engg physics notes

7/27/2019 Engg Physics Notes

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Notes Prepared by SCE, Department of Physics

UNIT 1

MODERN PHYSICS

Prepared by: 1Mr.Jagadeesh Gowda GV, HOD, Department of Physics, 2.Ms.Shashikala BS, Asst. Prof.,

3.Mr.Gnanendra DS, Asst. Prof., 4.Ms.Bharathi D, Asst. Prof., Department of Physics, Sapthagiri Collegeof Engineering, Bangalore, Karnataka India-560057.

Syllabus:

Introduction to Black Body Radiation. Energy distribution of spectrum in a Black body,

various laws of Black Body Radiation. Photoelectric effect & Compton Effect -Explanation

Wave-particle dualism. De-Broglie Hypothesis, Characteristics of Matter waves. Davisson

& Germer experiment Phase velocity, Expression for phase velocity, Group velocity,

Expression for Group velocity Relation between phase velocity & Group velocity, Group

velocity & particle velocity, De-Broglie wavelength_____________________________________________________________________________________

Black Body:A perfect black body is the one which absorbs all the

radiation that is incident on it, no matter what the radiationfrequency is. Conversely when a perfect black body is heated itemits radiation at all frequencies.

Fig (1-1)

A black body can be constructed using a hallowobject coated inside with lamp black and having a small hole as

shown in fig (1-1). The radiation entering through the hole is almost completely absorbed due to

multiple reflections.

Energy distribution of a Black Body Radiation Spectrum:In 1859 Kirchhoff formulated two laws about black body radiation which are as follows

1. A black body not only absorbs the entire radiation incident on it but it also a perfectemitter at high temperatures.

2. The radiation emitted by a black body depends only on the temperature not on the natureof the body


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The variation of radiation curves are shown in the graph (1-2). The following observations canbe madefrom the radiation curves.

(1)There are different curves for different visible Graph (1-2)Temperatures(2)There is a peak for each of the curves

which indicates that the EM wave ofthat wavelength corresponding to the Infra redpeak is emitted to the largest extent atthat temperature to which the curve 6000Kcorresponds Energy

(3)The peak shifts from curve to curve 4000Ktowards the lower wavelength side 3000Kas higher temperatures are considered

ultra violet 0.4 0.8 1.0 2.0Wavelength Laws of Black Body Radiation:In order to explain spectrum of black body radiation number of laws have been putforward, notable among them are as follows

Stefans Law of radiation:In 1884 Stefan and Boltzmann showed that the total energy radiated by a black body per unitarea per unit time is proportional to the fourth power of its absolute temperature i.e.,

E

E = Where the Stefan iss constant.This law is experimentally verified, but it does not explain the energy distribution in the

spectrum of Black Body Radiation.

Wiens Displacement Law:According to Wiens observation the wavelength of maximum intensity m is inversely

proportional to the absolute temperature of the emitting body because of which, the peaks of theenergy curves for different temperatures get displaced towards the lower wavelength side

i.e., m

m T = constant = 2.898x10

-3 mK

Wien showed that the maximum energy Em of the peak emission is directly proportionalto the fifth power of absolute temperaturei.e., Em T5 Em = constant x T5Wien also deduced the relation, between the wavelength of emission and the temperature

of the source as ,

Ed= C1 -5, -d


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Where Ed is energy per unit volume for wavelengths in the range and +dC1 and C2 are constants

Drawbacks of Wiens law:

Wiens law suits only for shorter wavelength region and high temperature value of thesource. It failed to explain the gradual drop in the intensity for radiations whose wavelengths arelonger than the ones corresponding to the peak value.

Rayleigh- Jeans Law:According to the principle of equipartition of energy, an average energy kT should be

assigned to each mode of vibration. But, the number of vibrations per unit volume, whosewavelengths are in the range and +d is given by 8-4d.Hence the energy per unit volume for waves whose wavelengths are in the range and +d is

given by,Ed = 8

-4d.

This equation is known as the Rayleigh- Jeans Law.Rayleigh- Jeans Law correctly predicts the fall of intensity of the radiation towards the longerwavelength side.

Ultraviolet Catastrophe:

As per the Rayleigh- Jeans equation, the radiant energy increases enormously with thedecreasing wavelength i.e., the black body is predicted to radiate all the energy at very shortwavelength side. But in practice, a black body radiates chiefly in the IR and Visible region of theEM spectrum, the intensity of radiation decreases down steeply for shorter wavelengths. Thus,

the Rayleigh- Jeans Law fails to explain the lower wavelength side of the spectrum. Thus thefailure of Rayleigh- Jeans Law to explain the spectrum beyond the violet region towards thelower wavelength side of the spectrum is particularly referred to as Ultraviolet Catastrophe.

Plancks Radiation law:Max planck assumed that radiation is emitted in packets or bundles of energy called

quanta or photons instead of continues waves. All photons associated with light of frequencyhave the same energy given by

E = hWhere h = 6.625x10-34JS (Plancks constant)

= frequency of a photonThe assumption in the derivation of Plancks law is that the walls of the experimental black body

consist of a very large number of electrical oscillator, with each oscillator vibrating with afrequency of its own. .in the Planck brought two special in his theory.They are,

(i) only that value of energy could be possessed by an oscillator, which is an integralmultiple of h.i.e., the allowed energy values are defined in the set nhwhere n = 0,1,2,3,.etc.

(ii) An Oscillator may gain or lose energy by emitting or absorbing respectively aradiation of frequency whose value is given by =

, where is the difference


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in the value of the energies of the oscillator before and after the emotion or absorptionhad taken place.

Based on the above ideas he derived an equation which successfully explained the entirespectrum of the blackbody radiation.

It is given by Ed =

0 {}1 d (since =

)

The following graph (1-3) predicts the observations of all the three laws of blackbody radiationspectrum

Plancks law

Rayleigh-Jeans lawWiens

law

Ed

Reduction of Plancks law to Weins law:

For shorter wavelengths, = is large

When is large, ,- is very large. ,-

,-

,-

=

,

-

Making use of this in Plancks equation,

Ed = 0 { }1 dEd= C1

-5,-dWhere C1 = and C2 =

The above equation is the expression forWiens Law.

Reduction of Plancks law to Rayleigh- Jeans Law:For longer wavelengths, = is smallWhen is small,

is very small.Expanding ,- as power series, we have ,-= 1 + (Since

is very small its higher powerterms could be neglected.)


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Therefore ,- = = Making use of this in Plancks equation,

Ed = 0

1 dEd = dThe above equation is the expression for Rayleigh- Jeans Law.

Photoelectric Effect:The phenomenon of emission of electron from the surface of a metal when a radiation of

suitable frequency incident on it.The emitted electrons are known as photo electrons. The movement of these electrons constitutesphotoelectric current.The apparatus used to study photoelectric effect is shown in fig(1-4).

Radiations

G Evacuated glass tubeW W Quartz window

G C Photosensitive cathodeA Anode

e-

e-

e- m Micro ammeter

C e-

e-

e-

A Ba Battery

_

Ba mThe apparatus consists of an evacuated tube which contains two electrodes. The electrode E isthe metal plate which emits electrons on being illuminated with light of suitable frequency. It iskept at a lower potential with respect to the other electrode C which collects the photoelectronsto constitute a small current in the external circuit.The observations in photoelectric effect are not explained by wave theory of light. They areexplained by Einstein using Plancks quantum theory.Following are the observations in the photoelectric effect:

1. There is no emission of photoelectrons below a certain frequency of incident light whichis called threshold frequency.

2.

There is no time lag between the arrival of light on a metal surface and the emission ofelectrons.3. The retarding potential required to reduce the current to zero is known as the stopping

potential. The stopping potential varies only with the frequency and does not dependupon the intensity.

4. Increase in intensity of light increases the current.According to Einsteins theory forphotoelectric effect, the incident light consists ofphotons of energyE= h


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When the photon strikes the metal surface, it transfers all its energy to an electron. Acertain amount of this energy is required to release the electron from the surface and theremaining energy appears as kinetic energy of the electron.

Therefore h = W +

The minimum amount of energy required to release the electron from the surface is

known as the work function. When incident photon has just sufficient energy to liberatethe electron with zero velocity, the incident photon frequency is the threshold frequency0.Therefore Work function is W0 = h0If W= W0 , the kinetic energy will be maximum.

