UNIT-I Modern Physicsqaqaq

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MODERN PHYSICS

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Radiationis the transmission of waves or particles through space or matter or any

medium without affecting it.

It is the emission of E.M.waves by matter when supplied with appropriate amount of energy.

(it depends on the Kirchoffs law of radiation).

Black body:A black body is a theoretical object that

absorbs 100% of the radiation that hits it. Therefore it reflects no

radiation and appears perfectly black.

In Real case no such black body exists, its only a concept but

nearly perfectly black body may be constructed.

A simple example of a black body radiator is the furnace.

Black body radiation: The E.M waves emitted by perfectly black body.

For perfectly black body, Emissive power = Absorptive power

Wein designed nearly black body to study black body radiation

Uniformly heated that body to higher and higher temperature the graph of intensity or power

density against wavelength is plotted and the observation are obtained as

1. At a given temperature the black body emits continuous range of radiations.2. There are different curves for different temperatures.3. There is a peak for each of the curves which indicate that the EM waves of that

wavelength corresponding to the peak are emitted to the largest extent at that

temperature to which the curve corresponds.

4. The peak shifts from curve to curve towards the lower wavelength side as highertemperatures are considered.

5. The area under the curve gives total energy emitted at a given temperature.


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{STEFAN'S LAW

P = Power radiated in W (J/s)= tefan!s "onstant #.$% x &'W m*+,

A = urface area of body (m-) = emperature of body (+)

herefore the Power radiated is proportional to ,for an identical body which explainswhy the area under the black body cures (the total ener0y) increases so much for arelatiely small increase in temperature.}

Based on these observations Wien stated that The wavelength of maximum intensity is inversely proportional to the absolute temperature of the emitting body.

Because of which, the peaks of the energy curves for different temperature get displacedtowards the lower wavelength side.

This is called Wiens displacement law.

.., 1Or

==2.89810

And maximum energy emitted is proportional to fifth power of absolute temperature.

Or

= Wien also deduced the relation between the wavelength of emission and the temperature

of the source as

= !"# $% &This is the equation of Wiens law in terms of energy.

Where, is the amount of energy in the wavelength range to ' and (are constantsDrawback of Wiens law

The law agrees with the experimental observations only at shorter wavelength and higher

temperature.


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The law fails to explain the decr

This law uses the Maxwe

be distributed throughout the Ebe emitted beyond the violet regi

since visible range is only small

According to the principle of eq

energy kT,where k is Boltzmann

Then 8)*is the number of ' .Therefore the energy radiated pe

PLAN

The probability of emission of ra

assume Black body surface comenergy value. This is the beginni

Assumptions made by Planck ar

1. The black body compose2. Energy of these oscillato

Where n =0,1,2,3............... , h is

3. An oscillator may lose orradiation of frequency +

MODERN PHYSICS

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ment of intensity at longer wavelength than the

Rayleigh Jeans law

lls law of equipartition of energy, by this total

spectrum. That is for black body most of radiaon and small amount of energy radiated in the

part of EM spectrum.

uipartition of energy, each mode of vibration ha

constant.

vibrations per unit volume in range of wavelen

r unit volume in wavelength range

and

' =8)*This is Rayleigh Jeans equation

distribution it explains gradual de

intensity at longer wavelength. T

Jeans law fails to explain energ

in shorter wavelength range, acc

theory all energy must be radiate

wavelengths. But it is not so. Th

of the Rayleigh Jeans law to exspectrum beyond the violet regio

lower wavelength side of the spe

known as Ultra violet Catas

KS LAW OF RADIATION

diation decreases as its frequency increases, thi

osed of oscillators and oscillators have only ding of quantum physics

of oscillators

s are integral multiple of -+i.e., -+the Plancks constant and +is the frequency ofgain energy by emitting or absorbing respectiv

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peak one.

energy must

tion shouldisible region,

s an average

ths and

is given by

for energy

crease of

is Rayleigh

y distribution

rding to this

at shorter

s the failure

lain thetowards the

trum is

trophe.

s lead to

crete set of

vibration.

ly a


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MODERN PHYSICS

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Hence the body can emit or absorb energy in discrete set of energy called quanta or

photon

= - +Based on these observation or ideas Planck derived the law for energy distribution ofblack body radiation which is in agreement with the experimental result.

= 8)- 1

/ $% 13 4 = 8)-

1/5 $% 13 ! 6 + =

% &This is calledPlancks radiation law.

