnetic property of the
material. At Hc2 flux fills the body completely and material losses its diamagnetic property
as well as superconducting property completely.
Between Hc1 and Hc2 material is said to be in vortex state. In this state though there is
flux penetration, material exhibits superconducting property. Thus flux penetration occurs
through small-channelised regions called filaments. In filament region material is in normal
state. As Hc2 the field strength increases width of the filament region increases at they
spread in to the entire body, and material becomes normal conductor as a whole. The value
of Hc2 is hundreds of times greater than Hc of soft superconductors. Therefore they are
used for making powerful superconducting magnets.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 103
Applications of Superconductivity
1. Superconducting Magnets:
We know that in ordinary electromagnet strength of the magnetic field produced depends
on the number of turns (N) in the winding and the strength of the current (I) flowing
through the winding. To produce strong magnetic field either N or I should be increased. If
N is increased size of the magnet increases and if I is increased power loss (I2R) increases,
which results in production of heat. Therefore there are limitations to increase N and I. If
superconducting wires are used for winding in electromagnets, even with small number of
turns strong magnetic fields can be produced by passing large current through the winding,
because there is no loss of power in superconductors. The type II superconductors, which
have high Hc and Tc values, are commonly used in superconducting magnets. Ex: Niobium-
tin, Niobium-aluminum, niobium-germanium and vanadium-gallium alloys.
The superconducting magnets are used in Magnetic Resonance Imaging
(MRI) systems, for plasma confinement in fusion reactors, in magneto-hydrodynamic power
generation, in Maglev vehicles, etc.
2. Maglev (Magnetically Levitated vehicles)
1. In these vehicles transportation is by setting afloat the carriage above the guide
way.
2. Utility of such levitation is that, in the absence of contact between the moving and
stationary systems, the friction is eliminated.
3. Great speed is achieved with low energy consumption.
4. The phenomenon is based on Meissner effect.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 104
5. The vehicle consists of superconducting magnets built into its base. There is an
Aluminium guide way over which the vehicle will be set afloat by magnetic
levitation. This is brought about by enormous repulsion between two powerful
magnetic fields, one produced by superconducting magnet inside the vehicle and
the other one by electric currents in the Aluminium guide way.
6. Guide way is divided into a number of segments provided with coils. The flow of
currents through the coils could be related to the position and instantaneous speed
of the vehicle.
7. The currents in the guide way not only produce the necessary magnetic field to
levitate the vehicle but also help in propelling the vehicle forward.
8. The vehicle is provided with retractable wheels. With these wheels the vehicle runs
on the guide way. Once levitated in air the wheels are retracted into the body,
while stopping, the wheels are drawn out and vehicle slowly settles on the guide
way by running over a distance.
3. SQUID (Superconducting Quantum Interference Device)
It is an ultra-sensitive measuring instrument used for the detection of very weak
magnetic fields. (~ 10-14T)
It is formed by incorporating two Josephson’s junctions (J1 and J2) in the loop of a
superconducting material.
An arrangement consisting of two superconductors separated by a thin oxide layer
(insulator) is known as Josephson’s junction. When the wavelength of the matter wave
generated by the cooper pair is greater than barrier width, cooper pairs from one
superconductor can tunnel through the barrier and reach the other superconductor.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 105
When the magnetic field is applied to this superconducting circuit, it induces a
circulating current which produces just that much opposing magnetic field as to exclude the
flux from the loop.
The flux remains excluded so long as the junction currents do not exceed a critical
value. (Where the critical current IC, which is the maximum current across the junction
under zero voltage condition)
But the circuit switches to resistive phase and thereby the flux passes into the loop.
Once, the current in either of the junctions or in both exceed the critical value, the loop acts
like a gate to allow or exclude the flux.
A mathematical analysis shows that, IC that is supported in the loop is a periodic
function of the applied magnetic flux (φ ).
It can be interpreted as the consequence of interference due to the phase difference
between the reunited currents.
The phase difference is caused by the applied magnetic field. Here both the
interference effects and the quantization effects in superconductivity state are involved.
Thus the device is named as SQUID.
Applications:
When the squid is brought under the influence of the external field, the flux through
the loop changes. It causes a change in squid current Is. The variation in Is will be periodic
in nature and hence induces an emf in an adjacent coil of an electric circuit that senses
changes in the flux.
Peridoic variation of the critical current IC with
the total flux through the area of the squid.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 106
It can be used to detect magnetic fields of heart and brain.
It can be used for logic operations in an electronic circuit as well as memory devices.
Temperature dependence of specific heat
The specific heat of the normal metal is seen to be of the form.
( ) 3TTTCn βγ ==
The first term in Eq. is the specific heat of electrons in the metal and the second
term is the contribution of lattice vibrations at low temperatures. The specific heat of the
superconductor shows a jump at Tc. Since the superconductivity affects electrons mainly, it
is natural to assume that the lattice vibration part remains unaffected, i.e., it has the same
value βT3 in the normal and superconducting states. On subtracting this, we notice that the
electronic specific heat Ces is not linear with temperature. It rather fits an exponential form.
( ) ( )TKATC Bes ∆−= exp
Temperature dependence of the electronic specific heat in the normal and
superconducting states
This exponential form is an indication of the existence of a finite gap is the energy spectrum
of electrons separating the ground state from the lowest state.
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 107
The number of electrons thermally excited across the gap varies exponentially with the
reciprocal of temperature. The energy gap is believed to be a characteristic feature of the
superconducting state which determines the thermal properties as well as high frequency
electromagnetic response of all superconductors.
(a) (b)
(a) Conduction band in the normal state.
(b) Energy gap at the Fermi level in the superconducting state E=10 eV.
Thermal Conductivity
The thermal conductivity of superconductors undergoes a continuous change between
the two phases and is usually lower in the superconducting phase suggesting that the
electronic contribution drops, the superconducting electrons possibly playing no part in heat
transfer.
The thermal conductivity of tin at 2 K is 34W cm-1 K-1 for the normal phase and 16W
cm-1 K-1 for the superconducting phase. At 4K, it is 55 W cm-1 K-1 (At 4K there is no
superconducting phase for tin as Tc=3.73K).
ENGINEERING PHYSICS PH14/24
Dept. of Physics, Jain University 108
Thermal conductivity of a specimen of tin in the normal and superconducting
state
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