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UNIT II
WAVES AND PARTICLES:A wave is spread out over a relatively large region of space and it cannot be said to
located just here and there
...
A wave is specified by its Frequency, Wavelength, Phase, amplitude or
Intensity etc
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Principles Of Quantum Mechanics
Particle: A particle is a representative unit of matter
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Wave: A wave is some disturbance traveling in space with time
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The time taken for one oscillation is time period
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Planck’s Quantum Hypothesis
Planck’s quantum equation is: E=h ν
Let there be ‘n’ number of oscillators
Let E1 = 1 X h ν, E2 = 2 X h ν ……………… E3 = 3 X h ν……………
...
En= nε
Also E0= 0
If there are n packets, then N=N0+ N1 +N2 + N3 +…………
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...
E = N0 E0+ N1 E1+ N2 E2+ N3E3+…………………
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...

The total no
...
]
let e- ε/KT = x ; then N= N0 [1+ x + x2 +…
...
=(1-x)-1]
where N is the total no
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 we know , E = N0 E0+ N1 E1+ N2 E2+ N3E3+…………………
E= N0 0+ N1 ε+ N2 2ε+ N3 3ε+…………………
 E=ε[N0 e- ε/KT +2 N0 e-2 ε/KT +……………
...
E= N0 ε x / (1-x)2-------------------[since 1+2x+3x2+………
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E= N0 ε e- ε/KT / (1- e- ε/KT )2 -----where E is total amount of energy
Average energy of each packet is:
av=E / N = N0 ε e- ε/KT / (1- e- ε/KT )2 / { N0 / (1- e- ε/KT )}
dividing the numerator and denominator of the above expression with e- ε/KT and solving
it ,we get av= ε / (e ε/KT -1)
 av = h ν / (e hν/KT -1)-------------------------------------------[since h ν = ε]
The De – Broglie Hypothesis
The energy exhibits wave particle duality
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e
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According to De-Broglie Electromagnetic waves behave like particles and particles like
Electrons will behave like Waves Matter Waves
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A particle of mass ‘m’ moving with velocity ‘v’ is associated with a group of waves
whose wavelength ‘λ’ is given by , λ = h /(mv)
...
In eqn(1); we get λ = h /√2mE-------(2)
2) when a potential difference ‘V’ is applied to an electron, then its energy is:E=eV
substituting the above eqn
...
(3)

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{m=9
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6 X 10-19 coulombs; h= 6
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26 /√V Ao----------eqn
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e
...
e
...
e
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So, in this
case new kind of matter waves are produced, These waves are called Matter Waves
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Proof:Kinetic Energy of particle is ½ mv2
According to Einstein relation = mc2
½ mv2 = mc2
v2 = 2c2
v = √2c
So v>c
Problems:
1) calculate the wavelength associated with an electron subjected to a potential
difference of 1
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25KV
 λ = 12
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26 / √(1
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34 Ao
2) What is the de-broglie wavelength for a beam of electrons whose energy is 45eV
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62X10-34 / √ (2*9
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6 X 10-19)
=1
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Let us consider a simple standing wave as shown:
ψ = A sin 2πx / λ—eqn
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Differentiate eqn
...
r
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’x’
 ∂ ψ /∂ x =A cos(2πx / λ) * 2π / λ—eqn(2)

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Differentiating eqn
...
r
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’x’
∂2 ψ /∂ x2 = -(A sin(2πx / λ) )* 4π2 / λ2 -----------eqn
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(3) reduces to : ∂2 ψ /∂ x2 = - (4π2 / λ2)* ψ
(or)
1 / λ2 = - ( 1/ 4π2 ψ)* ∂2 ψ /∂ x2-------eqn
...
E
...
E
...
E
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Hence , E=K
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+V
 E= (1/2)m v2+V
(1/2)m v2 = E-V------------------eqn
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(5) with ‘m’ on both sides:
 m2 v2 = 2m(E-V)--------eqn
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λ = h /(mv) and reducing the eqn
...
(7)
Equating equations (4) and (7)
(2m/ h2)(E-V) = - ( 1/ 4π2 ψ)* ∂2 ψ /∂ x2
(or)---REARRANGING THE TERMS
∂2 ψ /∂ x2 +(8 π2 m/h2) (E-V) ψ=0
(This is Shrödinger's Time Independent Wave Equation in one-dimension)
The above equation is summarized for three dimensions as:
∂2 ψ /∂ x2 +∂2 ψ /∂ y2 +∂2 ψ /∂ z2 +(8 π2 m/h2) (E-V) ψ=0

As we know that ∂2 ψ /∂ x2 +∂2 ψ /∂ y2 +∂2 ψ /∂ z2 = ▼2 ψ , Shrödinger's Equation
becomes
▼2 ψ+(8 π2 Uncertainity Principle
Heisenberg’s m/h2) (E-V) ψ=0

The position and momentum of a particle cannot be simultaneously measured with
arbitrarily high precision
...
There is likewise a minimum for the product of the
uncertainties of the energy and time
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∆ p ≥ h/4π----------(x= position and p=momentum)
∆ E
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Even with perfect instruments and technique,
the uncertainty is inherent in the nature of things
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Sol)Given :x= 4X10-10
We know that : ∆x
...
∆ p ≥ h
Substituting the values: ∆ p = 6
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65X10-24
2)An electron is confined to a box of length 10-9m
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Sol) )∆x
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∆ p ≈ h can be written as ∆x
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m ∆v≥h
∆v=h/(m∆x)
∆v=6
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732m/sec

Experimental Verification of matter waves
There are two experiments which give the proof of existence for matter waves
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Davisson&Germer experiment
2
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P
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P Thomson
independently demonstrated that streams of electrons are diffracted when they are
scattered from crystals
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Hence accelerated electron beam can be used for diffraction studies in
crystals
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Putting wave-particle duality on a firm
experimental footing, it represented a major step forward in the development of quantum
mechanics
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Davisson and Germer designed and built a vacuum apparatus for the purpose of
measuring the energies of electrons scattered from a metal surface
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The electron beam was directed at the nickel target, which could be rotated to observe
angular dependence of the scattered electrons
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It was a great surprise to them to find that at certain angles there was a peak in the
intensity of the scattered electron beam
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In an investigation, the electron beam accelerated by 54V was directed to strike the given
Nickel crystal are a sharp maximum electron distribution occurred at a angle of 500 with
the incident beam
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The spacing of planes in this Bragg’s planes by X-Ray
diffraction is 0
...

The experimental data above, reproduced above Davisson's article, shows repeated peaks
of scattered electron intensity with increasing accelerating voltage
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Using the Bragg law, the deBroglie wavelength
expression, and the kinetic energy of the accelerated electrons gives the relationship

In the historical data, an accelerating voltage of 54 volts gave a definite peak at a
scattering angle of 50°
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092 nm
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36, 14
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G
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Thomson experiment

In this experiment ,an extremely thin metallic film of gold ,aluminium etc is used as a
transmission grating to a narrow beam of high speed electrons emitted by a cathode C &
accelarated by an anode A
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The whole arrangement is enclosed in a vacuum
chamber
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The experimental result
and its analysis are similar to those of powdered crystal experiment of X-ray diffraction
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In any
electromagnetic wave system if A is the amplitude of the wave then the energy density
per unit volume is A2
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If ψ is the amplitude of matter waves then the particle density = ψ2
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According to “MaxBorn”,│ ψ│2 gives the probability of finding the particle in the
state
of ψ
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Particle in a Potential Box
Consider a particle present in a potential box of width a
...


