Classical and Quantum Theory of light.pdf

Quantum Theory of Materials
(MScE3111)
INTRODUCTION TO QUANTUM
THEORY LIGHT AND ITS PRINCIPLES
Filimon Hadish (PhD)
Material Sciences and Engineering
Adama Science and Technology University (ASTU)
Email: filimon.hadish@astu.edu.et
1

https://www.sapiens.org/archaeology/stone-
age-myths
https://www.profolus.com/topics/age-of-
enlightenment-legacy-and-criticisms
https://medium.com/lessons-from-history/socr
ates-answer-to-plato-
https://www.zeiss.com/corporate/int/newsroom/e
vents/international-day-of-light.html
https://www.sciencesource.com/archive/Image/Pre
historic-Man--Bronze-Age-Smelting-SS2783623.html
https://en.wikipedia.org/wiki/
John_Dalton
https://www.wired.com/2013/07/is-light-
a-wave-or-a-particle/
https://www.space.
com/16070-
christiaan-
huygens.html
https://thechemistrynotes.com/daltons-
atomic-theory/
https://en.wikipedia.
org/wiki/Isaac_Newt
on 2

• Semiconductor devices
 computers, cell phones, etc.
• Lasers
 CD/DVD players, bar-code scanners, surgical applications
• MRI (magnetic resonance imaging) technology
• Nuclear reactors
• Atomic clocks (e.g., GPS navigation)
Why is Quantum mechanics important?
3

4
• A flat monolayer of carbon atom
• Two-dimensional (2D) honeycomb
lattice
• Thickness ~ 0.34 nm
• C-C bond length ~0.142 nm
• Mechanical strength ~ 1TPa
• Electron Mobility ~ 250, 000 𝟐
/Vs
• Thermal conductivity 5000 W 𝟏 𝟏
Graphene Graphene quantum dots
(GQDs)
 Tiny nanoparticles ~2-10 nm
 Unique optical and electrical properties
Energy
LUMO
HOMO

5
Opening band gap of Graphene
Pyrrolic-N
Graphitic-N
C
Pyridinic
Pyrazole
B O
C
Sreeprasad & Berry (2013) Li et.al.(2016) Pullamsetty & Sundra (2016)
Substitutional doping
Higher sheet resistance (RSh) >
300 Ω/sq
Lower work function (WF) ~ 4.4 eV
Zero band gap
Non-emissive of light
Methods to open band gap
 Surface functionalization (charge transfer
doping)
 Chemical functionalization
 Quantum confinement

Synthesis Approach
Top down
Bottom Up
6

Graphene Quantum Dots
π - π*
n-π*
Photoluminescence
CB
(LUMO)
VB
(HUMO)
𝐛𝐠
7
PL Mechanism

Strong confinement
8 /13
Quantum Confinement
𝐸 , 𝐷 =
2ℏ 𝜒
𝑚∗𝐷
𝐸 𝐷 = 𝐸 + 𝐸 , 𝐷 = 𝐸 +
2ℏ 𝜒
𝑚∗
𝐷
+
2ℏ 𝜒
𝑚∗
𝐷
𝐸 𝑟 = 𝐸 +
ℏ 𝜋
2𝑟
1
𝑚∗ +
1
𝑚∗ − 𝐽 + 𝐸 + 𝐸 − 0.248𝐸∗
If r 𝟎 Consider 𝟎 ~ 2 to 50 nm
If r ≫ 𝒂𝟎 Here e- and h effective mass replaced
by exciton effective mass
If r 𝟎 KE>> Coulombic interaction
Weak confinement
No confinement
Energy

Wave theory
Refraction,
Diffraction
Reflection
laws 9

Waves transfer energy, momentum, and information, but not mass.
• Sound waves
• Light waves and
• Electromagnetic radiation
 Radio waves
 Microwaves
 Infrared
 Visible light
 Ultraviolet
 X-rays
 Gamma rays
• Water waves
• Seismic waves, a.k.a. earthquake waves
• Gravitational waves
• Waves in linear media (Standing wave)
 Plucking, bowing, or striking a guitar,
violin, or piano string
Materials wave
The physics hyper text bookhttps://physics.info/waves/
What are waves?
The General Characteristics,
Properties and Classification of Wave
10