Therefore h = h0 +

If stopping potential is V, then

Therefore eV = = h (-0)

The above equation predicts that there will be no emission of photoelectrons for0.These observations confirm the particle properties of light waves.

Compton effect:When x-rays are scattered by solid medium, the scattered x-rays will normally have samefrequency or energy. However Compton observed that in addition to the scattered x-raysof same frequency or wavelength, their existed some scattered x-rays of a slightly higherwavelength that is lower frequency. This phenomenon in which the wavelength of x-raysshows an increase after scattering is called Compton Effect.Compton explained the effect on the basis of the quantum theory of radiation.

Considering the radiation to be made up of photons, he applied the laws of conservation

of energy and momentum for the interaction of photon with electron

Consider an x-ray photon of energy h incident on an electron at rest as shown infig(1-5).

After the interaction, the x-ray photon gets scattered at an angle with its energychanged to a value h1 and the electron which was initially at rest recoils at an angle.It can be shown that the increase in wavelength which is known as Compton shift is

given by

Where is the rest mass of the electron.


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When =90o, This constant value is called Compton wavelength.

Thus Compton Effect is also experimental evidence in favor of the quantum theory of radiation.

Physical significance of Compton effect :

In Compton effect the Compton shift is explained on the basis that the x-ray photons collide with

electrons and, during the collision the energy exchange between the two occurs as though it is a

kind of particle-particle collision. Thus, it demonstrates the particle nature of x-rays which we

know is electromagnetic waves, or, it signifies the particle nature of waves in general.

Wave particle dualism:

The experiments based on interference, diffraction and polarization in light shows that lightbehaves as waves. While the Black body radiation, Photoelectric effect and Compton Effect

show that the light behaves as particle.

Therefore the light has dual property and is known as wave particle dualism.The dual nature was observed by de-Broglie & in his study, assuming that what is true withenergy (X- ray/light) is also true with matter, as they are interchangeable according to Einsteinstheory of relativity, put forward an hypothesis stating that Since nature loves symmetry, if

the radiation behaves as particle under certain circumstances, then one can even expect

that entities which ordinarily behaves as particles to exhibit properties attributable to only

waves under appropriate circumstances.This hypothesis is known as de-Broglie hypothesis.

According to which when a particle has a momentum p, its motion is associated with awavelength called de-Broglie wavelength given by,

=

Where h is Plancks constant and p is momentum of the particle.

de-Broglie wavelength of an accelerated electron: For an electron acceleratedunder a potential difference of V, the energy acquired will be eV. If m is the mass & v is thevelocity of the electron, then the energy equation for non-relativistic case can be written as,

eV = ..(1)If p is the momentum of the electron, thenP = mvSquaring on both sides, we have, p2 = m2v2

or mv2 =

Using the above equation, eqn(1) becomes


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eV =

or

From de-Broglie hypothesis wehave, =

..(2)

Since h, m & E are physical constants the value of will also be constant.By substituting the values of the constants in eqn(2) we have

* + m Davisson & Germer Experiment (Experimental study of matter waves)

Davisson & Germer in 1927 designed an apparatus to determine the wavelengthassociated with electrons. The experimental set up is as shown in figure(1-6).

Experimental set up:It consists of a tungsten filament (F), variable voltage source, Nickel metal (N) and an electrondetector (D). When the filament is heated the electrons are liberated from it by thermionicemission. These electrons are passed through narrow slits in order to get a fine beam of electrons.This electron beam was accelerated and directed at the nickel target which was mounted on asupport by the applied voltage. When the electrons hit the target metal they get scattered aroundthe metal. The electron detector was mounted on an arc so that it could be rotated to different

angles () to observe the scattered electrons from the metal surface. Keeping the acceleratingpotential constant the intensity of electrons was measured by varying the angle . Polar graphsshown in graph (1-7), below show the dependence of electron intensity on .


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From the graph (1-7) we makefollowing observation, In the plot of 40 V,the variation was found to be smooth

devoid of any maxima and minima. Whenthe experiment was repeated withaccelerating potentials of 44V, 48V, 54V,60V, 68V a slight but distinct maximawas observed at 44V. With higherpotential curves, the maxima appearedmore and more pounced up to the case of54V curve and there afterwards it declined& faded away.

Davisson & Germer came out with a proposition to account for the ionization currentbecoming maximum for particular values of V & .

Let us consider only the electrons that are scattered at an angle of 50

0

under anaccelerating potential of V = 54V. As per the de-Broglie hypothesis, = m..(1)

Davisson & Germer furthersuggested that, just as X rays undergodiffraction when incident on a crystal thewaves associated with the incident beamof electrons also under diffraction whileincident on nickel crystal obeying theBraggs Law

Braggs law is given by2dsin =

nWhere is the

glancing angle, d is the inter planardistanceThe inter planar spacing is obtained fromx-rays analysis is found to be d=0.91 .The glancing angle by applying Braggs equation, = 2d x

It is seen that the values obtained experimentally using Braggs equation and de-Broglie equationagreed well and provide evidence for the existence of matter waves.

Group velocity and Phase velocity:

The equation for displacement of a wave is given byy= A where y is the displacement along y axis at the instant t is the angular frequency


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k is the propagation constant or wave numberx is the displacement along x axis at the instant t. The phase velocity for a wave is given

by; Vphase =

If a point is imagined to be marked on a travelling wave, then it becomes a representative pointfor a particular phase of the wave, and the velocity with which it is transported owing to the

motion of the wave, is called the phase velocity Group velocity is the velocity with which the envelope enclosing a wave group is called wavepacket, formed due to superimposition of two or more travelling waves of slightly differentwavelengths, is transported. It is the velocity with which the energy transmission occurs in awave

Expression for Group velocity (Vg):Let us consider two travelling waves of same amplitude, but of slightly different wavelengths

and frequencies. The two waves can be represented by the following two equations.y1= A ..(1)y2= A (2)where, y1 and y2 are the displacement in the directions normal to the direction of propagation atthe instant tA is the common amplitude is the angular velocitiesK and (k+ is the wave numbersX is the common displacement at the instant t is the differences in angular velocity and wave numbersThe resultant displacement y due to the superposition of the waves is given by;Y = y1 + y2

y = A + A (3)Since Equation (3) is written as , -, -As the difference in frequency of the two waves is very small because y1 and y2 form part of a

group and k terms in the sinepart can be neglected as they appear in combination with 2 and 2kwhich are quite large compared to these quantities. The same cannot be done in cosine part as these termsdo not appear in combination with 2 and 2k. Hence one can write the above equatio n as , - (4)

From a look at the equations (3) and (4) one can conclude that under the approximationmentioned above, the wave component of the original reference wave and the resultant wave remains the

same while the amplitude changes from A to , -The amplitude of the resulting wave is as per the cosine component of equation (4) and it reachesits maximum when cosine component value becomes 1. That is , - , - From the above equation one can write

=


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Group velocity (Vg). Under the limiting case k tending to zero the group velocity Vg can be written as

------------- (5)Relation between Phase velocity (Vp) and Group velocity (Vg):

Equations for group and phase velocity are given by

Vg = and Vp =

Where is the angular frequency of the wave, and k is the propagation constant.Therefore, kVPhaseTherefore, Vg =

( )= ..(1)

We know that,

Differentiating we get, Or . / Substituting the above in eqn (1) This is the relation between Phase velocity and Group velocity.

Relation between group velocity(Vg) and particle velocityWe knowEnergy of a photon E = hor ..(1)We know or =

.. (2)

Further, = (3)dividing eqn (2) by (3) ..(4)We know that,E =

where p is the momentum of the particle.


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Using the above in eqn(4) But,

where

is the velocity of the particle.

(5)We know that, group velocity ..(6) from eqns(5) & (6)

Relation between Velocity of light(c), Group velocity (Vg) and Phase

velocity(Vp):

We know that, phase velocity

But, But, This is the relation of velocity of light with Phase velocity and Group velocity.