Mass=m
Ax=0

B x=a

Shrödinger's Wave Equation in one-dimension for the particle in the box is
∂2 ψ /∂ x2 +(8 π2 m/h2) (E-V) ψ=0------(1)
But V=0 ∂2 ψ /∂ x2 +(8 π2 m/h2) (E) ψ=0----(2)
Let (8 π2 m/h2)E =k2
Hence eqn(2) becomes ∂2 ψ /∂ x2 +( k2) ψ=0---(3)
The general solution for equations of the type (3) is: ψ=Asinkx+BcosKx------(4)

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Let us apply boundary conditions
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Substituting the above values in (4)
1
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0=Asinka+Bcoska
but B=0[from earlier case] ; hence , Asinka=0
which implies either A=0 or sinka=0
but A cannot be 0
...
What is Pauli’s exclusion principle?
Ans: No two electrons that have same set of four quantum numbers (n, l, ml ,ms) can
occupy same energy level
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How is the k vector related to momentum ρ of the electron?
Ans: p = h/λ = h/(2π/k) = (h/2π)
...

3
...

Ans: - ħ2/2m[∂2ψ/∂x2 + ∂2 ψ /∂y2 + ∂2 ψ /∂z2] = Eψ
4
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Ans: ψ has no physical significance
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5
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Derive the important conclusions from this equation
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Energy is quantized;
2
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smallest energy is for n = 1 i
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, E1 = Eo =h2/8ma2
6
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626X10-34 /m*v
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If the particle is at rest, then
the wavelength is infinity
...

7
...
What it says is that for a particle of momentum ‘p’ the associated wave has the
wavelength λ
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Is electron a particle or a wave? Explain
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Electric current is thought off as number of electrons crossing unit
area per unit time
...
Electron also exhibits wave
properties like interference and diffraction
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9
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Once we obtain the wave function by solving the
Schrodinger equation, we can extract information regarding the average values of
position, potential energy, total energy, momentum associated with it
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10
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The well-behaved solutions of time
independent Schrodinger equation are called the ‘eigen functions’, or eigen states and
the corresponding values of energy are called the eigen values
...
What properties are required to a well behaved eigen function and its derivative?
Ans: A well behaved eigen function and its derivative must possess following
properties: (a) It must be finite everywhere
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(b) It must be single valued
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(c) It must be continuous everywhere
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What is de Broglie’s hypothesis?
Ans: It tell us how to get wave associated with a free particle
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Fill in the blank questions:
1
...

2
...

3
...

4
...

5
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ANSWERS:
1
...
wave 3
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matter waves, de Broglie waves 5
...
The wave function for the motion of the particle in a one dimensional potential
box of lengths a is given by
ψn = Asin(nπx/a) , where A is the normalization constant
...
The energy of the lowest state in a one dimensional potential box of length a is
(a) zero
b)2h2/8ma2
c)h2/8ma2
d) h/8ma2
3
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Energy E of the photon of wavelength λ is
(a) hc/λ (b) hcλ
(c) c/λ
(d) hλ/c

5
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(a) v=h/p
(b) λ=h/p (c) λ =h/c
(d) v= p/h
ANSWERS
1
...
c

3
...
a

5
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Schrodinger’s theory tells us how to obtain the wave function associated with a
particle
...
Laws of classical physics can explain the motion of micro particles
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λ=h/p is called de Broglie wave length
...
∆ px
...

5
...

ANSWERS:
1
...
False 3
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False

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5
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If an electron is confined to one dimensional potential box of length L, the allowed
energy values are
given by En= _________
Ans: n2h2π2/(2mL2)
2
...

Ans: wave
3
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The waves associated with material particles are called matter waves or _______
Ans: de Broglie waves
5
...

Ans: Light
6
...
______ theory tell us how to obtain the wave function associated with a particle
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Laws of classical physics cannot explain the motion of _____ particles
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λ = h/p is called ______ wave length
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x
...
The energy of lowest energy state in a one dimensional potential box of length a is
______
Ans: h2/(8ma2)

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ELECTRON THEORY OF METALS

Classical Free Electron Theory of Metals (CFET)
1
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Because these valance electrons are responsible for the
electrical conduction of the solid, they are called Conduction Electrons
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The potential energy due to the ions is assumed to be constant everywhere
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3
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4
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MEAN FREE PATH / RELAXATION TIME/DRIFT VELOCITY:
We know from Ohm’s law:
V= iR → i = V/R
The current density (Current per unit Area)
J = i /A
Also, Electric field intensity E = V/l, where A, l are Area and Length of
the conductor
...
J = Eσ-------(1)
We want to express σ in terms of the Microscopic properties pertaining to
the conduction of electrons
...
Then the force acting on the electron due to the electric field is
FE = -eE
...


According to Newton second law F = ma
 - mV/τ = m * dv/dt

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At steady state / equilibrium FE = Ff
 -eE = mv/τ
v = (-eτ/m ) E →(1)
The above velocity of electron in equation (1) is known as drift velocity
...
of electrons per unit volume)
Therefore J = (-Ne) (-eτ/m E)
J= (Ne2τ/m) * E → (2)
Comparing (2) with (1)
We have σ = Ne2τ/m
σ is directly proportional to τ
...
e, mean free lifetime
...

Let us suppose that an electric field is applied till the time drift velocity is
established
...
The drift
velocity after this instant is governed by:FE = -e E ; FF = -m v/τ
m (dv / dt) = -e E – (m v)/τ ---------- If E = 0 , then
m dv/dt = -mv/τ  dv/dt = - v/τ ----------- (1)
 1/v dv = -dt/τ  ∫1/v dv = -∫dt/τ + C
So, log v = -t/τ + C
 log v = loge –t/τ + log C
 log v = log(e –t/τ
...
C , now at t = 0
vo = C therefore C = vo
v = vo e –t/τ
From (1) vd,t = v d,o e –t/τ
Since τis the time between 2 successive collisions, it can be expressed as

τ = (λ/vd)
λ = distance between 2 successive collisions, called mean free path of
collisions
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In
1928 Sommerfield, modified CFET in 2 important ways:
1) The electrons must be treated quantum mechanically
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2)The electrons must obey Pauli’s exclusion principle;that is ,no two electrons can have
the same set of quantum numbers
...
Thus this will affect the way the electron gas can
absorb energy from an external source,such as a heat source,and the way it responds to an
electric field
...

Elementary particles whose spin can take values that are integral multiples of ½ are
called fermions
...
Fermions obey Paulis’
exclusion principle
...

When mathematically developed this statement is known as Fermi – Dirac Statistics
Bosons are elementary particles whose spin can take values that are whole numbers or
zero
...
They do not obey Paulis’ exclusion principle
...


According to the Fermi distribution law,the probability F(E) that a given energy state E is
occupied at a temperature T is given by:

F (E) = 1/ exp[(E – EF)/KT + 1)]
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EF = Fermi Energy
1)

2)

If E > EF, the exponential term becomes infinite and F(E) =0
there is no probability of finding an occupied state of energy
greater than EF at absolute zero
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Energy that any electron may occupy at 00K
...

