4

UNIT-IV

QUANTUM PHYSICS

4.1Introduction

Max Planck put forward a revolutionary hypothesis that the molecules in a source does not emit energy continuously but in small discrete packets called quanta. A new theory which was base on planks work came to be known as quantum theory or quantum physics. Quantum physics explains the behaviour of matter and radiation at the microscopic level.

4.2Blackbody radiation

A blackbody is an object that absorbs all the energy that falls on it. When a perfect blackbody is heated, it emits radiation at all wavelengths and so it is a good emitter. The radiation emitted by a perfect blackbody is called blackbody radiation.

4.3Energy spectrum of blackbody radiationThe distribution of energy for various wavelength of radiations and its variation at different temperature was studies by Lummer and Pringsheim.

The experimental setup (Fig. 4.1) consists of a hot carbon tube which emits radiations in all directions. The radiations form this tube is made to pass through the slit S1 and is allowed to fall on mirror M1. The emergent parallel beam is focused on to the fluorspar prism. The rays get dispersed and fall on mirror M2. The rays are sent to the bolometer through the silt S2. The bolometer measures the intensity corresponding to the wavelength of radiations. By rotating the prism, the energy corresponding to different wavelength can be determined.

Analysis of energy spectrum

The energy distribution for different wavelength at various temperatures of the source is shown in Fig. 4.2. From the energy spectrum the following are observed.

1. For any given temperature the energy distribution is not uniform.

2. For any given temperature the intensity of radiation becomes maximum ((m) at a particular wavelength and then decreases with increase in wavelength.

3. As temperature increases the maximum wavelength ((m) decreases.

4. For all wavelengths, an increase in temperature causes increase in the energy emission.

4.4Laws of black body radiation 1.Stefan-Boltzmann Law

According to this law, the total energy (E) emitted by a hot body is directly proportional to the fourth power of temperature (T) of the body.

i.e. E ( T4

E = ( T4Where ( is Stefans constant and

Its value is 5.67 x 10-8 W / m2.K4

2.Wiens Law

According to this law the product of the wavelength corresponding to maximum energy ((m) and absolute temperature is a constant

i.e., (mT = constant

Wien also showed that the maximum energy Em is directly proportional to the fifth power of the absolute temperature.

i.e.,Em ( T5

Em = constant x T5Wien deduced the radiation law for the energy emitted at a given wavelength ( at a given temperature T1 as

Where C1 and C2 are constants.

Wiens law holds good only in the shorter wavelength region.

3.Rayleigh-Jeans Law

According to Rayleigh, the energy distribution in the thermal spectrum is given by

This law holds good in the region of longer wavelengths only.

4.5Planck Quantum Theory

Planck suggested the quantum theory of radiation based on the following as sumptions.

1. The black body radiation chamber is filled with electrons or simple harmonic oscillators.

2. The oscillators can vibrate with all possible frequencies.

3. The frequency of radiation emitted by an oscillator is same as the of the frequency of its vibration.

4. The oscillators cannot radiate or absorb energy continuously.

5. The oscillators emit energy only in the form of discrete packet of energy i.e quanta or photon.

Derivation

Let us consider a black body having N number of oscillators with total energy ET( Average energy of an oscillator E = ET / NIf there are N0, N1, N2, N3 . Nr oscillators with energy 0, E, 2E, 3E.rE respectively then we can write

Total number of oscillators N = N0,+ N1,+ N2,+ N3 . Nr (2) Total energy of oscillators ET = 0N0 + EN1 +2EN2 + 3EN3 + + rENr (3)

From Maxwells distribution formula the number of oscillators having energy rE is given by

Substituting the values of number of oscillators for various r we get,

From equation

Total number of oscillators

Similarly from equation

Total energy of oscillators

Substituting the values of N and ET from (6) and (8) in (1) we get,

The average energy of the oscillator

Substituting the value of E as hv in (9) we get

This is the average energy of an oscillator. If we assume that all the oscillators are within the wavelength range ( and ( + d(, then the number of oscillators per unit volume is given by

(Total energy per = No. of oscillators x Average energy of unit volume per unit volume an oscillator.