Deduction of Plancks radiation law to Wiens law

If is small or +is large then /5 $% is very large i.e., /5 $% 7 1 /5 $% 1 = /5 $%

Then Plancks radiation law becomes

= 8)- 1

/5 $% 3

Or = !"# $% &Where =8)-and ( = - % Hence Plancks radiation law can be reduced to Wiens law i.e., Plancks radiation law is

applicable in the range of shorter wavelength.

Reduction of Plancks radiation law to Rayleigh Jeans law

If is large or +is small then/5 $% = 1 ' - % ' !- % &( ' /5 $% : 1 ' - %

/5 $% 1 : - %


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Plancks law of radiation becomes

= 8)-

1

- %

=8)*Hence the Rayleigh Jeans law

i.e., Plancks law of radiation is applicable for longer wavelength of

experimental observed result of black body radiation.

PHOTOELECTRIC EFFECT

The phenomenon of emission of electrons

from certain metallic surfaces due to the

incidence of radiation of suitable

frequency.

The emitted electrons are called

photoelectrons.

It is observed by Hertz and perfectly

explained by Einstein by using Plancks

quantum theory.

Maximum energy of photoelectrons

depends on the frequency of radiation the

plot of maximum energy Emversus gives

a straight line and slope of straight line

gives Plancks constant. Extrapolation of

the straight line meets Em axis at .

The equation for the straight line is

= - + Arepresents the minimum energy

required for an electron to escape from the metal and is called work function of that metal.

Einstein considered the Plancks quantum theory i.e., energy emitted or absorbed by matter is

in discrete set of energy called quanta. Einstein called quanta as photons.

The energy of photon is

B/CDCE = -+


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The energy is transmitted in terms of electromagnetic radiation like light and light is

composed of particle i.e., particle nature of light is assumed by Einstein.

A beam of higher intensity has a higher density of photons or a beam of radiation of higher

frequency consists of photons of higher energy.

When the light incident on the metal surface particle particle collision takes place i.e.,

photon electron collision and photon delivers all its energy -+to electron.i.e.,

B/CDCE = - + = ' FG-4 4HI J K-L4F MH 4HI J L4 - LThe work function is equal to binding energy Ebof electron

Therefore - + = ' NAnd = = ( O( G-4 - J K-L4

O - PQ OLI J K-L4Hence - + = ' N or R- + NOr -+ = ( O( ' NPhysical significance of Photoelectric effect

The effect is explained on the basis of quantum theory i.e., particle concept oflight and one

to one interaction between photon and electron also photon transmit its all energy to

electron, thus photoelectric effect signifies the particle nature of light waves.


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COMPTON EFFECT

The phenomenon in which the wavelength of X rays scattered by an electron is greater

than that of incident rays due to exchange of energy between X ray photon and electron in

the target.

This effect is the evidence of existence of

photon by Arthur H Compton in 1923.

Compton made an experiment in which a

beam of X ray directed to graphite plate

and found that X ray scattered in all

directions, if the scattered ray appears in the

direction of incident one angle is taken as

zero and if it returns in the direction of

incident one angle is taken as 1800

.Scattered ray contains two wavelengths one

is same as that of incident one 0and another

of larger wavelength I. The difference

between these wavelengths i.e., (I - 0) is

known as Compton Shift.

The same experiment carried out with different material by replacing graphite and found that

Iis independent of target material and depends only on scattering angle .

The scattering of a photon by an electron is called Compton Scattering.

Compton used photon concept given by Einstein to explain this effect and succeeded. He

considered incident X ray as particle i.e., photon having energy

= - + = - Where h is Plancks constant, c is velocity of light and is the wavelength of incident X

ray.

If X ray is treated as composed of particle then it collides with an electron at angle

reducing its energy into

S = -+S = -S And electron recoils at an angle

Applying conservation of energy and conservation of momentum Compton derived an

equation for the change in wavelength or Compton shift as


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T = !S & = - !1UVWX&Where m is the mass of the electron and c is the velocity of light and

/5has the dimension of

length and is called as Compton Wavelength.

Physical Significance

We know that X ray is electromagnetic wave but to explain Compton shift X ray is

assumed to be a particle which collide with electron as a particle and exchanges its energy

with the electron.

Hence Compton Effect signifies particle nature of X ray.

Wave Particle Dualism

We know that electromagnetic radiation such as visible light, X ray etc are considered as

wave in physical optics to explain the phenomena like interference, diffraction, polarisation

etc.