Expression for De-Broglie wavelength (Derivation using Group velocity):

The Group velocity is given by, , where and and Thus

This leads to writing (particle velocity)Or, --------(1)Let m be the mass of the particle, vbe its velocity, and E be its total energy. If V is the potentialenergy of the particle, then

--------(2) h where is the frequency of the waves associated with the particle and h is thePlancks constant.

h --------(3)Let the particle be moving in a field of constant potential, that is ,Vis a constant. Thus whendifferentiated, eqn (3) becomes


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or, --------(4)

from eqns(1) & (4) Integrating,

Let the momentum of the particle be p, that is, p= also, postulating that the constant ofintegration is zero, we have, Or,

The above equation is the de Broglies equation.

Characteristics of matter waves:(1)Matter waves are the waves that are associated with a moving particle. The wavelength

of the wave is given by = (2)Matter waves are associated with charged as well as uncharged particles. Hence they are

not electromagnetic in nature.(3)The amplitude of the matter waves at a particular region and time depends on the

probability of finding the particle the same region and time.(4)Under the limiting case the relation between group velocity and the phase velocity is

given by , which leads to an interesting observation that matter waveshave the velocity more than that of light. This observation compel one to think that matterwaves are not the waves which can be physically felt but indicate only the probabilisticnature of these waves.

(5)The velocity associated with the electromagnetic radiation remains constant for all thewave lengths while the velocity of matter wave differs under different conditionsassociated with it.

(6)The wave and the particle properties are not exhibited simultaneously.


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Normalisation of wave function. Setting up of a one dimensional, time

independent, Schrdinger wave equation. Eigen values and eigen function. Application of

Schrdinger wave equationEnergy eigen values for a free particle. Energy eigen values of

a particle in a potential well of infinite depth.

The Uncertainty:

It is not possible for Simultaneous measurement of both the position and momentum (two

conjugate variables) of a particle with arbitrarily high precision. If one tries to measure theposition of a particle precisely there is an uncertainty in the measurement of momentum and vice

versa even with sophisticated instruments and technique, it arises from the wave properties

inherent in the quantum mechanical description of nature i.e. the uncertainty is inherent in the

nature.

Heisenberg's Uncertainty Principle:

Statement: It is impossible to measure two conjugate quantities simultaneously with

unlimited accuracy. Quantitatively the product of the uncertainties in the values of the two

conjugate quantities must be at least equal to or greater than

Heisenberg's derivation of the uncertainty relations, one starts with a particle moving all

by itself through empty space. To describe thus, one would refer to certain measured propertiesof the particle. Four of these measured properties which are important for the uncertainty

principle, are the position, momentum, energy, and the time. These properties appear as"parameters" in equations that describe the particles motion.

The uncertainty relations may be expressed as

Where,h Planck's constant Uncertainty in the position measurement.

Uncertainty in the momentum measurement

Uncertainty in the energy measurement. Uncertainty in the time measurement at the same time as the energy is measured.Uncertainty in the angular momentum measurement Uncertainty in the angle measurement at the same time as the angular momentum ismeasured

Applications of Uncertainty Principle:

1. The non-existence of free electron in the nucleus:


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The diameter of nucleus of any atom is of the order of 10 -14m. If any electron is confinedwithin the nucleus then the uncertainty in its position () must not be greater than 10-14m.According to Heisenbergs uncertainty principle,

(1)Where, = 10-14mOn substituting the values in eqn(1) We get Ns is being minimum uncertainty in the momentum of the electron which is equal to themaximum accurate value of momentum of the electron.i.e Ns .(2)According to the theory of relativity the energy of a particle is given by

.. (3)Where m0 is the rest mass of the particle and m is the mass of the particle with velocity v.Squaring the above equation (2) we get,

. (4)The momentum p of the body is given by or

multiplying by We get, .(5)Subtracting eqn(4) by eqn(3)We get , () Or ..(6)Substituting the value of momentum From eqn(2) and the rest mass as 9.11X10-31kgIn the eqn(6) we get,

The Kinetic energy of the electron becomes(6) [ ]Since the second term in the bracket is smaller in magnitude compared to the first term it isneglected. JOr The above value for the kinetic energy indicates that if an electron exist within the nucleus itmust have energy equal to or greater than . But the experimental results on decay show


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that the maximum kinetic energy of an electron is of the order of 3-4 MeV. Therefore theelectrons cannot exist within the nucleus.

Explanation of decay:When a radioactive nucleus under goes decay a nucleus is formed whose atomic

number increases by one unit but its mass number remains same.

i.e zXA

z+1YA

+ -1e0

+ i.e 0n11p1 + -1e0 + Where n, p, e- & represent neutron, proton, electron & neutrino respectively.An electron is emitted by the nucleus just when this process occurs, is called a -particle.

Kinetic energy of the -particles:The excess mass of the nucleus is converted in to kinetic energy of the emitted particles

as per Einsteins mass-Energy relation In decay since proton remains inside the nucleus theExcess mass converted to energy = kinetic energy of electron + kinetic energy of the neutrino (1)

If an electron are emitted with maximum kinetic energy the accompanying neutrino mustpossess minimum kinetic energy, since rest mass of neutrino is zero, the minimum kinetic energy

is almost zero. Practically it will be the kinetic energy of electron alone which accounts for the excess massconverted to energy as in eqn(1).A graph of number of electrons emitted in decay versus kinetic energy possessed by theemitted electrons is shown in graph.

Number of

-particle

Kinetic energy

2. Kinetic energy of the electron in an atom:In an atom electrons go round the nucleus in various orbits. Hence, the uncertainty in its

the position could be estimated and with this we can estimate its momentum and also the kineticenergy of electron.

We take an example of a H-atom in which the kinetic energy of an electron is estimated.We know that the radius of an H-atom is 0.5. The maximum uncertainty cannot exceed this value of 0.5.We can write, 0.5According to uncertainty principle Substituting the values,We get Kgms-1WKT, minimum uncertainty in the momentum of the electron which is equal to the maximumaccurate value of momentum of the electron.i.e Kgms-1


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But kinetic energy of the electron () Ji.e E = 3.8 eVActual kinetic energy of an electron in an H-atom is 13.6 eV. However the value of 3.8 eV

estimated on the basis of uncertainty principle, can be considered to be in reasonably goodagreement with the actual value.

Wave function:The function which describes the nature of matter wave is considered to be wave

function. The concept of wave function was introduced by Schrodinger in the matter waveequation.

Wave function is a variable whose periodic variations constitute matter wave.Wave function is represented by can be either positive, negative or complex. Hence is not a observable quantity. Usually consists of both real and imaginary parts.

The wave function is a complex quantity and can be written as Where A and B are real quantities.

Physical significance of Wave function:(1)Probability density:According to Max born interpretation, the probability of finding the

particle described by the wave function at a point (x,y,z) in space at time t proportionalto the value of|| . A large value of ||represents large probability of findingthe particle. The probability of finding the particle is zero at a point only if || atthat point.

The probability of finding the particle in a certain volume element dv = dxdydz is ||which is called the probability density if|| represents probability.Thus has no physical significance but || gives the probability of finding the atomicparticle in a particular region. the wave function is called the probability amplitude.

(2)Normalisation: As the particle has to exist somewhere in space, in the total probabilityof finding the particle is unity or 1.

i.e, || This type of mathematical equation for such condition is called normalization condition.A wave function which satisfies the above equation is said to be normalized equation.When the particle is bound to the limited regions the probability of finding at infinite

distance is zero. i.e, || is zero at x = Properties of the wave function:Physically acceptable wave function should satisfy the following conditions.

(1) is single valued everywhere: As probability is a single valued function, mustalso be a single valued function at every point in space.

Example:f1 At point x = P, the wave function has three


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f2 functions f1, f2 and f3. . f3 Since f1 f2 f3 the probabilities of findingthe particle has three different values at the samelocation which is not possible. Hence the wave

P x function is not acceptable.

(2) is finite everywhere: At x = Q, wave function is not finite. i.e, isInfinity. Thus it signifies a large probability ofFinding the particle at a single location x = Q which violates the uncertainity principle. HenceThe wave function becomes unacceptable.