This Plancks Law for black body radiation in terms of wavelength.

Case 1 For shorter wavelength

When ( is small v is large

Equation (13) represents Wiens radiation law.

Case 2 For longer wavelengths

Equation (14) represents Rayleigh-Jeans Law.

4.6. Compton Scattering

When a beam of monochromatic X-rays is scattered by a material of low atomic number such as carbon, the X-rays suffer a change in wavelength. The scattered radiation consists of two components.

1. A component having longer wavelength than the incident radiation.

2. A component having the same wavelength as the incident radiation. The change in wavelength is due to loss of energy of the incident X-rays. This phenomenon in which there is a change in wavelength of scattered X-rays is called compton effect.

Theory of Compton Scattering

Compton explained the scattering of X-rays on the basis of quantum theory of radiation. The photons of incident X-ray make inelastic collision with the looslly bound electrons of the carbon atom as shown in Fig. 4.3. The colliding photon exchanges some of its energy and momentum to the recoiling electron and the scattered photon. In this process the energy and momentum are conserved. The energy of the incident photon = hv Momentum of the incident photon = hv / c

Initial energy of the electron = Rest mass energy

= m0c2

Initail momentum of the electron = 0

The energy of the scattered photon = hv'

Momentum of the scattered photon = hv1 / c

The energy of the recoiled electron = mc2

Momentum of the recoiled electron = mv where v is the velocity of the recoiled electron.

Let and ( be the angles made by the scattering photon and the recoiling electron with the direction of the incident photon. By the principle of conservation of energy, Energy before collision = Energy after collision

hv + m0c2 = hv1 + mc2i.e., mc2 = h (v v1) + m0c2

(1)According to the principle of momentum,

Momentum before collision = Momentum after collision

( Momentum of incident photon+momentum of electron at rest = Momentum of scattered photon+momentum of recoiled electron.

X and Y component of the momentum can be written as

X component (

(2)

Y component (

(3)

From Equations (2) and (3) we get,

mvc cos ( = h(v-v1 cos )

(4)

muc sin (= hv1sin

(5)Squaring and adding (4) and (5)

m2v2c2 = h2 (v2 2vv1 cos + v2 cos2 + v12 sin2 )

= h2 (v2 2vv1 cos + v12)

(6)

Squaring equation (1) we get,

m2c4= h2 (v2 2vv1 + v2) + 2h (v v1) m0c2 + m20c4

(7)Subtracting (6) from (7), we have,

m2c2 (c2-v2) = -2h2vv1 (1-cos ) + 2h (v-v1) m0c2 + m20c4(8)The value of m2c2 (c2-v2) can be obtained from Einsteins relativistic formula as follows.

From equation (8) and (9)

The above relation shows that d is independent of the wavelength of the incident radiations as well as the nature of the scattering substance. It depends upon the angle of scattering only.

Case i

When = 0, cos0 = 1, (1-( =0, i.e., no scattering takes place along the direction of incident photon.

Case ii

This change in wavelength is called compton's shift or compton wavelength and it is a constant. Hence it can be concluded that when varies between 0 and (, the wavelength of scattered photon varies between ( and ( + 2h / m0c1 provided the incident photon is of very small wavelength.