To explain Compton Effect i.e., The phenomenon in which the wavelength of X rays

scattered by an electron is greater than that of incident rays due to exchange of energy

between X ray photon and electron in the target.

X ray are assumed to be particle as photon

Similarly to explain Photoelectric effect i.e., The phenomenon of emission of electrons fromcertain metallic surfaces due to the incidence of radiation of suitable frequency.

Light is considered as particle i.e., photon.

i.e. light behaves as both particle and wave

de Broglie hypothesis : Nature loves symmetry, if the radiation behaves as particle

under certain circumstances and as wave under certain other circumstances then entities

which ordinarily behave as particle should behave as wave under suitable condition.

It is given by Louis de Broglie in 1924.

Matter waves or de Broglie waves are waves associated with moving particle.

When a particle has a momentum p its motion is associated with a wave having wavelength

called de Broglie wavelength given by

= -KThe de Broglie hypothesis explains wave particle duality.


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MODERN PHYSICS

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De Broglie wavelength of electrons

Experiment held by Davisson and Germer and also by G.P.Thomson proved that electrons are

associated with waves.

Energy acquired by an accelerated electron under a potential difference of V is

= YAlso we know that kinetic energy of accelerated electron as

= 12 O(For non relativistic case energy equation is

Y = 12 O( ! 1 &Momentum of the electron is given by

K = OSquaring this equation

K( = (O(Or

O( = K( Using this equation in eqn (1)

Y = K(2Or

K = Z2Y

By de Broglie hypothesis

= -K

= -Z2Y


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= 1ZY [-

Z2\Substituting the values of constants

- = ] . ] 2 ] 1 0*^ , = 9.11 10H, =1.]02 10_We get = 1.22]ZY

In general, We know that = (by Einstein relationAnd

= - + =/5

by Planck quantum theory

Therefore - = (

= - =-K

4HI = 12 O( =K(2

Or

K = Z2Therefore

= -Z2

Characteristic properties of Matter Waves

1. Lighter is the particle, greater is the wavelength associated with it i.e., m is small islarge.

2. Smaller is the velocity of the particle greater is the wavelength associated with it v issmall is large.

3. When v = 0 then = `waves becomes indeterminateWhen v = then = 0matter waves are generated by motion of the particle

4. Matter waves are produced whether the particle are charged or uncharged i.e.,isindependent of charge. This shows that matter waves are not electromagnetic waves


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but they are new kind of waves.{ since electromagnetic waves are generated or

produced only by charged particles.}

5. Velocity of the matter wave is not constant but it depends on the velocity of particleVelocity of electromagnetic wave is constant.

6. Velocity of matter waves is greater than the velocity of electromagnetic waves. Thisshows that the matter waves are not physical waves.

Davisson Germers Experiment

Figure: Davisson-Germer experiment on diffraction of electron waves.

The experiments of Davisson and Germer were the first experimental evidence in support of

matter waves. These two American physicists performed experiment on the diffraction of

electron waves by a nickel target.

The electron beam from an electron gun is accelerated and collimated to strike a nickel

crystal. C is an ionization chamber for receiving the electron after they have been scattered by

a nickel crystal. The ionization chamber can be moved along a graduated circular scale so

that it is able to receive the scattered electrons at all angles between 20 to 90 and their

intensity is measured by the galvanometer current. The whole assembly is placed in a very

high vacuum. Graphs are drawn at various voltages and the pronounced maximum obtained

for 54 volt at

a= 50. (Fig. 1.6b)


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a+ + = 1802 = 180 -50 = 130

= 65

The interplanar distance for nickel is 0.1 nm.

Thus 2d sin =

= 2 X 0.1 X sin 65 = 0.167 nm (1)

By de Broglie's hypothesis,

Or

(2)

Thus we see 100% agreement in the value. See Eqn. (1) and compare with Eqn. (2).

Phase Velocity: The velocity with which the individual wave travels in medium called

phase velocity Vphase.Or The rate at which the phase of the wave propagates in space

and

YB/bcd = e We know that the equation for the travelling wave can be represented by

I=f Wgh!eP&Where A is the amplitude, is the

angular frequency = eO is thepropagation constant or the wavenumber; x is the distance between two

positions of mass which are in phase.

For uni-phase points !eP&must be same in the periodic wavei.e., !eP&

= 0or


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e [P\ = 0or

[P\ = e i.e.,

YB/bcd = e Group Velocity:It is thevelocity of wave packet or It is the

velocity with which energy

transmission occurs in a wave.