Q x(3) and its first derivatives with respect to its variable are continuous everywhere:

At x = R, wave function stops at A & restarts at B.f1 A Between A & B wave function is not defined. If it is

f2 B an acceptable wave function the state of the systemAt x = R cannot be ascertained.R x Also, when is not continues the first and second

derivatives of will not be finite.(3)For bound state, must vanish at infinity or || must vanish at

Infinity:

Time Independent Schrdinger wave equation:The general differential equation of a wave travelling in x- direction with velocity v

having the wave function is given by .(1) The general solution of the eqn (1) is given by .. (2)\Where is the angular frequency,k is the wave number associated with the wave & is aconstant.Differentiating eqn(2) with respect to t twice

we get,

(Since i2 = -1) Substituting in eqn(1),

We get (3)

but


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E3=9E0 n=2

E2=4E0

|

|

n=1 E1=E0 x=0 x=a x=0 x=a

x x

Energy Eigen values for a free particle:Free particle is not under the influence of any kind of field or force. Thus it has zero

potential i.e, V = 0.

Hence Schrdinger wave equation becomes ,

- WKT, in the case of a particle in a infinite potential well the condition V = 0 holds good onlyover a finite width a and outside this region V = .

Since for a free particle V = 0 holds good everywhere, we can extent the case of a particlein a infinite potential well to the free particles case, by treating the width of the well to beinfinity i.e, a = WKT, the equation for Energy Eigen values for a particle in a infinite potential well is (1)

where n = 1,2,3rearranging eqn(1) we get

here for particle with constant energy E but confined in the well n depends on a. Hence as , . In the limiting case when a = , the particle is no more confined in any sort ofwell but free, at which time it also follows that n= , which means a free particle can have anyany energy i.e, the Energy Eigen values are infinite in number. the permitted energy values are continuous and there is no discreetness in the allowed energyvalues of there is no quantization of energy. Thus a free particle is a classical entity.


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Radiation interact with such a system in terms of transitions, it may be from higher level to lowerlevel and vice versa.

According to Max Planck only those radiations whose frequency is equal to can

interact with matter. i.e, for a radiation

where h is Plancks constant.

There are three ways with which interaction of radiation and matter takes place.1.Absorption:In this process the system absorbs the incident radiation (photon) in its ground

state E1 and moves to upper (higher) energy level E2 , provided the photon energy is h or E2 E2

E = hE1 E1

The energies of the levels can be written asE1 + h E2

In such a condition the atom is said to have made transition to the higher energy level and isindicated as atom*.i.e, atom + photon atom*.

2.Spontaneous emission:When an atom in its excited energy state E2 makes a transition to the ground state E1 on

its own, without any external agency, it emits a photon of energy h or This is known asspontaneous emission of radiation.

E2 E2E = h

E1 E1The energies of the levels can be written as

E2 - h E1In such a condition the atom is said to have made transition to the ground energy level and isindicated as atom.i.e., atom* - photon atom.The photon may be emitted in any direction. Two such photons which are spontaneouslyemittedby two atoms under identical conditions may not have any phase similarities and also they arenot in same direction and hence that are incoherent.

3.Stimulated emission:

If a photon having energy h interacts with an atom in the energy state E2, the photonstimulates or forces the atom to undergo transition to ground state E1 giving raise to anotherphoton of the same energy h or this process is known as stimulated emission.

E2 E2E = h E = 2h

E1 E1


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In such a condition the atom is said to have made transition to the ground energy leveland is indicated as atom.i.e, atom* atom + (photon + photon)

The emitted photon has the same phase wavelength, direction and polarization as that ofthe stimulating photon. Hence the light produced by stimulated emission is coherent.

RELATION BETWEEN EINSTEINS COEFFICIENTS AND EXPRESSION

FOR ENERGY DENSITY:Consider a system of atoms having ground state energy state E1 and excited state energy

E2 with number densities N1 & N2 respectively.

If photons of frequency are incident on the system of atoms there will be inducedabsorption. The rate of absorption of photons will be proportional to the number density N1 ofthe atoms in ground state and the energy density Ein the wavelength range +dof incidentradiation,

The rate of absorption N1 E The rate of absorption = B12 N1 EWhere B12 is a constant known as Einsteins co-efficient of induced absorption.

Atoms in excited state E2 can come down to ground state either through spontaneousemission orthrough stimulated emission of radiation. In the case of spontaneousemission, the rate oftransition of atoms from E2 to E1

does not depend on the energy density of the incident radiationand is proportional only to the number density of atoms in the excited state i.e.,Rate of spontaneousemission N2 The rate of spontaneousemission = A21 N2

Where A21 is a constant known as Einsteins co-efficient of spontaneous emission.

In the case of stimulated emission, a photon of frequency is required to stimulate theatoms. Hence the rate of stimulated emission is proportional to the energy density Eand thenumber density N2 of the atoms in the excited energy state E2 i.e.,Rate of stimulatedemission N2 E The rate of stimulatedemission = B21 N2 EWhere B21 is a constant known as Einsteins co-efficient of stimulated emission.

In a state of thermal equilibrium , the rate of transition of atoms from E1 to E2 must beequal to the total rate of transition from E2 to E1. The rate of absorption = Rate of spontaneousemission + rate of stimulatedemission

B12 N1 E= A21 N2 + B21 N2 EDeviding each term by N1 ,we get

B12 E= A21 + B21 E = =


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= 0 1From Maxwell-Boltzmann law,

=

= = Comparing above eqn with the energy density eqn from Plancks law of black body radiation

Ed = 0 {}1 d

We get,

------------(1)

And ------------(2)Equation (1) indicates that the probability of induced absorption is same as the probability ofstimulated emission.From Equation (2), For large . As for large energy difference between the ground staste andexcited state, the probability of spontaneous emission is much larger than the the probability ofstimulated emission.

REQUISITES OF LASER SYSTEM

1.EXCITATION SOURCE FOR PUMPING: Pumping is the process of supplying energy tothe laser medium with a view to transfer into the state of population inversion.There are different pumping techniques like,Optical pumping (a light source such as a flash discharge tube is used to illuminate the activemedium) e.g.; solid-state lasers.Electrical discharge method (the electric field cause ionization of the medium & raises it to theexcited state) e.g.; He-Ne laser, argon laser etc.,Direct conversion method (conversion of electrical energy into light energy takes place) eg;semiconductor diode laser.

2. ACTIVE MEDIUM: in a laser medium only a small fractions of atoms of a particular typehave energy levels suitable for achieving population inversion. such atoms can produce morestimulated emission than spontaneous emission and cause amplification of light hence theseatoms are called active centers. The rest of the medium acts as host and supports active centers.The medium hosting the active centers is called an active medium or this is the medium whichwhen excited reaches the state of population inversion and promotes stimulated emissionsleading to light amplification.


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The energy levels whose life period is unusually long, compared to normal energy levelsare called meta stable states. Atoms excited to metastable state remain there for anappreciable time which of the order of 10-6 S to 10-3S which is 103 to 106 times the lifetimes of the normal energy levels.Example: same as above explanation along with diagram.

HELIUMNEON LASER:

Construction: It consists of a discharge tube with a diameter of 1 to 1.5 cm and of length 1m.the tube is filled with a mixture of He-Ne gases in the ration 10:1. Brewsters windows are sealedto the tube at both of its ends in order to get a plane polarized beam. Two optically plane mirrorsare fixed on either side of the tube out of which one is completely silvered and other is partiallysilvered so that some of the incident laser beam could be trapped by transmission.

Working: in He-Ne laser Ne atoms energy levels are suitable for laser action hence they are theactive centers & the pumping technique involved is electric discharge method.


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When a voltage of about 1000V is applied across the two electrodes, a glow discharge of thegases is initiated in the tube. During discharge many electrons are rendered free from the gasatoms. These free electrons accelerate towards anode at which time they begin colliding with Heatoms in their path since He is the majority gas present. This kind of collision is called 1st kindcollision where the He atoms are excited to the two energy levels 21S &23S which are metastable

states for helium. This process can be written as,e1+He e2+ He*Where e1 & e2 are the energy values of the electrons before and after the collision and He & He*are the helium atoms in the ground and excited states. Since 21S &23S are metastable states, theatoms remain there for a relatively long time, which leads to an increase of population in each ofthem.Now, for neon gas there is a close coincidence in energy between two of its excited statesdesignated as 3s and 2s with the two metastable states 21S & 23S of the He atoms. Because of thematching of energy levels resonant energy transfer takes place from helium to neon atoms. As aresult, the neon atoms get elevated to the 3s and 2s states, whereas the helium atoms return to theground state. This type of collision is called 2nd kind collision which can be represented as,

He*+Ne He+ Ne*Where Ne and Ne* are the neon atoms in the ground and excited states.Thus the population of 3s & 2s levels of neon increases rapidly which leads to populationinversion and also three main types of transitions become available between higher states tolower ones.