Experimental Verification of Compton Effect

Fig. 4.4 shows the experimental setup for study of Compton effect. Monochromatic beam of X-rays is allowed to fall on a carbon block B which act as scatterer. The Bragg's spectometer can freely swing in an arc about the scatterer. The scattering of X-ray photon can take place in different directions and their intensities can be measured by Bragg's spectrometer. The measurements are made for scattering angles 0, 45, 90 and 135. If a graph is ploted between relative intensities and wavelengths, then we observe that for each value of , there are distinct intensity peaks for two wavelengths, one of which corresponds to the incident radiation ( and the other has a higher value (1(Fig. 4.5). The peak corresponding to (1 is called modified peak. We can see that with increase of , (1 - (increases and the shift in wavelength ((. increases in accordance with the results obtained by Compton and so the Compton effect is verified experimentally. 4.7 Matter wavesThe predictions made on the basis of Compton and photoelectric effects established that electromagnetic radiation travel in form of tiny packets or bundles of energy. These packets of energy behaved mostly like a particle and were called photons. Subsequently the phenomenon of interference or diffraction of light or X-rays could be explained only when the electromagnetic radiation were assumed to be in wave nature. This resulted in the dual character of radiation. L.de Broglie in the year 1924 explained the dual nature of electro magnetic radiation. According to him, the electron or any other material particle must exhibit wave like properties in addition to particle nature. Hence any moving particle will have a wave associated with it.

4.7.1 de-Broglie wavelength associated with light or photon

From the theory of light, considering a photon as a particle, the total energy of the photon is given by, E = mc2where m-mass of the particle, c-velocity of light. Considering the photon as a wave, the total energy is given by,

E = hv

Where, h-plancks constant,

v-frequency of radiation.

Equation equation (1) and (2)

mc2 = hvBut, momentum = mass x velocity

i.e, p = mcequation (3) becomes

i.e, The wavelength of a photon

de-Broglie suggested that equation (5) can be applied both for photons and material particles if m is the mass of the particle and v is the velocity of the particle, than momentum p = mv

(de-Brogile wavelength4.7.2de-Broglie wavelength interms of energy

We know that the kinetic energy if the particle

Multiplying by m on both sides

But de-Broglie wavelength

( De-Broglie wavelength in terms of energy is

4.7.3de-Broglie wavelength interms of voltage

If a charged particle of charge e is accelerated through a potential difference V

The kinetic energy of the electron

The energy of the particle accelerated through a potential V is given by

E = eV

Equating (11) & (12)

Multiplying both sides by m

But de-Broglie wavelength

(De-Broglie wavelength in terms of voltage is

4.7.4de-Broglie wavelength interms of temperatureWhen a particle like neutron is in thermal equilibrium at temperature T, then theypossess Maxwell distribution of velocities.

The kinetic energy

is the root mean square velocity of the particle.

Also we know the energy

Where KB-Boltzman constant.

( Equating equations (15) (16)

Multiply both sides by m

But de-Broglie wavelength

(De-Broglie wavelength in terms of temperature is 4.7.5Properties of matter waves1.Lighter the particle, greater is the wavelength associated with it. 2.Smaller the velocity of the particle, greater is the wavelength associated with it. 3.These waves are produced in the particles even if the particles are charged or uncharged. This means that these waves are not electromagnetic waves but are new kind of waves. 4.The velocity of matter waves is not constant, since it depends on the velocity of the material particle. But the velocity of electromagnetic waves is constant. 4.8Schrodinger wave equation

Schrodinger in the year 1925 developed a mathematical theory to describe the wave nature of a particle based on de Broglie's ideas of matter waves. He incorporated the de Broglie wavelength in the general wave equation. There are two wave equations namely 1. Schrodinger time independent wave equation

2. Schrodinger time dependent wave equation

These equations are used to determine the electron energy levels in atoms and molecules. They also enable to find the location or state of the electron in a material.

4.8.1 Schrodinger time independent wave equation

Consi er a system of stationary waves to be associated with a particle. Let ( be the wave function which represents the displacement of the particle along x, y, z co-ordinates at any time 't' with a velocity 'v'. The classical differential equation representing the wave motion is given by

here ( is a function of x, y, z and t.