Thus, for temperature greater than 00K, the Fermi level may be
defined as the level where the probability of occupation is ½
...
As such the Fermi level changes with temperature
...
Each type of distribution function has a
normalization term multiplying the exponential in the denominator which may be
temperature dependent
...
At absolute
zero, the probability is =1 for energies less than the Fermi energy and zero for energies
greater than the Fermi energy
...
This is entirely consistent with the Pauli exclusion
principle where each quantum state can have one but only one particle
...
This density of states is the electron density of states,
but there are differences in its implications for conductors and semiconductors
...
In the case of the semiconductor, the density of states is of the

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same form, but the density of states for conduction electrons begins at the top of the gap
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Other possible scattering mechanisms are due to vacancies, dislocations and other lattice
imperfections
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Some
theories attribute the resistance due to scattering of conduction electrons by the positive
ion cores
...
This is a
common feature of all metals except those of which are super conducting in nature
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At low temperatures it
varies at T 5
...


The resistivity of the metals with impurities exhibits similar behavior
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BAND THEORY OF SOLIDS:
ORIGIN OF ENERGY BAND STRUCTURE IN SOLIDS

Bloch Theorem
The solutions of the Schrodinger equation for a periodic potential must be of the form:

 k  uk r eikr
where uk(r)=uk(r+T) is an amplitude function of the plane wave exp(ikr) and T is a
translation vector of the crystal
...
Wave functions of this form are called Bloch functions and they
are very useful in calculations because they allow us to concentrate on only one period of
the lattice to solve for the wave function of the electrons throughout the entire crystal
...
It can be shown that, for a 1D system, two and only two distinct values of k exist
for each and every allowed value of energy E
...
For a given E, values of k differing by a reciprocal lattice vector G give rise to
one and the same wavefunction solution
...
This leads to the concept of Brillouin Zones
...
For an infinite crystal, k can assume a continuum of real values in the range
specified in statement 2
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For a finite crystal, we adopt periodic boundary conditions
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The Kronig-Penney model
The Kronig-Penney model demonstrates that a simple one-dimensional periodic potential
yields energy bands as well as energy band gaps
...

The potential assumed in the model is shown in the Figure:

The periodic potential assumed in the Kronig-Penney model
...

The Kronig-Penney consists of an infinite series of rectangular barriers with potential
height, V0, and width, b, separated by a distance, a-b, resulting in a periodic potential
with period, a
...
The
actual derivation can be found in section
Solutions for k and E are obtained when the following equation is satisfied:

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This transcendental equation can be further simplified for the case where the barrier is a
delta function with area, V0b, for which it becomes:

with

This equation can only be solved numerically
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Energy band diagram of Solids
The CFE model and QMFET model assumes that a conduction electron in a metal has a
constant (Zero) potential and so is completely free to move about in the crystal but
restricted to the surfaces
...

But, it is suggested that the crystal be treated as an infinite array of lattice points
...

Inside a real crystal, there is a periodic arrangement of positively charged ions through
which the electrons move Ex: Fig
V=V0

V=V0

+

+

+

+

V=0
a
The potential of the electron at the positive ion site is zero and is maximum between
them
...

Let us study the motion of an electron in such a lattice and the energy state it can occupy
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In simple one – dimensional model, we are considering K as directed along the xaxis and in general K is to be treated as a vector and it is called as a propagation vector
...

For a vector (sinusoial wave) or a 3 – dimensional wave, Kroing & Penny
introduced a simple model for the shape of potential variation
...

1)
...
67 * 10 -27 kg,, confined to move
along the edge of an impenetrable box of length 10 -14 metre, plank’s constant = 6
...

En = n2h2/8ma2 ; Lowest Energy State, n= 1
Therefore E1 = 3
...

E1 = 3
...
6 * 10 -19 = 2
...
Calculate the energy difference between the ground state and the first excited state for
an electron in a one – dimensional rigid box of length 10 -8 cm
...
1 * 10 -31 Kg h = 6
...

En = n2 h2 / 8ma2 = n2 * (6
...
1 * 10 -31 * (10 -10)2
= 0
...

For ground state n = 1; E1 = 36 ev
First excited state, n =2 ; E2 = 148 ev
Hence, energy difference (E2 – E1) = 111ev
3
...
1nm
...
7 n2 ev
E1 = 37
...

4
...
167nm
5
...
Find a> how much
energy must be supplied to excite the electron from the ground state to the first excited?
En = n2h2/8mL2 ; E1 = 37ev
E2 = 148 ev; E2 – E1 = 111ev
Metals, insulators and semiconductors
Once we know the bandstructure of a given material we still need to find out which energy levels are
occupied and whether specific bands are empty, partially filled or completely filled
...
Therefore, they are not expected to contribute to the electrical
conductivity of the material
...
These unoccupied energy levels enable carriers to gain energy when
moving in an applied electric field
...

Completely filled bands do contain plenty of electrons but do not contribute to the conductivity of the

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material
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In order to find the filled and empty bands we must find out how many electrons can be placed in each
band and how many electrons are available
...
Therefore, the minimum number of states in a band equals twice the number of
atoms in the material
...

In order to find the filled and empty bands we must find out how many electrons can be placed in each
band and how many electrons are available
...
Therefore, the minimum number of states in a band equals twice the number of
atoms in the material
...

To further simplify the analysis, we assume that only the valence electrons (the electrons in the outer
shell) are of interest
...


Figure
...
Shown are: a) a half filled band, b) two overlapping bands, c) an
almost full band separated by a small bandgap from an almost empty band and d) a full band and an empty band
separated by a large bandgap
...
3
...
This situation occurs in materials
consisting of atoms, which contain only one valence electron per atom
...
Materials
consisting of atoms that contain two valence electrons can still be highly conducting if
the resulting filled band overlaps with an empty band
...
No
conduction is expected for scenario d) where a completely filled band is separated from
the next higher empty band by a larger energy gap
...

Finally, scenario c) depicts the situation in a semiconductor
...
This yields an almost full band below an almost empty band
...
The
almost empty band will be called the conduction band, as electrons are free to move in
this band and contribute to the conduction of the material
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They accelerate in an applied electric field just like a free electron in vacuum
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The presence of the periodic
potential, due to the atoms in the crystal without the valence electrons, changes the properties of the
electrons
...
Because of the
anisotropy of the effective mass and the presence of multiple equivalent band minima, we define two types
of effective mass: 1) the effective mass for density of states calculations and 2) the effective mass for
conductivity calculations
...

The effective mass of a semiconductor is obtained by fitting the actual E-k diagram around the conduction
band minimum or the valence band maximum by a paraboloid
...
In this section we first describe the different relevant band minima and maxima,
present the numeric values for germanium, silicon and gallium arsenide and introduce the effective mass for
density of states calculations and the effective mass for conductivity calculations
...
In addition there are three band maxima of interest close to the valence
band edge
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What do you understand by the Fermi energy EF?
Ans: At absolute zero the Fermi level divides the occupied states from the unoccupied
states
...

2
...

3
...

Ans: FE = 1/[1 + exp(E – EF/kT)]
4
...