1. Transition from 3s to 3p level gives rise to radiation of wavelength 33912 which is ininfrared region.

2. Transition from 3s to 2p level gives rise to radiation of wavelength 6328 which is invisible region.

3. Transition from 2s to 2p level gives rise to radiation of wavelength 11523 which is alsoin infrared region.

Fallowing the 3 transitions described earlier, the atoms from 3p and 2p levels undergospontaneous transitions to 1s level. But, 1s level is a metastable state for neon, because of whichits population increases rapidly. This in turn adversely affects the population inversionconditions for 3s & 2s levels. This problem is counteracted by considering the collision of 1slevel atoms with the walls of the container.Thus starting from the stage of excitation of helium atoms to the stage of neon atoms coming tothe ground state, the cycle works continuously and hence He-Ne laser is referred to as acontinuous- wave laser.

Energy level diagram:Resonant energy transfer

Metastable 3S21S 20.61 eV 20.66 eV

3.39m

23S Metastable 2S 632819.82 eV 19.78 eV

1.15m 3P


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When the junction is forward biased by applying suitable voltage the electrons are injected fromthe n type region and holes are injected from the p type region into the junction. A current begins

to flow and at low forward current level, the electron-hole recombination causes spontaneous

emission of photons and the junction acts as an LED.

As the forward current trough the junction is increased the intensity of the light increaseslinearly. However, when the current reaches a threshold value the carrier concentrations in thejunction will rise to a very high value. As a result the conduction band or the upper energy levelsare having a high electron population while the valence band or the lower energy levels arevacant. Therefore the condition of population inversion is attained in the narrow junction regionwhich is then named as inversion oractive region. At this stage, a photon released by aspontaneous emission may trigger stimulated emissions over a large number of recombinationsleading to the buildup of laser radiation of high power (around 9000 ).Energy level diagram:

EC

EVEF

EFEC

p-type n-type EVdepletionregion

CHARACTERISTICS OF LASER:

1.Directionality: In laser ,the active medium is in a cylindrical laser cavity. Any light that istravelling in a direction other than parallel to the cavity axis is eliminated and only the light thatis travelling parallel to the cavity axis is selected and reinforced. Light propagating along theaxial direction emerges from the cavity and becomes the laser beam. Thus, a laser emits lightonly in one direction.2.Monochromaticity: If light coming from a source has only one frequency(single wavelength)

of oscillation, the light is said to be monochromatic and the source a monochromatic source.Light from traditional monochromatic sources spreads over a wavelength range of 100 to 1000. On the oter hand, the light from lasers is highly monochromatic and contains a very narrowrange of a few angstroms (< 10 ).3.Coherence: If the phase difference between different wave fronts of the propagating light ismaintained same then the light is said to be highly coherent.There are two types of coherence, namely a) temporal coherence and b) spatial coherence.


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Temporal coherence is the correlation of the phase characteristics of the propagating light wavesat the same reference plane located in their path, but at different instants.Spatial coherence is the correlation of the phase characteristics of the propagating light wavesat different points in space at the same instant of time.

4. Light intensity: Because of thephase

correlation that exits continually between thephotons issued from a laser source, the laser light ishighly intense.5. Focussability: Since laser light is highlymonochromatic and also a highly collimated beam, itcan be brought to a sharp focus by a lens. However,because of diffraction effects the beam cannot comeexactly to a point focus but will be focused over aspot.

APPLICATIONS OF LASERS:Lasers find variety of applications in various fields due to their properties which are

much different from the ordinary light. Few applications like laser welding, cutting, drilling,

measurement of pollutants in the atmosphere and holography are discussed here.

LASER DRILLING:

In case of a conventional drilling a drill bit is held to the specimen with large amount of forcebetween them apart from the force used to hold the drilling machine and the specimen. Thisbrings about large amount of mechanical friction and hence most of the energy gets wasted inproducing of heat, sound and other energies. The heat energy produced creates an unevendistribution and in the process either the specimen gets distorted or spoiled hence, proper

lubricant is essential. Apart from this the specimen also puts the force on the drill bit resulting into deformation of drill bit. This leads to non-uniform drills created by the same drill bit. Apartfrom this if the drill is needed to be very small in diameter then having a drill bit with highmechanical strength of that size becomes impossible but in the todays world of miniaturizationthus becomes essential. These practical problems can be overcome by using high power lasers asthere is no mechanical contact between the laser and the specimen and the highly directionallaser can also be concentrated on a narrow space. Drilling can be done at any angle and throughvery hard and brittle materials. However the drilling through lasers needs a high precession bothin space and time. Typical drilling through laser is as described below.

High energetic pulses of 10-4 to 10-3 s duration are made to fall on the material. The heatgenerated due to these pulses evaporates the material locally, thus a hole will be been formed.

Nd-YAG laser is used for metals and the CO2 laser is used for metallic and non-metallicmaterials. The pulsed laser exposure is controlled through a time controller, as any variation inthis will result in not only the variation in depth but also in width. Further, to avoid accumulationof molten metal in the drill a high-pressure inert gas preferably argon gas is passed. A typicalschematic diagram is shown in figure. The drills created through Lasers are highly uniform.Since lasers can be concentrated on a small space very small drills can be created.

LASER WELDING:


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that happens at lower temperature is directly proportional to the concentration of the material.This is achieved usually by making a continuous light to fall on the substance known asAbsorption technique. By measuring the absorption and the wavelength the quantitative andqualitative analysis of the elements present are done.

But in case of air pollution the quantity is so small that even this method finds it difficult

to identify using an ordinary light source. Thus to achieve the accuracy, lasers are used as asource of light because of their important properties like monochromatic, high radiation densityand highly concentrated energy so that it can even interact with the smallest percentage ofelement present in the atmosphere. Pollutants can also be identified using Raman Back-scatteringmethod. In this method the laser light is made to pass through the sample and the spectrum isrecorded for the transmitted light. As the laser is a monochromatic source the spectrum isexpected to give single intense line corresponding to the frequency of laser light. But also itconsists of several lines very close to the intense line on either side of it. It is because of Ramanscattering. These spectral lines are called side bands. The side bands in the Raman spectrumobserved at certain frequencies correspond to oscillating frequency of the molecule plus or minusthe incident frequency. Each molecule has different oscillating frequency and hence the different

molecules produce side bands at different frequencies. With the help of these side bands one canfind the type of pollutant present in the sample.

HOLOGRAPHY

When a photograph of an object is taken all that we see is the amplitude variation but nopath difference. This leads to a two dimensional view. In view of this, there is a necessity that atechnique has to be developed, such that when photograph is taken it should have the informationabout amplitude variation and path difference between the two points. This is what is known asHolography. The information about the path difference between the two points is nothing but theinformation regarding the phase difference between the two rays coming from two differentpoints, That is, to have a holography what that we need is the information regarding amplitude

variation and phase variation. The only experiment which can give us both these information isan interference experiment and hence the techniques of holography has to essentially dependupon the interference phenomena.

In principle holography can be had using multiple wave lengths of source but a singlecolor will give a better information regarding the phase difference and hence use ofmonochromatic source of light is preferable. Hence, a laser as a source becomes very importantin holography because not only it is monochromatic but also it has both the phase and temporalcoherence.PRINCIPLE

The idea of recording the whole information (Holography) of a three dimensional objectson a two dimensional photographic film was conceived by Dennis Gabor in the year 1948. The

principle is, superimposing the wave that is scattered by the object with another coherent wavecalled reference wave. These two waves produce the interference pattern on the recordingmedium which is the characteristic of the object. This is known as recording process. Theinterference pattern which was recorded consists of amplitude distribution and the phase of thewaves scattered from the object hence it is called hologram.

Construction of a hologram and reconstruction of the image are as explained below.


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Consider two coherent waves P and Q aremade to fall on the photographic plate with differentangles of incidence as shown in the Figure.Superposition of these two waves results in

interference fringes on the photographic plate whichcan be observed after developing the film. Thesefringes carry the information regarding phase andintensity of the beam.

If once again the coherent waves P and Q are made to fall on the film at the samepositions then these waves pass through fringe pattern and the waves P / and Q/ can be observedfrom the transmission side these waves appear as if they are the continuation of the waves P andQ respectively (Figure).