Based on the medium through which they propagate.
mechanical waves
Require a material medium.
Eg. Sound wave (cannot travel through a vacuum)
electromagnetic waves (emr)
Propagate through the electric and magnetic fields in space. Eg. Light. Emr
waves can propagate through transparent materials as well as through vacuum.
gravitational waves
Propagate through the gravitational field (matter or empty space).
matter waves
quantum mechanics of particles like electrons and quarks as a wave.
Classification of waves
11

Based on the type of disturbance (relative direction or shap)
Traverse wave: disturbance is perpendicular to propagation.
Eg. light, emr waves, gravitational waves, matter waves etc.
Longitudinal: disturbance parallel to propagation.
Eg. sound wave, shock wave etc.
Complex waves: circular or elliptical
Eg. ocean waves (gravity waves), tsunamis (tidal waves)
Torsional waves: a twist Eg. bridges, skyscrapers, airplane wings, wires
Cont.
12

Complex waves
Torsional waves
13

Based on appearance
Traveling waves: Propagating waves but, actually that do not appear to be
going anywhere.
Standing waves: do not appear to be propagating (They are also
called stationary waves.)
Based on duration
Episodic
Periodic
Cont.
14

Assignment
1. What are the limitation of Classical Mechanics (CM)?
2. In what way the Quantum mechanics (QM) improves the CM?
15

Christiaan Huyghens (1690) theory of
wave propagation, ‘elementary
wavelets’.
He argued that the known properties
of light, such as refraction, reflection,
& propagation in straight lines, could
be understood by assuming that light
as a wave in some invisible medium.
Isaac Newton, (1704)
For various reasons he favoured a
particle theory of light – the
explanation of light propagation
in straight lines, except at
interfaces, was then easily
understood. He argued that light
are composed of particles .
16

Rectilinear Propagation, Reflection, Refraction, Dispersion, Interference, Diffraction,
Polarisation, Double refraction, Doppler’s effect, Photoelectric effect.
Properties of wave
Refraction of a plane wave
Based on Huygens principle: Let PP′ represent the surface separating medium 1 and medium
2, as et 𝑣 and 𝑣 represent the speed of light in medium 1 and medium 2, respectively.
Assume a plane wavefront AB propagating in the direction A′A incident on the interface at
an angle i. Let 𝜏 be the time taken by the wavefront to travel the distance BC.
where i and r are the angles of incidence and
refraction, respectively. Thus equating above eqs.
and
if 𝑟 < 𝑖 (if the ray bends toward the normal),
This prediction is opposite to the prediction
from the corpuscular model of light (not
correct). 17

Now, if c represents the speed of light in vacuum, then
and
Are known as the refractive indices of medium 1 and medium 2, respectively. In terms
of the refractive indices,
Snell’s law of refraction
Interms of wave lengths i.e. taking 𝜆 and 𝜆 the wave length of the light in m-1 and
m-2 respectively, and 𝜆 = 𝐵𝐶 and 𝜆 = 𝐴𝐸
or
The above equation implies that when a wave gets refracted into a denser medium
(𝑣 > 𝑣 ) the wavelength and the speed of propagation decrease but the frequency
𝑉(= ) remains the same.
Remark (rule): When wave changes medium
a) Frequency does not change
b) Speed changes
18

Refraction at a rarer medium
Taking into account
i.e.
Thus, if 𝑖 = 𝑖 then sin 𝑟 = 1 and 𝑟 = 1
and 𝑟 = 90 . Obviously, for 𝑖 > 𝑖 ,
there can not be any refracted
wave. The angle 𝑖 is known as the
critical angle and for all angles of
incidence greater than the critical
angle, we will not have any refracted
wave and the wave will undergo what
is known as total internal reflection.
19

Consider a plane wave AB incident at an angle i on a reflecting surface MN. If 𝑣
represents the speed of the wave in the medium and 𝜏 represents the time taken by
the wavefront to move from the point 𝐵 to 𝐶 then the distance
𝐵𝐶 = 𝑣𝜏
In order to construct the reflected wave-front we draw a sphere of radius 𝑣𝜏
from the point A, Let CE represent the tangent
plane drawn from the point 𝐶 to this sphere. Obviously
AE = 𝐵𝐶 = 𝑣𝜏
Reflection of a plane wave by a plane surface
This is the law of reflection
Consider the ∆EAC and ∆BAC are
congruent, thus, the angles i and r
would be equal.
20

Dr. Shao-Ju Shih lecture on X-ray Properties and Diffraction Geometry, 2016
X-ray and X-ray diffraction
Target
Tungaten Filament
Cathode (-) Anode (+)
X-ray
X-ray Tube
The Photoelectric Effect
21