From equation(l) we get Where ( is the Laplacian operator

The solution of equation (2) is of the form Here(o is the amplitude of the stationary wave at any point considered. Differentiating (3) with respect to 't' Substituting (4) in (2)From de Broglie's hypothesis we have ( = h / mvSubstituting in (6) we get

If E is the total energy of the particle and V is its potential energy, then its kinetic energy is given by

mv2 = E - V ( ( Total energy = kinetic energy+ potential energy) mv2=2(E - V)

m2v2 =2m(E - V) This is schrodinger's time independent wave equation. 4.8.2 Scrodinger's time dependent wave equation

Schrodinger's time dependent wave equation can be arrived from time independent wave equation by applying some modification. We have from equation (3) Differentiating with respect to time

Schrodingers time independent wave equation is

Substituting the value of E( from (11)

This is schrodingers time dependent wave equation.

4.9Physical significance of wave function (1.The wave function ( hasno direct physical meaning. It is a complex quantity representing the matter wave of electron. 2.It connects the particle nature and its associated wave nature statistically.

3.( is a measure of the probability of finding the particle at a particular position. It does not give the exact location of the particle.

4./(/2 = (* ( is real and positive and has physical meaning.

5.The probability of finding out a particle in a particular volume d is given by

Where (* is the complex conjugate of (.

4.10.Particle in a one dimensional potential boxConsider a particle of mass 'm' moving along x axis in a one dimensional box of length L as shown in Fig 4.6 The walls of the box are of infinite potential and so the particle cannot penetrate out from the box. The potential energy V of the electron inside the box is constant and can be taken as zero for simplicity . ( the boundary conditions are V(x) = 0 for 0 ( x ( L

V (x) = ( for 0 ( x ( LSince the particle cannot exist outside the box the wave function ( =O when 0 ( x ( LThe wave function ( is to be determined within the box. The schrodinger time independent wave equation is Within the box V = 0

Where

Equation (2) is a second order differential equation and the solution for this equation is given by

( = A sin kx + B cos kx

Where A and B are arbitrary constants and their values can be obtained by applying the boundary conditions

At x = 0, (= 0

Substituting in (4) we get, B = 0

At x = L, (= 0

Substituting in (4) we get, 0 = A sin kL

A (0

(Sin kL = 0

Substituting the value of B and k in (4) we get,

Squaring equation (5) we get

Comparing (7) and (3)

The energy of the particle

For every value of en' there is an energy level and corresponding wave function. Each value of En is known as Eigen value and the corresponding value of ( is called as Eigen function. The value of A in equation (6) can be obtained by applying the normalization condition. Since the particle is inside the box of length L, the probability that the particle is found inside the box is unity The normalized wave function of the particle is Particle in a three dimensional box

The solution of one dimensional potential box can be extended for a three dimensional potential box. In a three dimensional potential box the particle can move in any direction and so they have three quantum numbers nx, ny and nz corresponding to the three co-ordinate axes x, y and z respectively. If we assume that the sides of the box are same then the eigen functions are given by Where

The eigen value is given by

Where

The three integers n1, n2 and n3 are called quantum numbers and are required to specify each energy state. For a particle inside the box, ( cannot be zero and so no quantum number can be zero.

DegeneracyFor several combination of quantum numbers we have same energy eigen value but different eigen functions. This condition is called degeneracy. The combination of quantum numbers (1 1 2), (1 2 1) and (2 1 1) give the same eigen value but different eigen functions. These levels are also called three fold degenarate state. For various combination of quantum numbers if we have same energy eigen value and same eigen function then that condition is known as non-degenatrate state.

The states such as (1 1 1) (2 2 2) etc are called non-degenerate states. 4.11Electron Microscope

Electron microscope is an instrument used for magnifying small objects so that their minute parts are observed. This can be used for both the physical and chemical analysis.

The electron microscope is base on the following principles

Particles such as electrons possess wave like properties and have shorter wavelength . Like light being focussed by lenses, electrons can be focussed by suitable electric and magnetic fields.

The basic reason for utilizing the electron microscope is its superior resolution, resulting from the very small wavelengths as compared to other forms of radiation like X-rays and neutrons. Electron microscopes giving magnifications more than 2,00,000 X are found common in research laboratories. The difference types of electron microscope are

1. Transmission electron microscope (TEM) 2. Scanning electron microscope (SEM) 3. Field emission electron microscope

In TEM and SEM, the electron beam emerging from the electron gun acts as source of beam. However in field emission electron microscope, the specimen itself acts as a source of radiation.