If one of the incident waves say Q is blocked and only onewave P is made to fall on the film. This wave gets diffracted bythe fringes that are present on the film and the emergent wave getssplit in to two parts. One of the part gets deviated from its path andemerges as if it is the continuation wave Q (Q/) which is notpresent there and the other part emerges as P/ Figure 16. In this

way in holography, the arrangement is done to get one of thecoherent waves reflected from the object called object beam on tothe photographic plate when the reference beam is made to

incident directly on the photographic plate in the sameposition as explained above. The reference beam getsdiffracted by the hologram and produces secondarywavelets. These secondary wavelets superimpose oneach other and produce maxima in certain directionsand generate a real and virtual images of the object. Onthe transmission side one can view the real image andon the other side virtual image of a three dimensional

object Figure.


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RECORDING:Recording of the image of an object can be done in two ways(i) wave front division technique and(ii) Amplitude division technique

In both the techniques the object and a mirror are placed one next to the other as shown in figure.

The expanded laser beam is then directed on this arrangement in which a part of the laser beam isincident on the mirror and the rest is incident on the object. A photographic plate is placed insuch a position that it receives the light reflected from both the mirror and the object. The part ofthe light reflected from the mirror and is incident on the photographic plate will be in the form ofplane wave fronts and this is called the reference beam.

When laser is incident on an object every point ontheobject scatters the incident light all around. Due to sucha scattering spherical wave fronts proceed from eachpoint on the object, a part of each of which will be

incidenton the photographic plate and this is called the objectbeam.The photo sensitive surface responds to the resultant effectof interference between the spherical wavelets of theGABOR ZONE PLATEobject beam and the plane waves of the reference beam.Thus the interference effect is recorded on the plate and itconsists of concentric circular rings or zones that marksuccessive regions of constructive and destructiveinterference. The ring pattern is called Gabor Zone Plate

pattern.

RECONSTRUCTION OF THE IMAGE:When the reference beam is made to incident directly on the photographic plate in the same

position as explained above. The reference beam gets diffracted by the hologram and producessecondary wavelets. These secondary wavelets superimpose on each other and produce maximain certain directions and generate a real and virtual images of the object. On the transmission sideone can view the real image and on the other side virtual image of a three dimensional object.

Since the beam is incident on the entire hologram each and every zone plate participates in thisprocess to regenerate both a real and virtual image of the object point by point.

APPLICATIONS1. HOLOGRAPHICINTERFEROMETRY.

The one of the important testingprocess is non destructive testing ofspecimen which basically depends uponinterferometric methods. The nature ofholographic process to give objectiveanalysis of a specimen when illuminated


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with a reconstruction wave allows us to have interference leading to a method called doubleexposure holographic interferometry. This uses the photographic process between the objectwave and a reference wave of a specimen which is under stress alternatively to create ahologram. The holograms so created give us two object waves when illuminated withreconstructive wave, the first one for an unstressed object and the other one for the stressed

object. When these are overlapped again they gives us the interference pattern leading to theinformation about the qualitative and quantitative stress distribution. Since interference isinvolved in the whole process the slightest variation in the stress distribution can beanalyzed.

There is another type of holography known as variant holography which gives us what isknown as real time interferometry in which a hologram is formed of an unstrained object andthe image produced by the hologram is superimposed on the reconstructed object of astrained specimen. If the object undergoes any strain, fringes will be formed and this can bestudied as function of time. Such type of studies is used in study of vibrating objects likemusical instruments etc.2. MICROSCOPY

The very principle of holography which involves the measurement of path variationgives us a tool in which depth measurements are involved of microscopic level which areconstant or transient in nature. The ordinary microscopic studies involving interference canbe used for non transient variations but are inadequate for transient microscopic event.Therefore to study the transient phenomena in a certain volume hologram comes as handywhich is recorded over a time and event and can be studied later through a hologram as it isfrozen in this dynamic holography. Such studies can be utilized in the analysis of cloudchamber, rocket engine and aerosols etc.3. CHARACTER RECOGNITION

This is used in optical image in which the recognition of a character is essential. Thisinvolves what is known as cross correlation of character and image, the difference of which

is measured through holography process.4. PRODUCTION OF HOLOGRAPHIC DIFFRACTION GRATINGS.Two laser beams one as reference and another as a sample having a constant phase

difference are superimposed such that they produce interference. The laser source used willbe plane polarized and hence alternatively bright and dark lines of uniform thickness will beformed. Using this, holograms are produced which are nothing but diffraction gratings. Sucha uniform formation of gratings is impossible as all other methods involve mechanicalapproach which needs a very high control.5. INFORMATION CODING

In the present era of quantum computing which involves optical coding requires theinformation to be stored in a compact space and in more secured manner. This is achieved

through virtual storage of information through a hologram. The information can bereconstructed and acquired through this hologram.


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UNIT-5SUPER CONDUCTIVITYANDOPTICAL FIBERS

Prepared by: 1Mr.Jagadeesh Gowda GV, HOD, Department of Physics, 2.Ms.Shashikala BS, Asst. Prof.,3.Mr.Gnanendra DS, Asst. Prof., 4.Ms.Bharathi D, Asst. Prof., Department of Physics, Sapthagiri Collegeof Engineering, Bangalore, Karnataka India-560057.

Syllabus:Introduction to Super conductivity, Concept of critical temperatures & critical field,

effect of magnetic field. Miessner effect, Type-1 & Type-2 Super conductors & their

applications. BCS theory, Concept of cooper pair &their motion in the lattice. High

temperature superconductors Applications: Maglev vehicle, SQUID, Super conducting

magnets. Problems. Introduction to Optical fibers, Total internal reflection, Critical angle,

structure of Optical fiber, Concept of numerical aperture, Expression for numerical

aperture Propagation mechanism in Optical fibers. Acceptance angle, condition for

propagation, V number. Types of Optical fibers (a) Step index (b) Graded index, Modes of

transmission (1) Single mode (2) Multi mode. Attenuation mechanism, attenuation

coefficient. Applications: Block diagram discussion for point- to-point communication.

Problems on Optical fibers.

---------------------------------------------------------------------------------------------------------------------

INTRODUCTION TO SUPERCONDUCTIVITY:

The resistance offered by certain materials to the flow of electric current abruptly dropsto zero below a threshold temperature. This phenomenon is called Superconductivity and thethreshold temperature is called critical temperature.Temperature dependence of resistivity of a metal:

All metals are good conductors of electricity. These conductors have loosely bound electrons intheir outermost shells. These electrons are known as free electrons.During the flow of current in a metal, the electrons leave the atoms to which they were bound toand move in a general direction i.e., the fields direction. Because of the loss ofelectrons, theatoms become positive ionic cores. Due to the thermal excitation, the atoms will always beoscillating about fixed positions in the framework of the metal i.e., lattice. These vibrations arecalled lattice vibrations.The resistance of a metal to the flow of current is caused by the scattering of the conductionelectrons by the lattice vibrations. When the temperature increases, the amplitude of the latticevibrations also increases there by increasing the resistance. The resistance decreases withtemperature and reaches a minimum value at T=0K. The residual resistance at T=0K is due to theimpurities in the metal.The variation is expressed by the Matthissens rule, Where is the resistivity of the given metal


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is the residual resistivity is the temperature dependent part of resistivity.Temperature dependence of resistivity of a Superconductor:The resistance of a superconductor in the non-superconducting state decreases with decrease inthe temperature as in the case of a normal metal up to a particular temperature Tc. At Tc ,the

resistance abruptly drops to zero. Tc is called the critical temperature and signifies the transitionfrom normal state to the super conducting state of the material under study. The criticaltemperature is different for different superconductors.Ex: Mercury loses its resistance completely and turns into a superconductor at 4.2K.BCS Theory of Superconductivity:According to the BCSTheory an electron moving through crystal lattice creates a slightdistortion in the lattice. This is because of the coulomb forces between the negatively chargedelectrons and the positively charged lattice. If this distortion persists for a long time then apassing electron can be affected by it. The effect of this phenomenon is that current is carried ina superconductor not by individual electrons but by bound pair of electrons called cooper pairs.The BCS theory is based on a wave function in which all the electrons are paired. The interaction

between one of the electrons in a pair and the lattice does not affect the total momentum of acooper pair. As a result the flow of electrons continues indefinitely.The lattice vibrations are quantized in terms of phonons. Therefore the process described abovecan be called as electron-lattice-electron interaction via the phonon field. During this interactionthere is a reduction of energy. The interaction between two electrons may be considered to be acase of attraction between the two. Cooper has shown that the attractive force between the twoelectrons is a maximum when the two electrons have equal and opposite spins and oppositemomentum. At temperatures below the critical temperature the attractive force between the twoelectrons will exceed the force of repulsion. This leads to the formation of Cooper pairs. aCooper pairs can be defined as a pair of electrons bound together by the interaction between twoelectrons with opposite spins and opposite momentum in a phonon field.