Properties
/Phenomenon
Wave Particle Dual nature
Corpuscular Quantum
Reflection Y Y Y Y
Refraction Y Y Y Y
Diffraction Y N N Y
Interference Y N N Y
Photoelectric
effect
N N Y Y
22

Wave surface: When a point source of light is situated in air then its waves travel in
all possible directions. If 'c' is the velocity of light in air then each wave covers a
distance 'ct' in time t and reaches the surface of a sphere.
Characteristics of wave
Wavefront: "The locus (set of all points) of the medium at which the waves reach
simultaneously such that all points are in the same phase is called a wavefront".
• i.e. a surface of constant phase
There are three type of wavefront.
Spherical wavefront: A wavefront in the form of spherical
surfaces is called spherical wavefront. It is obtained from a
point source of light up to a finite distance.
Cylindrical Wavefront: It wavefront in the form of cylindrical
surface is a cylindrical wavefront It is obtained from an extended
light source.
Plane Wavefront: A wavefront in the form of plane surface is
called plane wavefront. It is obtained by keeping point source at a
focus of a convex lens or at a large distance from the point
source.
P. Kshetrapal ://physicswithpradeep.files.wordpress.com
videoplayback (1).mp4
Underwater atomic bomb test, 1946.mp4
23

Wave normal: A normal drawn on the surface of the
wavefront at any point in the direction of propagation
of light is called a wave normal.
• The ray of light shown in a plane wavefront or
spherical wavefront is a wave normal.
• Wavefront transfers light energy in the direction
perpendicular to its surface and is represented by a
wave normal.
Note: Every point on the given wave front acts as a source of new
disturbance called secondary wavelets. Which travel in all directions with the
velocity of light in the medium.
• Wave front always travels in the forward direction of the medium.
• Light rays is always normal to the wave front.
• The phase difference between various particles on the wave front is zero.
Secondary wave front: a surface touching
these secondary wavelets tangentially in the
forward direction at any instant gives the
new wave front at that instant.
24

Important terminology of waves
Phase : The argument of sine or cosine in the expression for displacement of a wave is
defined as the phase. For displacement 𝜓 = 𝐴𝑠𝑖𝑛(𝜔𝑡); term 𝜔𝑡 = phase or instantaneous
phase
Phase difference (Phase angle) : The
difference between the phases of two waves at a
point. eg. if 𝜓 = 𝑎 sin 𝜔𝑡 and 𝜓 = 𝑎 sin(𝜔𝑡 + 𝜙)
so phase difference = 𝜙, (Δ𝜙 = )
Path difference ( ) : The difference in path length’s of
two waves meeting at a point. ∆𝑥 = 𝜙
Constructive waves (interference)
Destructive waves (interference)
In which two waves of the same wave length interact (crest + crest,
trough + trough)in such a way that they aligned, leading to a new wave
that is bigger than the original wave
𝜓(𝑥) = 𝐴𝑐𝑜𝑠(
2𝜋
𝜆
𝑥)
A
𝜓 𝑥 = 𝐴𝑠𝑖𝑛(2𝜋𝑥/𝜆)
If two waves are not perfectly aligned, then the crest of one wave
dragged down by the trough of the other wave. This resulting,
combined wave crests that are shorter than the crest of either
original waves, and troughs that are shallower than either of the
incoming waves. 25

Traveling Waves
• The shape of the disturbance in the whip is called the wave profile and is
usually symbolized ψ(x).
• The wave profile for the traveling wave shows where the energy is located at
a given instant (information about how much energy is being transmitted)
• The medium (wave travels) itself undergoes no permanent displacement.
• Merely undergoes local oscillations as the disturbance passes through.
traveling Waves
Quantum Chemistry 3rd edt. By Lowa and Peterson (2006)
26

One of the most important kinds of wave in physics is the harmonic wave,
for which the wave profile is a sinusoidal function.
( = 0 when = 0, and the argument of the sine function goes from 0 to 2 ,
encompassing one complete oscillation as goes from 0 to .)
2
3
1
27

• is similar to of Eq. (2) except for being displaced.
• Compare the two waves at the same time, we find to be shifted to the left
of by .
• For odd i.e. , then is shifted by , 3 ... and the two
functions ( and ) are said to be exactly out of phase.
• For even i.e. the shift is by …, and the two waves
are exactly in phase.
• is the phase factor for relative to .
• Alternatively, we can compare the two waves at the same point in x, in which
case the phase factor causes the two waves to be displaced from each other
in time.
28