5
...

FILL IN THE BLANK QUESTIONS:
1
...

2
...

3
...

4
...
Free mean path determines the _______of an electron
...
n,l
...
mobility
...
Eo,Ef

3
...
n=N/V =π/3[8n/h2]3/2 EF3/2

MULTIPLE CHOICE QUESTIONS:
1
...
Order of resistivity of silver is
(a) nano-ohm m
b)milli ohm m

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c)ohm m

d) micro ohm m

3
...
The elementary particles whose spin can take values that are whole numbers or
Zero
...
Current density =
a)charge density * drift velocity
c)drift velocity / change density
ANSWERS:
1
...
a

3
...
a

b)change density / drift velocity
d) drift velocity * density

5
...
In free electron theory of metals, the electron is assumed to be moving in a
uniform (zero) potential field
...
The potential experienced by an electron in passing through a crystal is taken as
perfectly periodic with the period of the lattice
...


According to Kronig – Penney model, the potential experienced by an electron is
considered as linear array of square potential wells of finite depth
...
Terminal velocity of an electron is in opposite direction to the applied electric
field
...
When the electron is free, the effective mass is not the true mass
...
True

2
...
True

Dr
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Srinivas M
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...


4
...
False

UNIT-II
Statistical Mechanics
The subject which deals with the relationship between the overall behavior
of the System and the properties of the particles is called Statistical
Mechanics
...
Statistical
Mechanics can be applied to classical Systems such as molecules in a gas as
well as photons in a cavity and free
Electrons in a metal
...

Macrostates
Let us consider that the compartments represent some property of the
particle such as energy, momentum and velocity
...
Particles with
energy E1 occupy compartment 1 and the particles with energy E2 occupy
compartment 2
...

So, there are 4 distributions possible: (0,3), (1,2), (2,1), (3,0)
Each compartment distribution of a system of particles is known as a
macrostste
...
(n-1,1) and (n,0)
...

“Any state of a system as described by actual observations of its
macroscopic statistical
properties is known as a macrostate”
...
M
...
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...
Tech
...
D
...
of Microstates
Compartment I

Compartment II

0,3

0

a,b,c

1

1,2

a
b
c

b,c
c,a
a,b

3

2,1

a,b
b,c
c,a

c
a
b

3

3,0

a,b,c

0

1

Each distinct arrangement is known as the microstate of the system
...

In the above example, forn3 particles there are 8 micro states
...

“The state of a system specified by the actual properties of each individual
component
in detail permitted by the uncertainity principle is known as microstate”
...

A small volume element in the position space is given by: dV = dx dy dz
Similarly, The three dimensional space in which the momentum of the
particle is completely specified by the three momentum coordinates(dpx, dpy ,
dpz,) is known as the momentum space
...

A small volume in phase space is given by: dτ= dx dy dz dpx dpy dpz
i
...
dτ= dV d Γ
Thus a volume element dτ in phase space is the product of a volume element
dV in position space and volume element d Γ in momentum space
...

The total no
...
M
...
Sc (Tech)
...
Tech
...
D
...
In other words, a statistical ensemble is a probability
distribution for the state of the system”
...

The notional size of ensembles in thermodynamics, statistical
mechanics and quantum statistical mechanics can be very large indeed,
to include every possible microscopic state the system could be in,
consistent with its observed macroscopic properties
...

The concept of an equilibrium or stationary ensemble is crucial to
some applications of statistical ensembles
...
In fact, the ensemble will not evolve if it
equally contains all past and future phases of the system
...


Dr
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Srinivas M
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, M
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...


Types of Ensemble:
1
...
The system is assumed to be isolated in the sense that
the system cannot exchange energy or particles with its environment, so the
energy of the system remains exactly known as time goes on
...

2
...
The system is said to be closed in the
sense that the system can exchange energy with a heat bath, so that various
possible states of the system can differ in total energy
...

3
...
The system is said to be open in the sense that the system can
exchange energy and particles with a reservoir, so that various possible
states of the system can differ in both their total energy and total number of
particles
...


Statistical Distributions
Statistical Mechanics determines the most probable way of distribution of
total energy E among N particles of a system in thermal equilibrium at
absolute temperature T
...
Maxwell-Boltzmann Statistics
2
...
Fermi-Dirac Statistics
Maxwell-Boltzmann Statistics (Assumptions)
1
...

2
...

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3
...

4
...

5
...

6
...

Maxwell-Boltzmann distribution function is given by: f (E) =

1
AeE/KT

Bose-Einstein Statistics (Assumptions)
Bose-Einstein Statistics is obeyed by those particles which are
indistinguishable and have integral spin
...

1
...

2
...

3
...

4
...
of Bosons can occupy a single cell in phase space
...
Bosons do not obey the exclusion principle
...
The number of phase space cells is comparable with the number of
Bosons
...
Energy states are discrete
...


Fermi-Dirac Statistics (Assumptions)
Fermi-Dirac Statistics is obeyed by those particles which are
indistinguishable and have half integral spin
...

1
...

2
...
They obey Pauli’s exclusion principle
...
They have weak interaction between the particles
...
Uncertainty principle is applicable
...
M
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...
Tech
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D
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In a system of Fermions, the presence of a particle in a certain state
prevents any other particle from being in that state
...


1+e (E-EF)/KT
Where EF is the Fermi Energy

Photon gas

A photon gas is a gas like collection of photons, which has many of the same
properties of a conventional gas like hydrogen or neon including pressure,
temperature and Entropy
...
The most common way of equilibrium distribution is by
the interaction of the photons with matter
...

Comparison of Maxwell-Boltzmann, Bose-Einstein, Fermi-Dirac
Statistics
Points Compared Maxwell-Boltzmann Bose-Einstein
Statistics
Classical
Applicable
Nature of particles Identical &
distinguishable
Examples
Molecules of a gas

Quantum

Fermi-Dirac
Quantum

Identical &
indistinguishable

Properties of
particles

Any spin
wave functions do
not overlap

Identical &
indistinguishable
i)Photons in a cavity Free electrons in
conductors
ii)phonons in a
solid
Spin=0
...
2
...
4…
...

Overlap of wave Overlap of wave
functions
functions

Distribution
function

f (E) = Ae-E/KT

f(E)=

1

f(E)=

AeE/KT-1

Concept of Electron gas
1) The valence electrons of the constituent atoms of a metal move about

freely through the volume of a metallic specimen
...

2) Without external electric field, there are no internal forces
...
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4) The valence electrons are responsible for the conduction of electricity by
metals , and for this reason these free electrons are termed as “conduction
electrons” as distinguished from the other electrons of the filled shells of the
ion
cores
...

Therefore, to calculate the number of electrons in an energy level at a given
temperature ,it is must to know the number of energy states per unit volume
...
the energy levels appear continuum inside the space of an atom,
therefore, let nx, ny, and nz (n2=nx2+ ny2+ nz2) in the three dimensional space
...
e
...
Hence the number of energy states in any volume
in units of cubes of lattice parameters
...
Hence, the number of
energy states
available in the sphere of radius n is :
1
8

4 πn3
3

NOTE: ђ=(h/2π)

Dr
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Srinivas M
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, M
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, Ph
...