When the electrons flow in a material in the form of cooper pairs they do not encounterscattering the resistance factor simply vanishes the conductivity becomes infinity the result issuperconductivity.

Meissner Effect:A superconducting material kept in a magnetic field expels the magnetic flux out of its bodywhen it is cooled below the critical temperature and thus becomes perfect diamagnetic. Thiseffect is called Meissner Effect.Consider a superconducting material above its critical temperature a primary coil and asecondary coil are wound on the material. The primary is connected to a battery and a plug keyK. the secondary coil is connected to the ballistic galvanometer. When the key K is pressed the

primary circuit is closed and the current flows through the primary coil which sets up a magneticfield in it. The magnetic flux immediately links with the secondary coil. This amounts to achange in flux across the secondary coil and hence a momentary a current is drawn through theBG which shows the deflection. Since the primary current is steady the magnetic flux will alsobecome steady and the flux linkage with the secondary becomes constant as there is no furtherchange in the flux linkage in the secondary coil ,the current will no more be driven in thesecondary circuit. Now the temperature of the superconductor is decreased gradually as soon asthe temperature croses below the critical temperature the BG suddenly shows the deflectionindicating that the flux linkage with the secondary coil has changed.


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Critical Field:The strength of the magnetic field required to just switch a material from superconducting stateto normal state is called critical field.Temperature dependence of resistivity of Critical feild:If T is the temperature of the superconducting material, Tc is the critical temperature, Hc is the

critical field and H0 is the field required to turn the superconductor to a normal conductor at 0K ,then the relation for critical field is given by ,= --------------- (1)Under the influence of a magnetic field whose strength is greater than , the material can nowbecome superconducting however low the temperature may be.Types of superconductors:

Type I superconductors(soft superconductors)

Type I superconductors is perfectly diamagnetic in the super conducting state. It possesses anegative magnetic moment.

Immediately after the applied magnetic field H exceeds the critical field HC the materialloses its diamagnetic property and becomes normal. The magnetic flux will penetrate throughoutthe body. The resistance of the material rises from zero to a value as applicable to a normalconductor. The critical field of Type I superconductors is very low. The low Hc value excludesthem from being used for applications.

Type II superconductors(hard superconductors)

Type II superconductors has two critical magnetic fields. When the applied magnetic fieldstrength is less than the critical value the magnetic flux is completely expelled from the bodyand the body behaves like a perfect diamagnetic. When the field strength cxceeds themagnetic flux partially fills the body. As the field is increased more and more of the fluxpenetrates the body. The diamagnetic property of the material gradually decreases. When thefield strength exceeds the second critical value the flux fills the body completely. Thediamagnetic property vanishes and the body becomes a normal conductor. Between the lowercritical field and the upper critical field the material is said to be in a mixed state calledthe Vortex state. In this state there is a flux penetration but the material retains the zeroresistance. Hence it is still a super conductor in this state. As the field increases the body

becomes a normal conductor. The upper critical field is many more times the lower criticalfield hence they find use in the buildup of devices which work in high magnetic fields suchas super conducting magnets.


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The atoms / molecules are electrically neutral. But when atoms or molecules are brought closer

together, a repulsive force operates between the similar charges in the atoms or molecules. An

attractive force operates between the dissimilar charges. The ultimate force, holding the particles

together in solids, is the resultant of attractive and repulsive forces (figure 1)

ttr

ctiv

f

rc

li

f

r

r0

Inter atomic/inter molecular distance

r0 = the equilibrium distance between atoms or molecules


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There appears at least distance, at which the particle cluster is the most stable. This minimum distance

between the particles is the equilibrium distance (r0). The arrangement of particles in crystals is decided

by the nature of bond between the particles and the value of the equilibrium distance. The atoms /

molecules in the solids are held together either by (1) Ionic bonds, (2) Covalent bonds, (3) Metallic

bonds or (4) Molecular bonds.

Crystal Structure: In 1848, A French crystallographer Bravais was the first person tointroduce the concept of space lattice (which is a mathematical concept) to describe the crystal

structures.

SPACE LATTICE: A space lattice can be generated by putting infinite number of points in space in such a

way that the arrangement of points about a given point is same as at any other point. Each lattice point

represents the location of an atom or particular group of atoms of the crystal. Intersection of any two

lines in the figure 2 is a lattice point.

Figure 2

Basis:A set of an identical atom/s which is correlated to lattice points is called basis.

Crystal structure: A crystal structure is formed when the basis is substituted in the space lattice


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i.e. Lattice + Basis = Crystal structure (figure3).

Bravais Lattice: The Bravais Lattice has infinite number of lattice points in it. If a set of identical

atoms / molecules are substituted in the space lattice then the lattice is said to be Bravais lattice. The

surroundings of any atom/molecule is same as any other atom/molecule in the lattice. Otherwise it is

said to be non-Bravais lattice. Below are the some figures representing both Bravais lattice and non-

Bravais lattices.

+ =

Lattice + Basis = Crystal Structure

+ =

Fig 4(a)

/+ =


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BASIC VECTORS:

To represent the position of lattice points a coordinate system is required.

A coordinate system is used to represent the position of lattice points in space lattice. The periodically

repeating arrangement of all lattice points in space can be described by the operation of parallel

displacement called a translation vectorR .

Figure 5

Let a and b be two vectors having equal magnitudes and oriented along AB and AC respectively as

shown in the figure 5. With a and b as coordinate vectors, the position vector R of any lattice point

can be written as,

R = n1a + n2b

Where, n1 and n2 are integers, whose values depend upon the location of lattice points with respect to

the origin.


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TYPES OF UNIT CELL:

1. Simple cubic (sc)

2. Body centered cubic (bcc)

3. Face centered cubic (fcc)

4. Base centered (bc)

Simple cubic: (sc)

When a unit cell contains points only at the corners, the arrangement is the simplest and the cell is

referred as simple cubic or simple primitive or p-type cell. There is no lattice point inside the cell.

Body centered cubic: (bcc)

There is one point at each of the eight corners and one lattice point at the center of the cell. Totally

there are 9 lattice points in this unit cell. It is also called as I type cell.


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Face centered cubic: (fcc)

There is one lattice point at each of the eight corners and one lattice point at the center of each of the

six faces of the cubic cell. Totally there are 14 lattice points in this structure. It is known as F-type cell.

Base centered cell : (bc)

There is one lattice point at each of the eight corners and one lattice point at each of top and bottom

faces. There are totally 10 lattice points in this cell. It is also called as C- type cell.

CRYSTAL STSTEMS:The classification of crystals into seven crystal system follows from the symmetry of the primitive cells.

The systems can be distinguished from one another by the angles between the axes and the intercepts

of the faces along them. Geometrical considerations show those seven sets of three axes called

crystallographic axes, which are sufficient to construct all crystal lattices. This leads to the classifications

of all crystals into seven crystal systems.

The seven basic crystal systems are;


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1. CUBIC2. TETRAGONAL3. ORTHORHOMBIC4. TRIGONAL OR RHOMBOHEDRAL5. MONOCLINIC6.

TRICLINIC7. HEXAGONAL

(1) Cubic:a = b = c

= = = 90

simple cubic(sc)

body centered cubic(bcc)

face centered cubic(fcc)


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abc

= = 90

Simple base centered

(5) Trigonal or Rhombohedral:

a = b = c

= = 90

Rhombohedral system

(6) Triclinic:

abc

90


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Triclinic system

(7) Hexagonal :

a = bc

= = 90, = 120

Hexagonal system

LATTICE PARAMETERS:

To completely illustrate the crystal structure, the basic minimum parameters required are:


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(1) The inter atomic or intermolecular molecular distance in x-, y- and z- direction (a b and c)

(2) The angles (, and ) between the x and y, y and z and z and x axes.