4.12Scanning electron microscope (SEM)The scanning electron microscope enables to analyse the specimen in 3 dimension

Principle

Electrons from the electron gun interacts with the specimen and emits wide spectrum of electromagnetic waves and these rays are used to analyse the physical and chemical properties.

Construction

Fig. 4.8. shows a scanning electron microscope. The essential parts of a scanning electron microscope are

i. An electon Gun.

ii. A specimen holder.

iii. Electron lenses (condenser lens, objectives lens, intermediate lens, projector lens).

iv. Vacuum pumps.

v. Scanning coils to scan the spot across the specimen.

vi. A scan generator driving both the scanning coils and CRT output.

vii. A detector amplifier system to modulate the brightness of the CRT.

i.Electron GunThe common source for electrons in a scanning electron microscope is the electron gun. When a positive potential is applied to the anode, the filament cathode gets heated uniformly and a stream of electrons are produced. The electrons, being negatively charged are attracted by the positively charged anode. They are accelerated down the column. In order to condense the electrons a negative potential ((500V) is applied to the wehnelt cap. Now electrons are collected in the space between the filament and the wehnelt cap. This collection of electrons is called as 'space charge'.

The electrons at the bottom of the space charge (nearest to the anode) can exit the gun area through the small < 1 mm) hole in the wehnelt cap. These electrons are then send down the column to be later used for imaging. WorkingThe source at the top represents the electron gun producing a stream of monochromatic electrons. The stream of electrons is condensed by the first condenser lens. It is usually controlled by the 'Coarse probe current knob'. It works in conjunction with the condenser aperture to eliminate the high-angle electrons from the beam. The beam is then constricted by the condenser aperture eliminating some high angle electrons. The second condenser lens further focuses the electrons into a thin tight coherent beam and is usually controlled by the 'fine probe current knob'. A user selectable objective aperture further eliminates the high angle electrons from the beam.

A set of coils 'scan' or 'sweep' the beam in a grid fashion (like a Television), dwelling on points for a period of time determined by the scan speed (usually in the microsecond range). The final lens, the objective focuses the beam onto the part of the specimen desired.

The beam strike the specimen and dwells for a few micro seconds, so that interactions occur inside the specimen and are detected with various instruments.

The primary electron beam interacts with the specimen and produce back scattered high energy electrons. Secondary electrons are created by an electron crossing near the atom and infusing one of its electron with extra energy. These secondary electron escapes from the atom with some energy. They are collected by the scintillator and converted into light signals.

The light signals are converted into electrical signals and then amplified using the photo multiplier. The signal is sent to the CRT where the image the formed.

Applications 1.The surface feature of an object and its texture can be analysed.

2.The morphology of the specimen surface can be studied.

3.The composition of various elements in the specimen can be studied.

4.It is used to find out the disease causing agents like virus.

5.Specimens of large thickness can also analysed.

4.13 Transmission electron microscope

PrincipleThe principle of transmission electron microscope is based on the following

(i)Electrons exhibit wave properties like light rays but have much shorter wavelength. (ii) Electrons can be focussed by electric and magnetic fields just as light rays can be focussed with the help of lenses. Components of transmission electron microscope (TEM)

The essential components of a transmission electron microscopes are

(i) Electron gun (ii) Specimen holder (iii) Electron lenses (condenser lens, objective lens, intermediate lens, projector lens) (iv) Vacuum system (v) Viewing screen or photographic plate Electron gun

Electron gun consists of a tungsten 'hair pin' filament, which when heated emits electrons due to thermionic emission. This filament is given a negative potential (100 kV with respect to the rest of the microscope. These electrons are then accelerated via a controlling Wehnelt electrode through a hole in the anode. Thus the electron gun produces a practically collimated stream of electrons of energy 100 keV and electron beam size 50 to 100 (m, which diverges over a small solid angle on the exit from the gun.