The density of states refers to the number of quantum states per unit energy
...
So, what is the importance of
the density of states? Consider the expression g ( E )dE
...

N (E)  

E

E

g ( E )dE

Thus g ( E )dE represents the number of states between E and dE
...
e
...

From the Schrodinger equation, we know that the energy of a particle is
quantized and is given by
k2 2
E
2m

The variable k is related to the physical quantity of momentum
...
Therefore, k must also have direction components kx , k y , and kz
...
M
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Tech
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D
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An electron can
only exist in the well, and the wave function is given by
 ( x)  A cos(kx)  B sin(kx)
where:
n
k
a

where a is the width of the barrier
...

Therefore, the wave function is only valid for all integers greater than zero
...
Below is a plot of the
a
kx 

valid solutions in k-space
...
M
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Tech
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D
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The red dots are those between
the intervals of k and k +dk
...
The unit cell is the smallest shape
which can be repeatedly be used to construct a lattice as in a diamond
crystal, for example
...
In k-space, the interval is simply k and k +dk
...
Hence, a shell is
created which encloses a certain amount of quantum states in an
infinitesimal interval
...
Since we want to find
the density of states in an infinitesimal interval of energy, the shape used for
the boundaries of the interval must represent equal energies
...
M
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V  4 k 2 dk

Alternatively, we can find the volume by subtracting the volume inside the
inner sphere from the volume inside the outer sphere (of radius k  dk )
...

4
4
3
3
V    k  dk     k 
3
3
4
   k 3  3k 2 dk  3kdk 2  dk 3  k 3 
3
because dk 2 and dk 3 have no real meaning,
the volume is
4
   3k 2 dk   4 k 2 dk
3

Even though the k-space image displays valid wave equations solutions for
both positive and negative integers of kx , k y , and kz , the wave function
should only be valid for all positive values of kx , k y , and kz
...
e
...
The revised shell volume is
then,
1
V    4 k 2 dk
8
1
  k 2 dk
2

The number of quantum states in an interval of dk is found by dividing the
volume of the shell by the volume of a single state (i
...
the volume of the
unit cell)
...
Once again, the
relationship between k and E is
Dr
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Srinivas M
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, M
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...


2mE

k

2

1  2m  2mE 
dk   2  2 
2


m  2mE 
 2 2 



1

1

2

dE

2

dE

Substituting the results into the density of states equation will give the
density of states in terms of energy
...


  m  m  E  E  dE
1

g ( E )dE 

a

3

2 2

2
3

a
 2
2

2

1

2

 m
1

a 3  2m 
g ( E )dE  2  2 
2 


2

3

2

3

3

2

1

4

22 2

1

2

EdE

2

EdE

Problems:
1
...
5
eV above Fermi energy will be occupied
...
At what temperature we can except a 10% probability that electron in silver have an
energy which is 1% above the Fermi energy? The Fermi energy of silver is 5
...
(Ans: T = 290
...
calculate the velocity of an electron with kinetic energy of 10 electron volt; what is the
velocity of a proton with kinetic energy of 10 electron volt?
( Ans: cp= 19
...
Find the drift velocity of free electron in a copper wire of cross sectional area 10 mm2
when the wire carries a current of 100A
...

(Ans: 7
...
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Dielectrics are classified into two categories:

Polar

Nonpolar

When dipole moment is present (finite)
Dipole moment ‘μ’=q * d




when no dipole moment is present(0)

Best Dielectric example is wax
...

Capacitance = Q/V = Farads

C1
C2

Dr
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Srinivas M
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││
V
C1 is a capacitor without any dielectric material between the plates
...
85X10-12 farad/meter
...
An air filled in parallel plate capacitor has a capacitance of 1
...
The separation
between the plates is doubled and wax is inserted between them
...
Find the dielectric constant
Sol)Given: C1=1
...
5 pf;
C2= kAεo/2d = 3pf
k = C2/C1 or  C2/C1 = Aεo/2d / (Aεo/d) k = 2*3/1
...
A parallel plate capacitor has a capacitance of 50 pf
...
35m2
...
6 then
find
the new capacitance
...

d = Aεo /C1 d= (0
...
85X10-12;
substituting the values , we get d= 0
...
6
but , k = C2 / C1 5
...
A parallel plate capacitor of capacitance 15 pf has a potential difference of
15Vbetween plates
...
What is the energy of capacitor before
and after the slab is introduced
Sol)

Given : C=15pf;V=15 V; k = 5
E1 = 1/2 C1V2
1/2 X 15 X 15 X 15 X 10 -12
E1= 1
...
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43 X10-9J
POLARIZATION:
A dielectric material gets polarized by the application of an electric field
E
...
It also varies, very sensitively
with the time variation of the field E
...

Suppose a dielectric contains ‘N’ molecules per unit volume
...
This dipole moment induced per molecule is
proportional to the applied filed E
...

When E is switched off, P disappears
...
But finally P attains a steady state value under the applied static field E
...
It tells
how much P can be produced in a material by a given field E
...

1) 2 condensers of 20μF and 30μF are connected in series across a 200V d
...
Find the equivalent capacitance, the charge on each plate of the
condensers & the potential difference across each condenser
...
M
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Tech
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D
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The electronic distributions
about the nuclei are distorted by an imposed electric field
...


R

Expression for electronic Polarisability:
Let an atom of a dielectric be placed in a D
...
electric field E
...
The nucleus wills move towards the
field direction and the electron cloud will move in the opposite direction of the field
...
Finally, a new equilibrium
will be reached when these two forces are equal and opposite
...
M
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Tech
...
D
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...



...
e
...

Total negative charge in the sphere of radius x ,is Qe=-Ze(4/3πx3)/ 4/3πR3
 Qe=-Ze(x3/ R3)-----(2)
The coulomb attractive force between the nucleus with charge QP=+Ze and the electron
cloud at a distance x from the center of the nucleus is given by:
F=(1/4πεo)(QeQP/x2) = -(Ze/R3)x3(Ze)/( 4πεox2)
 F=(1/4πεo)[-Z2e2x/R3]-----------------------------(3)
Also the force between the nucleus and the electron cloud is given by
F=qE=ZeE--------------------- (4)
Under equilibrium conditions, these two forces are equal and opposite
...
e
...

1)
...
55 Ao
...
M
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...
Tech
...
D
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55 Ao
αe = 4πEor3
= 4 * 3
...
85 * 10-12 *(0
...
49 * 10-42 F/m2

2)
...
The space between the
plates is 1mm and filled with a dielectric of relative permittivity εo
...
Find:
1
...
The charge on the capacitor
3
...
The potential gradient
Sol)
...
D
...
85 X10-12 / (1 X 10-3)
C=1
...
239 X 10-7 X 300 Q=371
...
8585X10-5C/m2
E = 1
...
85 X 10-12)
E = 3 X 105 N
The Ionic Polarization:
The polarization produced by the relative displacement of the ions is
called the ionic polarization
...
The combined structure of opposite ions in many of these
materials is such that the molecules are neutral and without a net dipole moment
...
This process is
resisted by binding forces and stops when 2 opposing forces balance each other
...
This is in addition to the electronic polarization produced in
all kinds of materials
...
In most materials αi<< αe as
the ions are heavier than the electrons
...