Standing Waves
• A string will have ends, and clamped, so that it cannot oscillate when the
disturbance reaches them.
• Under such circumstances, the energy pulse is unable to progress further.
• Rather travel back along the string in the opposite direction.
• The reflected wave is now moving into the face of the primary wave, and the
motion of the string is in response to the demands placed on it by the two
simultaneous waves.
• When primary and reflected waves have the same amplitude and speed
the above equation can be rewrite as
4
5
29

standing wave—a wave that does not appear to travel through the medium,
but appears to vibrate “in place” (in between 0 and L).
Between the nodes, sin(2πx/λ) is finite.
• The cosine function oscillates between plus and minus unity. i.e.
oscillates between + and – value of sin(2πx/λ).
• On the standing wave function x is responsible the maximum displacement
of and the t-dependent part governs the motion of the medium back and
forth between these extremes of maximum displacement.
Cont.
• The first part of the function depends only on the x.
• vanish whenever sine function vanishes independent of t and
• it is an indication in a place where no vibration occurs (nodes).
30

• The profile is often called the amplitude function and is the
frequency factor.
6
where
Cont.
Equation 4 often written as
Energy of standing wave—at the node and clamped endpoints of the string the
spring never moves.
• KE energies are zero at all times.
• PE is also zero because the string never displaced from equilibrium positions.
• TE = KE + PE = 0 at the segments
• The maximum KE and PE in SW can be attained where greatest average
velocity and displacement from the equilibrium position record.
• These places exists at wave peaks and valleys (called the antinodes)
31

Properties of wave (The Classical Wave Equation)
It helps to predict and understand what kind of wave exist as the medium
(system) disturbed.
• Consider a string under a tensile force T . When the string is displaced from
x to x +dx , this tension is responsible for exerting a restoring force.
• Tension exerted at either end of this segment can decomposed
Parallel component: stretch the string (assumed to be unstretchable),
Perpendicular component: acts to accelerate the segment from the rest
position.
• For small angle α the horizontal component is nearly equal in
length to the vector T . i.e. approximately
at x +dx
32

But the slope is also given by the derivative of 𝜓, and so we can write
At the other end of the segment the tensile force acts in the opposite direction, and we
have
The net perpendicular force on our string segment is the resultant of these two:
The difference in slope at two infinitesimally separated points, divided by dx, is by
definition the second derivative of a function. Therefore,
If the string has mass 𝑚 per unit length, then the segment has mass 𝑚𝑑𝑥, and Newton’s
equation 𝐹 = 𝑚𝑎 may be written
Equation 10 represents the wave equation for motion in a string of uniform density under
tension T.
7
8
9
10
11
Cont.
33

, , , 12
For three-dimensional media eq. -11 can be improved as
where 𝛽 is a composite of physical quantities (analogous to m/T ) for the particular
system.
It is obvious that eq.-11 is a time-dependent differential equation
Consider a standing waves that can be separated into a space-dependent amplitude
function and a harmonic time-dependent function. Then
and the differential equation becomes
13
14
15
dividing by cos(ωt),
Cont.
34

Eq.-15 is the classical time-independent wave equation for a string and one solution of
this second differential equation is
16
Comparing eq.-16 with eq.1 we get 2𝜋/𝜆 = 𝜔 𝑚/𝑇
Then, substituting 2𝜋/𝜆 into eq.-15
17
The corresponding three dimensional classical time-independent wave equation is
18
where 𝜆 depends on the elasticity of the medium
19
where the left hand side coefficient of eq.-19 is called Laplacian and is represents
20
35

Boundary conditions
Suppose that the string is clamped at 𝑥 = 0 𝑎𝑛𝑑 𝐿. This means that the string cannot
oscillate at these points. i.e
To solve the 2nd order differential equation of eq.-17 and satisfy the above boundary
conditions , one must recall
To determine the constants 𝐴 and 𝐵 let set x=0
21
22
Since 𝑠𝑖𝑛 0 = 0 and 𝐵𝑐𝑜𝑠 0 = 1 then, 𝐵 = 0
∴for first boundary condition forces B to be zero eq.-21 reduced to
23
Moving to second boundary condition, at 𝑥 = 𝐿, gives
24
Possible solutions for the above equation are
36

The 2nd option gives the relation
, 25
Substituting this expression for 𝜆 into eq.-23 gives
, 26
Solutions for the time-independent wave equation in one dimension with boundary
conditions 𝜓 0 = 𝜓 𝐿 = 0.
The above sketch is a classical wave equation and is different but closely related to
the wave equation in quantum mechanics (quantization of energy)
37