The electrons coming out of the electron gun are focussed by a set of condenser lenses to illuminate the specimen.

Specimen holder

The specimen is held in a special holder. The specimen is introduced through an air lock because the interior of the microscope must be highly evacuated. The holder is fitted to a stage which can be moved or tilted so that the specimen can be studied at different orientations. The specimen is usually surrounded by a cooled anticontamination shield. Electron lensAll the lenses used in electron microscope are of magnetic type. These are energised by highly stabilized direct current sources. Fig. 4.9 shows an electron lens of magnetic type called magnetic lens. A non uniform magnetic field along the axis of short solenoid behaves like a lens. In such a field, the electron beam undergoes divergence or convergence depending on the potential gradient at different points of the magnetic field. An electron passing through this non uniform magnetic field gets focussed. The focal length of the magnetic lens can be varied by changing the current in the coil.

The three important defects such as spherical aberration, chromatic aberration and astigmatism considered in lens design are related to the shape of the magnetic field i.e., variation of magnetic field with the position in the lens pole piece. By designing a proper pole piece, spherical aberration and chromatic aberration are corrected. Using a compensating magnetic field astigmatism is corrected.

Vacuum system

The main limitation of the electron beam is that it must pass through a vaccum because air molecules would scatter the beam. Vaccum is also needed to prevent high voltage discharge between the tungsten filament/shield and anode since it is one of the major cause of filament failure. Filaments are extremly sensitive to oxidation and must be protected. Vaccum eliminates the presence of contaminant gases that are broken down under high energy electron bombardment and generate corrosive radicals which destroy the fine structure of the specimen.

Vaccuum system consists of vaccuum pump valves and switches for evacuating primarily the path way of electron beam and also the chambers such as specimen, camera and gun column that need evacuation. The electron microscope requires a vaccum more than 10-6 torr for its operation.

Viewing screen

The electron beam travels through the specimen. Depending on the density of the material present, some of the electrons are scattered and get disappeared from the beam. At the bottom of the microscope the unscattered electrons hit a fluorescent screen, which gives rise to a "shadow image" of the specimen with its different parts displayed in varied darkness according to their density. The image can be studied directly by the operator or can be photographed. The electrons are typically detected by a photographic film or a charge coupled device (CCD).

Working

An electron beam emitted from the gun is made to pass through the centre of the doughnut magnet shaped electromagnet called condenser lens. The electrons get deflected to form a parallel beam which strike the object to be magnified. It should be noted that the electrons will be transmitted more through the transparent part of the object and less through comparatively denser portions. The transmitted beam will thus have the likeness of the object transversed by it.

The second electromagnet called objective lens causes the electron beam to diverge to produce enlarged image of the object. The third electromagnet called projector lens focusses the electron beam from part of the enlarged image on the fluorescent screen producing still greater magnification. The image obtained on the fluorescent screen is made visible by scintillation for direct view. It can also be obtained on a suitable photographic plate for a permanent record. Sharp focussing is obtained by adjusting the intensity of the magnetic fields produced by electromagnets. Uses1. It is used in the investigation of atomic structures and structure of crystals in detail. 2. It has been used in the study of structure of textile fibres, purification of lubricating oils, composition of paper and paint surfaces of metals and plastics.

3. In biology it is used to study the presence of virus.

4.14.Difference between optical microscope and Transmission electron microscope Optical Microscope Transmission Electron microscope

1.The lenses used are made of glass.The lenses used are electromagnets.

2.The focal length of the lenses are fixed.The focal length of the lenses can be varied by changing the current through the coil.

3.The objective lenses can be varied for difference magnificationThe objective lenses is fixed and the magnification is altered by changing the focal length of projector lens.

4.Depth of field is small.Depth of fixed is large.

5.The source is normally placed at the bottom.The source is normally placed at the top.