The angle between the x and y axes is taken as , the angle between the y and z axes is taken as ,

angle between the Z and x axes is taken as are called the crystal parameters(figure 7 represents cubic

structure).

Therefore, the acceptable way to study structure of crystals is in terms of inter-lattice distances

and the inter-planar angles, i.e. in terms of the crystal parameters.

Crystal Systems:

Bravais demonstrated mathematically that, in 3 dimensions there are only 14 different types of

arrangements possible theoretically for Bravais lattices in seven crystal systems.

The 14 Crystal lattices are represented in table1 and the crystal systems with unit cells in table2.

Table 1 Seven Crystal Systems and 14 Bravais lattices

Crystal SystemNo. of

lattices

14 BLUnit Cell Coordinate Description

1 Triclinic 11. Simple cubic OR

primitive

abc

90

2 Monoclinic 2

2. simple cubic

abc = = 90

3. Body centered

3 Orthorhombic 4 4. Simple cubic abc
http://www.coolphysics.com/crystals/bravais/triclinic/triclinic.htmhttp://www.coolphysics.com/crystals/bravais/monoclinic/monoclinic.htmhttp://www.coolphysics.com/crystals/bravais/orthorhombic/orthorhombic.htmhttp://www.coolphysics.com/crystals/bravais/orthorhombic/orthorhombic.htmhttp://www.coolphysics.com/crystals/bravais/monoclinic/monoclinic.htmhttp://www.coolphysics.com/crystals/bravais/triclinic/triclinic.htm


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5. Base Centered = = = 90

6. Body Centered

7. Face Centered

4 Tetragonal2 8. Simple cubic a = bc

= = = 909. Body Centered

5 Trigonalor

(Rhombohedral)

110.

Simple cubica = b = c

= = 90

6 Hexagonal 111.

Simple cubica = bc

= = 90, = 120

7 Cubic 3

12. Simple cubic

a = b = c

= = = 9013. Body centered

14. Face centered

Directions and Planes in a Crystal Lattice:
http://www.coolphysics.com/crystals/bravais/orthorhombic/base_centered_orthorhombic.htmhttp://www.coolphysics.com/crystals/bravais/orthorhombic/body_centered_orthorhombic.htmhttp://www.coolphysics.com/crystals/bravais/orthorhombic/face_centered_orthorhombic.htmhttp://www.coolphysics.com/crystals/bravais/tetragonal/tetragonal.htmhttp://www.coolphysics.com/crystals/bravais/tetragonal/body_centered_tetragonal.htmhttp://www.coolphysics.com/crystals/bravais/trigonal/trigonal.htmhttp://www.coolphysics.com/crystals/bravais/trigonal/trigonal.htmhttp://www.coolphysics.com/crystals/bravais/hexagonal/hexagonal.htmhttp://www.coolphysics.com/crystals/bravais/cubic/cubic.htmhttp://www.coolphysics.com/crystals/bravais/cubic/cubic.htmhttp://www.coolphysics.com/crystals/bravais/hexagonal/hexagonal.htmhttp://www.coolphysics.com/crystals/bravais/trigonal/trigonal.htmhttp://www.coolphysics.com/crystals/bravais/tetragonal/body_centered_tetragonal.htmhttp://www.coolphysics.com/crystals/bravais/tetragonal/tetragonal.htmhttp://www.coolphysics.com/crystals/bravais/orthorhombic/face_centered_orthorhombic.htmhttp://www.coolphysics.com/crystals/bravais/orthorhombic/body_centered_orthorhombic.htmhttp://www.coolphysics.com/crystals/bravais/orthorhombic/base_centered_orthorhombic.htm


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In crystal system, it is necessary to refer to the crystal planes, and directions of the straight lines joining

the lattice points in a space lattice. A notation system which uses a set of three integers (n1, n2 & n3) is

adopted to describe both the positions of planes or directions within the lattice.

Figure9

A resultant vector R which joins Lattice points A and B (figure9) can be represented by an Equation.

R = n1a +n2b +n3 c , If n1= n2 = n3 =1,thenR =a + b + c


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CRYSTAL PLANES AND MILLER INDICes

Figure10

It is possible to define a system of parallel and equidistant planes which can be imagined to pass

through the crystal structure are called as Crystal Planes. The position of a crystal plane can be

expressed in terms of three integers namely Miller indices.

If x, y and Z are the starting co-ordinates for a plane then the Miller indices of the plane are

obtained by the following procedure:

(1) Consider the x,y and z co-ordinates of the lattice points of the plane lying on the x, y and the zdirections of a reference frame

(2) Take the reciprocals of these x, yand z co-ordinates values.

(3) Reduce the reciprocals into integers, by multiplying each reciprocal with LCM of thedenominators.

(4) Then, if possible simplify these resulting numbers.


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These simplified numbers, derived from the x, yand z co-ordinates of the lattice points on the

plane are named as the h, k and l values of the plane and is called the Miller indices of the plane.

All the planes parallel to this plane will have the same indices. So the hkl values for a plane also

represent a family of all the parallel planes. The miller indices which is the set of parallel planes

is written as

Example: Given that,

X= a5

6Y= b

4

3Z= c

2

3

Taking the ratio of intercepts with the basis vectors, we obtain

c

z

b

y

a

x,,

2

3,

4

3,

5

6

Taking reciprocals of the three fractions

3

2,

3

4,

6

5

Multiplying throughout by least common multiple *LCM+ 6 for the denominator, we have the

Miller indices

(5 8 4)

Which is read as five eight four

If a plane is oriented parallel to a coordinate axis, its intercept with the coordinate is taken as

infinity, since the reciprocal of infinity is zero, the corresponding Miller indices value will also be

zero.

Thus the Miller indices is a set of 3 lowest possible integers whose ratio taken in order is the same as

that of the reciprocals of the Miller integers of the planes on the corresponding axes in the same

order. Similar to the case of representation of directions in the space lattice, any given Miller indices

set represents all parallel equidistant crystal planes for a given space lattice. Owing the rotational

symmetry, certain planes which are not parallel to each other become in distinguishable from the


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k = b/y therefore, y=b/k

l= c/z therefore z= c/l

Writing the values of x, y and z in the above trigonometric equations we get,

Cosha

d

OA

OP

/

Coskb

d

OB

OP

/

Coslc

d

OC

OP

/

From solid geometry, Cos2 + cos2 Cos2 =1

Substituting the values of the trigonometric relations we get

12

22

2

22

2

22

c

dl

b

dk

a

dh

12

2

2

2

2

22

c

l

b

k

a

hd

d2

hkl =

2

2

2

2

2

2

1

c

l

b

k

a

h

Then, the interplanar distanced is given by

dhkl = 2/1

2

2

2

2

2

2

1

c

l

b

k

a

h

For cubic lattice a=b=c then,


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222 lkh

adhkl

EXPRESSION FOR SPACE LATTICE CONSTANT:

Density is a macroscopic property. Basically it is the mass per unit volume. In case of crystals, mass

of atoms packed in a conventional unit cell per unit volume of the cell gives the density of the

crystal.

Density of a crystal = = .(i)Here m is the mass of atoms packed in the conventional unit cell of the crystal. V is the volume of

the unit cell. Mass of atoms packed in the conventional unit cell of the crystal. V is the volume of the

unit cell. Mass of an atom in the structure is given by the ratio of the atomic weight or Molecular

weight (M) to the Avogadro number (NA). Mass of atoms contained in the conventional unit cell is

then the number of atoms in the unit cell times the mass of an atom. If there are n atoms in the

conventional unit cell, then, the mass of atoms in the unit cell is given by

Na

nMm

..(ii)

From equation ( i ), mass of atoms in the unit cell in terms of the density of the crystal is given by,

From (i) Vm

From (ii)VN

nM

A

From this equation, the density of the crystal is


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3/mkgVN

nM

A

In case of a cubic lattice the volume of the unit cell V = a3

Therefore3

3/mkg

aN

nM

A

The lattice constant a =0

3/1

AN

nM

A

Coordination Number (N) & Atomic packing factor(APF)

The number of atoms at equal and least distance from a given atom in the structure is the

coordination number. It can be taken as the first nearest neighbors of an atom in the structure.

Atomic packing fact

engg physics notes

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