Is Light wave? Or particle? Or Both?
• According to the results of the double slit experiment, when photon passes through
the slit the outcome of the experiment goes parallel with the behavior of the
photon.
• If the photon passing slit is identified then, the photon will behave as a particle.
• If not identified, the photon will behave as if it were a wave when it is given an
opportunity to interfere with itself.
• The double-slit experiment is meant to observe phenomena that indicate whether
light has a particle nature or a wave nature.
Double slit experiment
A single particle,
both slits are open
Many classical particles,
both slits are open
Many atomic particles,
both slits are open
Many particles,
one slit is open
Young double slit experiment (?3d animation ) , physics.mp4
38

(MScE3211)
HUYGENS’ WAVE THEORY OF LIGHT and
POLARIZATION
39

Postulates on which Huygens’ wave theory are given as follows:
• The source of light emits light in the form of waves.
• Light waves are like sound wave, which are longitudinal in nature.
• Light waves move with constant speed in a homogeneous medium.
• Different colours of light are due to different wavelengths of light
waves.
• When light waves enter in our eyes we feel the sensation of vision.
• Light waves travel through vacuum due to presence of a hypothetical
medium called as luminiferous ether (dragging medium).
Huygens’ wave theory of light
Wardha jankidevi bajaj college of science, https://ncert.nic.in/textbook/pdf/leph202.pdf
40

Wave surface: possible direction of wave travels from point of source
Characteristics and properties of wave
Wavefront: "The locus (set of all points) of the medium at which the waves
reach simultaneously such that all points are in the same phase"
41

Secondary wave front: a surface touching these secondary wavelets
tangentially in the forward direction at any instant gives the new wave front
at that instant.
Constructive waves Destructive waves
Cont.
42

Phase: The argument of sine or cosine in the expression for displacement of a
wave
Phase difference (Phase angle):
• and
•
• so is phase difference
Path difference ( ) : Path length difference of two waves meeting at a
point.
Cont.
43

Refraction
Snell’s law of refraction
Cont.
http://www.gvp.cz/~vinkle/mafynet/fyz_5_6lete/vlnova_optika/WaveOptics.pdf
44

Refraction at a rarer medium
Taking into account
i.e.
Where = Critical angle
• If then and and . Obviously,
• For , there can not be any refracted wave.
• The wave will undergo what is known as total internal reflection.
Cont.
45

Reflection
The law of reflection
Consider the EAC and BAC are congruent, thus, the angles i and r
would be equal.
Cont.
46

Advantage
 It is helpful to explain phenomena such as reflection, refraction, interference and
diffraction.
 As per the wave theory of light, velocity of light in optically denser medium is less than
the velocity of light in a rarer medium, which is correct (𝑣 > 𝑣 ).
Limitation
 Assumed the presence of hypothetical ether medium but experiment proved that there
is no ether or drag.
 Rectilinear propagation of light is not explained by this wave theory.
 Could not explain phenomena such as Compton effect and polarization of light.
 Could not explain bending of wave through an obstacle.
 Assumed that light waves are longitudinal in nature but experiment proved that they are
electromagnetic transverse waves.
Wardha jankidevi bajaj college of science, https://ncert.nic.in/textbook/pdf/leph202.pdf
47

• If light is a wave, interference effects occurs , where one part of
wavefront interact with another part.
• One way to study this is to do a double-slit experiment:
Young’s Double slit Experiment
file:///C:/Users/User/Desktop/Advance%20Physical/new%20documents/27%20Wave%20optics.pdf
48

Cont.
• Depending on the path length difference, the wave can interfere
constructively (bright spot) or destructively (dark spot).
Constructively interferences (bright spot)
Destructively interfernces (dark spot).
49

The maxima (except the central one) position depends on , the first- and
higher-order fringes contain a spectrum of colors.
50

Cont.
The maxima (except the central one) position depends on , the first- and
higher-order fringes contain a spectrum of colors.
51

The Visible Spectrum and Dispersion
prism
Cont.
52

Atmospheric rainbows are created by dispersion in tiny drops of water.
Multiple Slit Diffraction
Can be created by passing light
through a large number of evenly
spaced parallel slits, called a
diffraction gratin.
Cont.
53