E
+
-

Dr
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Srinivas M
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x1

x2

μ= e(x1 + x2)

F

F

F α x1
F = k1 x1
k1 = m1 ω2
x1 = F/k1
F/(m1 ω 2 )

F α x2
F = k2 x 2
k2 = m2 ω 2
x2 = F/k2
F/(m2 ω 2)

F=eE
x1= eE/m1 ω 2

x2 = eE/m2 ω 2

x1+x2 = eE/ ω 2 [ 1/m1 + 1/m2]
μ= e2E/ ω 2 [1/m1 + 1/m2]
But μ α E  μ= αiE
Therefore: αi E = e2E/ ω 2 [1/m1 + 1/m2]


αi = (e2/ ω 2) [1/m1 +1/m2]

ORIENTATIONAL POLARIZATION:
Some molecules carry dipole moment even in the absence of electric field
...
The polarization due to such alignment is called orientational
polarization and is dependent on temperature
...

The resultant polarization dueto the 3 types of polarization
...
M
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Tech
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D
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e
...
55Ao
αe = 4πEoR3
αe =0
...
7 * 10 25
N α E = (k-1) E0E
→Nα /Eo +1 =k
→k = 1
...


FREQUENCY DEPENDENCE ON TEMPERATURE:
Any neutral sample of a material contains a very large number of equal
and opposite charges and neutral particles
...
The frequency ‘fo’ (or) corresponding angular
frequency ωo = 2πfo is called natural frequency
...
M
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...
Tech
...
D
...
Then the
system will execute damped harmonic motion
...
Frequency of the
damped oscillations is nearly same as the natural frequency
...
This
phenomenon is called resonance and ωo is called as resonating frequency of the
external field
...
e
...

The polarizations verses electric field curve exhibit hysteresis, which is similar
to the BH curves exhibited by Ferro magnetic materials
...

Eg: BaFiO3 (Barium Titanic Oxide)
PROPERTIES:
1
...
The spontaneous polarization vanishes above a particular temperature
called Curie Temperature (Tc)
...
The dielectric constant depends on field strength
...
Above ‘Tc’ the dielectric constant varies with temperature according
relation
k = C/(T – θ ) where C→ Curie constant

When an Electric Field is applied to a specimen the polarization
increases
...
In order
to reduce the polarization is zero
...
This field is called Coercive field
...


Dr
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Srinivas M
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, M
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...


PIEZO ELECTRICS: Development of electric chargers i
...
, electric
polarization when subjected to stress is called ‘Piezo electrics
...
These materials also exhibit the Inverse Piezo Electric Effect that is these substances get strained when
subjected to electric field
...
They can be classified as
natural & synthetic
...
Synthetic group consists of ADP (Ammonium
Dinitrogen Phosphate),LS – (Lithium Sulphate ),DHT – Dipotassium Tartarate
...
The dielectric constant of Argon at 0o C and 1 atm pressure is 1
...

Calculate the polarisability of the atom if Argon contains 2
...
000345
We know k – 1 / k + 2 = Nα/3εo
(1
...
000345 +2) = 2
...
85 X10 -12
X3)
 α = 1
...
A solid elemental dielectric with density of 3 X 10 28 atoms / m3 shows
an electronic polarisability of 1 X 10 -40 Fm2
...
85 X 10 -12 X 3)
k = 1
...
They
add up vectorially and modify the original field produced by external sources
...

It is denoted by ‘μ’ and is similar to permittivity ‘ε’ used in electricity
...
I unit of B is Weber /m2 and magnetic field strength H is ampere/meter
‘μ’ is defined as constituent relation of the medium
...
Its S
...


Dr
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Srinivas M
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...


The wave equation derived from Maxwell’s equations shows that the speed of the
wave ‘v’ in any medium is given by
V = 1/√εμ
For vacuum
c= 1/√εoμo
By measuring c & εo through experiment, μo can be calculated
c= 3
...
85 X 10-12 F/m
μo= 1
...
The material is then said to be magnetized and acquired a
magnetic moment
...
Its MKS unit is Ampere/meter
...
The field is described by a
vector called the magnetic induction or flux density B
...

If a positive test charge qo moving with a velocity ‘v’ through a point in a
magnetic field experiences a sideways force F , then the magnetic
induction B at that point is defined by
F= qo(vXB)where v and B are vectors

Dr
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Srinivas M
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, M
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, Ph
...


The magnitude of the magnetic induction lBl is therefore defined by
MAGNETIZATION:
Suppose we apply a Magnetic field H to a body, then it will be
magnetized
...

The extent of which a body is magnetized by H is described by another vector quantity, called magnetization and denoted
by M
...
It has the same dimensions as H but not
as B
...


M=χH
Where χ is called the Magnetic susceptibility
...

χB is analogous to χe, the electrical susceptibility
...
It is
similar to the fact that the electric nature of material is characterized by the electric
susceptibility χe (or)
...
Both positive and negative charges are present but their magnitude and
distribution is such that the net charge is zero
...
The net magnetic moment depends on the electronic configuration
of atom
...

However, a piece of such material contains a very large number of dipoles whose
orientation is perfectly random
...
Its magnetic character can be revealed by external fields
...

On the bases of nature of orientation of the neighbouring atomic dipoles, the
materials are classified as Ferromagnetic, anti-ferromagnetic and ferromagnetic materials
...
M
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...
Tech
...
D
...
They are:
1
...
Magnetic moment due to the spin of electron
3
...
MAGNETIC MOMENT DUE TO ORBITAL MOTION:
ee-

er

e-

ee-

ee- e-

we know
i = -e/T
Linear velocity v = displacement / Time
 T = displacement / velocity (angular velocity)
T = displacement / ω = 2π/ ω
i = -e ω /2π
Magnetic moment μ= i * A
μ=-e ω /2π * π r2
μ=-e ω r2 /2
Again, we know that angular momentum L = mv * r
v=rω
 L = mr2W
r2 ω = L /M
The net magnetic moment μ = eL/2m
 μ= (-e/2m)2
L is angular moment
e/2m is called as gyro magnetic ratio
L = l (ħ)
ħ = h/2π
lh/2 π = L
μ= -e/2m * hl/2 π →-(eh/4 πm)l
(eh/4πm )is called Bohr – magneton
1 Bohr magneton = 9
...
MAGNETIC MOMENT DUE TO DUE TO SPIN OF ELECTRONS:
μs = (eh / 4πme)1/2 me → mass of electron
3
...
M
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...
Tech
...
D
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LANGEVIN’S THEORY OF DIAMAGNETISM:
WE KNOW



i = e/T
T= 2π/ω
i = e ω /2 π

According to Lenz’s law:∫B
...
dt = m
...
M
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Sc (Tech)
...
Tech
...
D
...

All atoms possessing odd number of electrons exhibit paramagnetism
...
There are number of compounds with
even number of electrons including molecular oxygen that exhibit paramagnetism
...

Examples: O2, Al, Tungsten etc
...
They are randomly
oriented and distributed
...
With in the
field we can treat the material sample as an assembly of magnetic dipoles
...
The magnetic dipole will orient
themselves along field direction
...