Wave packetis the envelope (or

cover or packet) enclosing wave

group formed due to superposition

of two or more waves of slightly

different wavelengths.

Theory, Expression for Group velocity

Let us consider two travelling waves of same amplitude and of different wavelengths and

frequencies. i.e.,

I = f W g h ! e P & ! 1 &I( =fWghi!e ' j e& ! ' j & P k ! 2 &

Where y1and y2are displacements of two waves in the direction normal to the direction

of propagation of wave at the instant t, A is the amplitude, e e ' jeare the


14/19

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angular velocities, ' jare the wave numbers, x is the commondisplacement at the instant t.

The resultant displacement of these two waves due superposition

I = I ' I(From equations (1) and (2) above equation becomes

I=f Wgh!eP&'f Wghi!e ' j e&!'j&PkUsing the identity

W g h ' W g h M = 2 U V W [ M2 \ Wgh [ ' M2 \

I = 2 f U V W lmnojnn( p moj( p Pq Wgh lmnojnon( p mojo( p Pq

I = 2 f U V W r[je2 \ [j2 \ Ps Wgh r[

2 e ' j e2 \ [

2 ' j 2 \ Ps

je j 4 LL ,2e ' je : 2e 2 ' j : 2I = 2 f U V W r[je2 \ [

j2 \ Ps Wgh!eP&!t&

Comparing equations (1) and (3) we get

2fUVW r[je

2\ [j

2\ Ps =KLQ ..,KLQ J 4QL GO

Amplitude of individual waves is constant but amplitude of resultant wave is

not constant but varies as wave.

[je2 \ [j2 \ P = 0

4 [je

2\ = [j

2\ P


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mPp = [jej\ =

e

OuvCwB =e

Relation between Group velocity and Phase velocity.

We know that phase velocity

xyz{|} = ~ Or ~ = xyz{|} . ! 1 &

& group velocity xy = ~ Or xy = !xyz{|} & [from eqn. (1)]Or xy = xyz{|} ' xyz{|}

Or xy = xyz{|} ' mp mxyz{|} p . !2&We know that = (

= (#

=

#

(

Or mp = m( p m #

(p = mp = . . . ! t &

rom equations (!) & (")#


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xy = xyz{|} ' !&!xyz{|} &

Or xy = xyz{|} !xyz{|} &

Relation between Group velocity and Particle velocity

Particle velocity:The rate at which particle is moving is called particle velocity.

We know that group velocity

OuvCwB = e ! 1 &And angular frequency

e = 2 ) + = 2 ) - ! 6 = - + &

e = [2)

-\ ! 2 &

WV = 2) = 2)

K- !6 =

-K&

= [2)- \ K ! t &Dividing equations (2) and (3) we get

e = K ! &

But we know that

= K(2

K =

2K2 =

K


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Using the above equation in (4) we have

e =

K

But K = OBbvD5de =

OBbvD5d = OBbvD5d ! &Comparing equations (1) and (5) we get

OuvCwB = OBbvD5dThe above equation tells us that the de Broglie wave or matter wave associated with

particle travels with a velocity equal to the velocity of the particle itself.

Relation between Velocity of light, Group velocity and Phase velocity

We know,

= And

e = 2 ) + = 2 ) - = 2) = 2) K-

= =2) -2) K-

= K

V = (

=

= (

= (


18/19

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Expression for de Broglie wavelength using Group velocity

According to de Broglie hypothesis a moving particle is associated with wave and group

velocity of that wave is given by

OuvCwB = e ! 1 &Where

e = 2 ) + = 2) e = 2) + = 2) !1&

Substituting these in equation (1) we get

OuvCwB = 2) +2) !1&

OuvCwB = +!1&

4 !1& = +OuvCwBBut

OuvCwB = OBbvD5d [1\ =

+OBbvD5d =

+O ! 2 &

Let m is the mass of the particle, v is the velocity of the particle and Vis the potential energy

then the total energy E of the particle is given by

= 12 O( ' Y ! t &But by the quantum theory

= - + ! &From equations (3) and (4)

-+ =12 O

(

' Y


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MODERN PHYSICS

Differentiate above equation assuming particle moving in constant potential i.e., V= constant

Then

- + = O O

+ = m- p O O !&Substituting dfrom eq

n(5) in (1)

[1\ = m- p O

Integrating 1

= m

-

p O'Let constant = 0 and p=mv

1 =K-

4 = -KThis is the required equation.

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UNIT-I Modern Physicsqaqaq