Double Vs Multiple Slit Diffraction
For monochromatic light:
• The maxima can be produced at the
same angles, but those for the
diffraction grating are narrower and
hence sharper.
• The maxima become narrower and the
regions between darker increses as
the number of slits is increased.
54

Assignment
Discuss detailed mechanism of Thin Film Interference
55

Polarization
• A wave’s oscillations have a definite direction relative to the direction of
propagation of the wave.
• Waves having such a direction are said to be polarized.
• An EM wave, such as light, is a transverse wave
• The electric and magnetic fields are perpendicular to the direction of
propagation
56

Cont.
consider the transverse waves in the ropes
• Vertical slits pass vertically polarized waves and block horizontally
polarized waves
• Horizontal slits pass horizontally polarized waves and block vertically
polarized waves
vertically polarized horizontally polarized
57

Cont.
Un-polarized light
Polarizing filters: polarizing slit for light, allowing only polarization in one
direction to pass through
Axis of a polarizing filter: is the direction along which the filter passes the
electric field of an EM wave
58

• Only the component of the EM wave parallel to the axis of a filter is passed.
• If the angle between the direction of polarization and the axis of a filter θ .
• If the electric field has an amplitude E, then the transmitted part of the
wave has an amplitude
• The intensity of a wave is proportional to its amplitude squared,
• Thus the intensity I of the transmitted wave is related to the incident wave
by
Malus’s law
• where is the intensity of the polarized wave before passing through the
filter.
Malus’s Law of polarization
Cont.
59

The effect of rotating two polarizing filters
All of the polarized light is passed by the
second polarizing filter.
As the second is rotated, only part of
the light is passed.
When the second is perpendicular to
the first, no light is passed.
Cont.
60

A polarizing filter transmits only the component of the wave parallel to its
axis, , reducing the intensity of any light not polarized parallel to its
axis.
Cont.
61

Polarization by Reflection (Brewster’s law)
Cont.
Brewster’s law
Where is known as Brewster’s angle
Useful to show reflected light is completely polarized at an angle of reflection
62

Electromagnetic waves
(Maxwell’s Electromagnetic theory of light)
Introduction to quantum mechanics by solo Hermeline
• According to this theory light is electromagnetic waves.
• Experimentally it is observed that velocity of light is equal to velocity of
electromagnetic waves.
• EM waves travel even in vacuum
• EM waves are transverse waves consisting of varying electric and magnetic
fields that oscillate perpendicular to the direction of propagation.
• There are specific directions for the oscillations of the electric and
magnetic fields .
• In EM wave the electric and magnetic fields, forces that is perpendicular to
fields and motions.
63

Maxwell unified electricity and magnetism
Cont.
64

Cont.
65

References
1. http://www.gvp.cz/~vinkle/mafynet/fyz_5_6lete/vlnova_optika/WaveOptics.pdf for
wave -optics lecture note
2. https://ncert.nic.in/textbook/pdf/leph202.pdf for leph 202
3. file:///C:/Users/User/Desktop/Advance%20Physical/new%20documents/27%20Wave
%20optics.pdf for 27 wave optics
4. Introduction to quantum mechanics by solo Hermeline
66

(MScE3211)
Particles Nature of Light
67

• Newton proposed that a source of light emits many minute, elastic, rigid
and massless particles called corpuscles.
• Corpuscles means a tiny piece of anything (something)
• These particles travel through a transparent medium at very high speed in
all direction along a straight line.
• These corpuscles enter our eyes and produce the sensation of vision.
• Due to different sizes of the corpuscles, they produce different colors.
• These light particles are repelled by a reflecting surface and attracted by
transparent materials
Newton's corpuscular theory of light
Jankidevi Bajaj college of sciences, Wave theory of light
68

Newton's corpuscular theory and Reflection
Nyambuya G. G., Dube, A. and Musosi, G., Salvaging Newton’s 313 Year Old Corpuscular Theory of Light (2017)
https://www.researchgate.net/publication/319839980
69

Newton's corpuscular theory and Refraction
• The speed of light is different in different transparent materials
• The speed of these corpuscles was much larger in a denser medium than in
a less dense one.
Nyambuya G. G., Dube, A. and Musosi, G., Salvaging Newton’s 313 Year Old Corpuscular Theory of Light (2017)
https://www.researchgate.net/publication/319839980
70