FERROMAGNETISM:
1
...
A ferro magnetic rod when suspended in a uniform magnetic
field aligns itself along the direction of the field
...
The Relative permeability is very high
...
The Magnetic susceptibility is +ve and very high
4
...
Above a certain temperature a
ferromagnetic becomes paramagnetic
...
The susceptibility of a ferromagnetic material increases with magnetic
field upto a certain value and then decreases
...
Ferromagnetism is given to a existence of, what are known as magnetic
domains
...
e
...
They exhibit the property of hysteresis
...
M
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...
Tech
...
D
...
:
1
...

These domains are spontaneous magnetized
...
With in each domain the spontaneous magnetization is due to the existence of a
Internal molecular field
...

HYSTERESIS:

The Hysteresis Loop and Magnetic Properties
A great deal of information can be learned about the magnetic properties of a material by
studying its hysteresis loop
...
It is often referred to as the B-H
loop
...


The loop is generated by measuring the magnetic flux of a ferromagnetic material while
the magnetizing force is changed
...
As the line demonstrates, the greater the amount of current applied
(H+), the stronger the magnetic field in the component (B+)
...
The material has reached the point of
magnetic saturation
...
" At this point, it can be seen that some magnetic flux remains in the material
even though the magnetizing force is zero
...

(Some of the magnetic domains remain aligned but some have lost their alignment
...
M
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This is called the point of coercivity on the curve
...
) The force required to remove the residual magnetism from the material
is called the coercive force or coercivity of the material
...
Reducing H to
zero brings the curve to point "e
...
Increasing H back in the positive direction will return B
to zero
...
The curve will take a different path from
point "f" back to the saturation point where it with complete the loop
...

1
...
In other words, it is a material's
ability to retain a certain amount of residual magnetic field when the magnetizing
force is removed after achieving saturation
...
)
2
...
Note that residual magnetism
and retentivity are the same when the material has been magnetized to the
saturation point
...

3
...
(The value of H at
point c on the hysteresis curve
...
Permeability, - A property of a material that describes the ease with which a
magnetic flux is established in the component
...
Reluctance - Is the opposition that a ferromagnetic material shows to the
establishment of a magnetic field
...


Types of magnetic material
Most of magnetic materials of industrial interests are ferromagnetic materials
...
As shown in the magnetization curve,
ferromagnetic materials with the demagnetized state does not show magnetization
although they have spontaneous magnetization
...
Within the magnetic domains, the
direction of magnetic moment is aligend
...
In the demagnetized state, total magnetization is cancelled because of the

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random orientation of the magnetizations in magnetic domains
...
When all magnetic domains are wiped
away and magnetizations are all aligned to the direction of the magnetic field,
magnetization is saturated
...

When domain wall can easily migrate, the ferromagnetic material can be easily
magnetized at low magnetic filed
...
Since soft magnetic materials can be demagnetized at low magnetic field,
coercivity Hc is low
...
For
ferromagnetic materials to be soft, their magnetocrystalline anisotropy and
magnetostriction constant must be low
...

When domain wall is difficult to migrate, magnetization of the ferromagnetic material
occurs only when high magnetic field is applied
...
These materials are called hard magnetic materials, and are suitable for
applications such as permanent magnets and magnetic recording media
...
Since large magnetic field is required
to demagnetize, their coercivity Hc is usually high, but coercivity is highly sensitive to
the microstructure
...
An electronic current produces a magnetic field
...
Some materials are easily magnetized when placed in a weak magnetic field
...
When field is turned off, the material is demagnetized
...
These materials are called soft magnetic materials (or) Temperary magnets
...
Some other materials are magnetized with difficulty: they require a strong
Magnetic field
...
But once the materials are magnetized they retain their magnetization when
Field turns off
...
They are permanent magnetic materials
...
Diamagnetic materials have a weak, negative susceptibility to magnetic
fields
...
Diamagnetic materials are slightly repelled by a magnetic field and the
material does not retain the magnetic properties when the external field is
removed
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In diamagnetic materials all the electron are paired so there is no
permanent net magnetic moment per atom
...
Diamagnetic properties arise from the realignment of the electron paths
under the influence of an external magnetic field
...
Most elements in the periodic table, including copper, silver, and gold, are
diamagnetic
...
The magnetic moment, intensity of magnetisation and magnetic

susceptibility are all negative while magnetic permeability has value less
than1
7
...
The magnetic susceptibility is independent of temperature
Paramagnetic materials
1
...

2
...

3
...

4
...

5
...
Attracted by a strong magnet
7
...
Ferromagnetic materials have a large, positive susceptibility to an external
magnetic field
...
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2
...
They get their strong magnetic properties due to the
presence of magnetic domains
...
In these domains, large numbers of atom's moments (1012 to 1015) are
aligned parallel so that the magnetic force within the domain is strong
...

4
...

5
...

Components with these materials are commonly inspected using the
magnetic particle method
...
The magnetic moment, intensity of magnetisation and magnetic
susceptibility are all positive and quite large and magnetic permeability is of
the order of hundreds and thousands
...
The magnetic susceptibility decreases with rise of temperature
...
Antiferromagnetism, type of magnetism in solids such as manganese
oxide (MnO) in which adjacent ions that behave as tiny magnets (in this case
manganese ions, Mn2+) spontaneously align themselves at relatively low
temperatures into opposite, or antiparallel, arrangements throughout the
material so that it exhibits almost no gross external magnetism
...
In antiferromagnetic materials, which include certain metals and alloys in
addition to some ionic solids, the magnetism from magnetic atoms or ions
oriented in one direction is canceled out by the set of magnetic atoms or ions
that are aligned in the reverse direction
...
This spontaneous antiparallel coupling of atomic magnets is disrupted by
heating and disappears entirely above a certain temperature, called the Néel
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temperature, characteristic of each antiferromagnetic material
...
)
4
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The Néel temperature for manganese oxide, for
exam ple, is 122 K (−151° C, or −240° F)
...
Antiferromagnetic solids exhibit special behaviour in an applied magnetic
field depending upon the temperature
...
At higher temperatures, some atoms
break free of the orderly arrangement and align with the external field
...
Above this temperature, thermal agitation
progressively prevents alignment of the atoms with the magnetic field, so
that the weak magnetism produced in the solid by the alignment of its atoms
continuously decreases as temperature is increased
...
Ferrimagnetism, type of permanent magnetism that occurs in solids in
which the magnetic fields associated with individual atoms spontaneously
align themselves, some parallel, or in the same direction (as in
ferromagnetism), and others generally antiparallel, or paired off in opposite
directions (as in antiferromagnetism)
...
The magnetic behaviour of single crystals of ferrimagnetic materials may
be attributed to the parallel alignment; the diluting effect of those atoms in
the antiparallel arrangement keeps the magnetic strength of these materials
generally less than that of purely ferromagnetic solids such as metallic iron
...
Ferrimagnetism occurs chiefly in magnetic oxides known as ferrites
...
The oxygen ions are not magnetic, but both
iron ions are
...
The
iron(III) ions are paired off in opposite directions, producing no external
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magnetic field, but the iron(II) ions are all aligned in the same direction,
accounting for the external magnetism
...
he spontaneous alignment that produces ferrimagnetism is entirely
disrupted above a temperature called the Curie point, characteristic of each
ferrimagnetic material
...