Advantages: (Newton's corpuscular theory )
• It explains the rectilinear propagation of light.
• It could explain the reflection and refraction of light separately.
Drawbacks:
• Fails to explain simultaneous phenomenon of partial reflection and refraction on the
surface of transparent medium such as glass or water.
• Fails to explain optical phenomena such as interference, diffraction, polarization etc.
• The theory stated that velocity of light is larger in the denser medium than in the rarer
medium but later experiment conducted “Jean L´eon Foucault” proved it wrong. It was
found the light travels faster in rare medium rather than denser one.
• Experiment proved that mass of the source of light is constant. But, the theory stated
that mass of the source of light should decrease as the particles are emitted from the
source.
71

• One that perfectly absorbs all the radiation (electromagnetic radiation/emr )
falling on it and then reemits all of that radiation.
• At high temperatures, an appreciable proportion of the radiation is in the
visible region of the spectrum,
• As temperature raised a proportion of short-wavelength blue light is
generated
• Beyond which the emission includes increasing amounts of UV
Black Body
Atkins’ Physical Chemistry 8th and 9th edt. By Peter Atkins and Julio De Paula
72

• In the 19th century explanation of black-body radiation was a major challenge,
and was beyond the capabilities of classical mechanics.
• Lord Rayleigh thought the electromagnetic field of a black body as a
collection of oscillators of all possible frequencies.
• Based on equitation calculation, along with Jeans they develop the Rayleigh-
Jeans law
Rayleigh-Jeans equation (CM)
where (rho), the density of states,
Rayleigh-Jeans equation predicts
• Oscillators of very short wavelengths ( ) or high frequencies such as Uv, X-
ray and -rays strongly excited at room temperature.
• Further the equation implies total energy density
is infinite at all temperature above zero.
73

• The paradox result can be interpreted as
• Large amount of energy can be radiated in the high frequency region of
EMR at room temperature called Ultraviolet catastrophe.
• Do not support the experimental observation.
• Thus, according to classical mechanics (CM), even cool objects should
radiate in the visible and ultraviolet regions.
• So objects should glow in the dark.
Experimental increases with T.
Rayleigh-Jeans law
Experimental
Cont.
74

Planks distribution (QM)
• Radiation is emitted when a solid after receiving energy goes back to the most
stable state (ground state).
• The energy associated with the radiation is the difference in energy between
these two states.
• That is contrary to equitation basis of Rayleigh formula and introduces the
discrete value of energy called “quantization of energy”.
• And is found in good agreement with experimental values.
• When T increases, the average energy ( ∗
) is higher and intensity
increases.
∗
• In his assumption, allowed energies of emr of frequency are integer
multiples of :
i.e. , where h is Planck's constant ( )
• Thus,
/
Planck's distribution
75

/
• For short ,
and / faster than
0; therefore 0 as 0 or
• Hence, the energy density approaches
zero at high frequencies, in agreement
with observation.
• For long , , and the
denominator in the Planck distribution can
be replaced by
The expression fits the experimental curve very well at all wavelengths.
/
Substituting reduces planks distribution
into Rayleigh Jeans Law
( / )
76

The Photoelectric Effect
• Evidence for the particle-like character of light (radiation) comes from the
measurement of the energies of electrons produced in the photoelectric effect.
Albert Einstein 1905
• Light energy is quantized
• Photon is a quantum or packet of energy
• This effect is the ejection of electrons from metals when they are exposed to
ultraviolet radiation.
• The experimental characteristics of the photoelectric effect are as follows:
1. No electrons are ejected, regardless of the
intensity of the radiation, unless its frequency
exceeds a threshold value characteristic of the
metal.
2. The KE of the ejected es increases linearly with
the frequency of the incident radiation but is
independent of the intensity of the radiation.
3. Even at low light intensities, electrons are ejected
immediately if the frequency is
above the threshold.
77

• Φ work function, the energy required to remove an electron from the metal.
• Photo ejection cannot occur if ℎ𝜈 < Φ because the photon brings insufficient energy:
The energy of the photon is
more than enough to eject an
electron, and the excess energy is
carried away as the kinetic
energy of the photoelectron
(the ejected electron).
The energy
of the
photon is
insufficient
to drive an
electron out
of the
metal.
KE of
ejected
electron
Calculate the value of Planck's constant given that the following kinetic energies were
observed for photoejected electrons irradiated by radiation of the wavelengths noted
𝜆 /𝑛𝑚 320 330 345 360 385
𝐾𝐸/𝑒𝑉 1.17 1.05 0.885 0.735 0.511
Class work
Cont.
78