Superconductivity
At very low temperatures, the electric and magnetic properties of some
materials such as lead, mercury and some oxides radically change
...
This phenomenon, which was discovered a
hundred years ago, is quite an impressive illustration of quantum physics on
a human scale: the many free electrons of the material merge into a quantum
wave which spreads across very large distances
...
Superconductivity also enables spectacular
feats of levitation
...
)
Superconductivity occurs in a wide variety of materials, including simple
elements like tin and aluminum, various metallic alloys, some heavily-doped
semiconductors, and certain ceramic compounds containing planes of copper
and oxygen atoms
...

Superconductivity does not occur in noble metals like gold and silver, nor in
ferromagnetic metals
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There also exists a class of materials, known as unconventional
superconductors, that exhibit superconductivity but whose physical
properties contradict the theory of conventional superconductors
...
)
There is currently no complete theory of high-temperature superconductivity
Properties of superconductors
Most of the physical properties of superconductors vary from material to
material, such as the heat capacity and the critical temperature at which
superconductivity is destroyed
...
For instance, all superconductors have exactly zero
resistivity to low applied currents when there is no magnetic field present
...

Zero electrical resistance
Suppose we were to attempt to measure the electrical resistance of a piece
of superconductor
...
If we carefully account for the resistance R of
the remaining circuit elements (such as the leads connecting the sample to
the rest of the circuit, and the source's internal resistance), we would find
that the current is simply V/R
...

According to Ohm's law, this means that the resistance of the
superconducting sample is zero
...

In a normal conductor, an electrical current may be visualized as a fluid of
electrons moving across a heavy ionic lattice
...
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This is called the Meissner effect
...

The Meissner effect is sometimes confused with the "perfect diamagnetism"
one would expect in a perfect electrical conductor: according to Lenz's law,
when a Changing magnetic field is applied to a conductor, it will induce an
electrical current in the conductor that creates an opposing magnetic field
...

The Meissner effect is distinct from perfect diamagnetism because a
superconductor expels all magnetic fields, not just those that are changing
...
When the material is cooled below the critical temperature,
we would observe the abrupt expulsion of the internal magnetic
field, which we would not expect based on Lenz's law
...
In Type I superconductors, superconductivity is
abruptly destroyed when the strength of the applied field rises above a
critical value Hc
...

In Type II superconductors, raising the applied field past a critical value Hc1
leads to a mixed statein which an increasing amount of magnetic flux
penetrates
the material, but there remains no resistance to the flow of electrical current
as
long as the current is not too large
...
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The mixed state is actually caused by
vortices in the electronic superfluid, called "fluxons" because the flux
carried by these vortices is quantized
...

Type I superconductors are those superconductors which loose their
superconductivity very easily or abruptly when placed in the external
magnetic field
...


After Hc, the Type I superconductor will become conductor
...
Type I superconductors are also known as soft superconductors because of this
reason that is they loose their superconductivity easily
...

d) Example of Type I superconductors: Aluminum (Hc = 0
...
0054)
Type II superconductors:
a)
...
As you can see from the graph of intensity of magnetization (M) versus
applied magnetic field (H), when the Type II superconductor is placed in the magnetic
field, it gradually looses its superconductivity
...


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b) The state between the lower critical magnetic field (Hc1) and upper critical magnetic
field (Hc2) is known as vortex state or intermediate state
...

c)
...

c) Type II superconductors obey Meissner effect but not completely
...


Applications of Superconductors
HIGH-ENERGY APPLICATIONS:
Superconducting inductors that must transport relatively high current densities in high
magnetic field have a variety of applicatrons:
1
...

2
...

3
...

4
...

5
...

6
...

7
...


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SUPERCONDUCTING MAGNETS:
Twisting of superconducting Strands:
An eddy current may be induced in adjacent strands of superconductor that are
embeded in a normally-conducting substrate
...
The eddy current is reduced by a practice
familiar to electrical engineers who wish to reduce external magnetic fields around lines
carrying alternating currents, that is, twisting of the individual pairs of superconductors
...


Josephson effect
When two superconductors are separated by a very thin insulating layer,
quite unexpectedly, a continuous electric current appears, the value of which
is linked to he characteristics of the superconductors
...
Since then, this superconductorinsulator-superconductor sandwich has been called a “Josephson junction”
...
If the
electric insulator separating the two superconductors is very thin (only a few
nanometres), then the wave can somehow spill out of the superconductor,
which enables the electron pairs to go through the insulator thanks to a
quantum effect called tunneling effect
...
Each
superconductor is characterized by a quantity called phase, with a subtle
signification
...

SHORT ANSWER QUESTIONS:

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1
...

2
...

3
...

Ans: αe = 4πεo R3 farad –m2 where R is the radius of the atom
...
What is the effect of dielectric on a capacitor?
Ans: Capacitance increases by ε’times, where ε’ is the dielectric constant of the
diectric
...
What is meant by complex dielectric?
Ans: The dielectric which is partly polarisable and partly conductive
...
What is meant by lossy dielectric?
Ans: The dielectric, which is partly conductive and partly polarisable
7
...

Ans: αi = e2/ωo2 [1/m1 + 1/m2] where ωo is the natural frequency; m1 and m2 are the
masses of the positive and negative ions respectively
...
Define dielectric loss
...

9
...

Ans: Nαe / 3εo = ε’ -1/(ε’ +2)
Where N = number of molecules per unit volume
...

ε' = dielectric constant
αe = electronic polarisability
10
...

Ans: E4 = P/3εo or E4 = (E/3)(ε’-1)
where E4 = electric field strength at a point produced by the polarization on the cavity
surface
...
Write Clausius equation
...
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D
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of molecules per unit volulme
ε' = dielectric constant
εo = permittivity of free space
12
...

13
...

14
...
electronic 2
...
ionic 4
...
interfacial
15
...

Ans: It is the dipole moment induced per unit field strength resulting from shifts of
Electron cloud relative to nucleus
...
Write down the macroscopic expression for polarization P
...

17
...

18
...

19
...


FILL IN THE BLANK QUESTIONS:
1
...

2
...


3
...


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4
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5
...

6
...
____________ Polarization depends strongly on temperature
...
In some dielectric crystals, it is found that mechanical strain may produce an
electrostatic charge on the faces of the crystal, which is known as ______
9
...

ANSWERS:
1
...
Induced 3
...
Ionic
5
...
Natural frequency 7
...
piezoelectricity

9
...
Materials, which exhibit electric polarization even in the absence of the applied
electric field, are known as ________materials
...
Dielectric constant, ε’ =?
a) εo/ε
b)ε*εo c)ε/εo d)ε-εo
3
...
The vectors D,E and P are related by the equation
...
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D
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Clausius equation can be written as
a) P = N εo α E
b)P = N α E c) E = N α P d) P = ε’(εo -1)E
6
...
If E is effective field between the plates of a parallel plate capacitor with a
dielectric medium and Eo is the field without the dielectric medium, then the
dielectric constant is given by
a) Eo/E b)E/Eo c)Eεo/Eo
d)εoEo/E
8
...
c field
across it
...
The imaginary part of the complex dielectric constant is a measure of absorption of
______________ that takes place in the material
...
The plot between dielectric constant ε’ and dielectric loss ε” is called
...

ANSWERS:
1
...
c

3
...
a

5
...
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a

7
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a

9
...
b



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