• The Compton Effect is an incoherent and inelastic scattering of a photon by an elastic
collision with electron in which both relativistic energy and momentum are conserved
• Compton scattering is the process whereby photons gain or lose energy from collisions
with electrons.
• An energetic photon produced from 𝛾 or 𝑥 − 𝑟𝑎𝑦𝑠 when collide with electron at rest
the energy and momentum of both photon and electron changes relatively.
• Here both photon and electron treated as relativistic particles.
• i.e consider the relativistic mass of an object is given by
The Compton Effect
1
multiplying both sides by and rearrange we get
2
http://www.magadhuniversity.ac.in/download/content/Dr.Shilpicomptoneffect
79

• Equation 2 is a relativistic momentum (𝑃) related to relativistic energy(𝐸) and rest
energy (𝐸 )
• Thus, the magnitude of momentum and energy of massless particle (photon) is
related as
Which means
In the interaction, the 𝑥 −ray is
scattered in the positive x and y
directions at an angle 𝜙
with momentum of magnitude
and energy is
http://www.magadhuniversity.ac.in/download/content/Dr.Shilpicomptoneffect
80

 From the law of conservation of energy
. . . .
the energy of the incident ,
the rest mass of the electron
the energy of the scattered ,
the total energy of the electron
i.e. 3
 From law of conservation of momentum.
Total Momentum Before = Total Momentum After
. . . .
where 𝑃 and 𝑃 are the momenta of the scattered gamma ray and electron after
interacting
and
4
5
 For an electron at rest, its initial momentum is zero and has no x and y components.
 For an incident x-ray photon moving in the positive x direction and interacting with an
electron at rest, the initial x-component of momentum is p and the y-component is
zero so that
http://www.phys.utk.edu/labs/modphys/Compton Scattering Experiment
81

All the above equations leads to the equation that relates the energy of a scattered
photon 𝐸 to the energy of the incident photon 𝐸 and the scattering angle 𝜃
Another expression of equation 6 is that
6
8
7
 Equation 8 is know as Compton equation and it describes the phenomenon known as
Compton Effect.
 The term Δ𝜆 gives the change in photon wavelength due Scattering with a free
electron and it is called Compton Shift
 The term 𝜆 = is called Compton Wavelength of the scattering particle (Here
electron).
http://www.phys.utk.edu/labs/modphys/Compton Scattering Experiment
82

1. Dr. Shao-Ju Shih lecture on X-ray Properties and Diffraction Geometry, 2016
2. http://www.phy.cam.olemiss.edu/cremald/PHYS315/Chapter-3.pdf
83
X-ray and X-ray diffraction
 X-rays are light quanta on energy in the KeV
range. X-rays were first produced by Roentgen
(1895) by accident.
 He discovered that mysterious and very
penetrating rays could be generated by
directing a beam of electrons into a metal
Target
Tungaten Filament
Cathode (-) Anode (+)
X-ray
X-ray Tube

+
K
L
M
K X-ray
Secondary electron
Backscattered-electron
(absorbed electron)
Cont.
Continuous Spectrum
• No X-rays are observed below a minimum
wavelength.
𝐸 = ℎ𝑐/𝜆 = 𝐸
• spectra are produced by de-accelerating
electrons.
• As the electron looses energy in the metal
through collisions with atoms X-rays are
emitted.
Discrete X-ray Lines are produced when the
electron excites one or more atomic
transitions (from ground to excited state)
1. Dr. Shao-Ju Shih lecture on X-ray Properties and Diffraction Geometry, 2016
2. http://www.phy.cam.olemiss.edu/cremald/PHYS315/Chapter-3.pdf
84

The Davisson-Germer Electron Diffraction Experiment
𝐶𝐵 + 𝐵𝐷 = 2𝑑𝑠𝑖𝑛 𝜃
2𝑑𝑠𝑖𝑛 𝜃 = 𝑛𝜆
By measuring the number
of 𝑒 s scattered at
different angles with
respect to the incident
beam one can find out
whether it has a pattern
similar to that of X-ray
diffraction.
• The diffraction patterns of
X-rays can be explained by
Braggs planes (both beam
same phase and integral
multiple of 𝜆)
• Distance travelled by the
beam from the lower
plane is CB + BD.
• Path difference between
two beams is
• The distance of the
point from the origin is
proportional the
number of electrons
• Highest intensity of
Scattered 𝑒 s
recorded at a 54eV of
energy is when
𝜙 = 50 .
http://madhu.uccollege.edu.in/wp-content/upload/sites/187/2015/09Davisson-Germer
85

(MScE3211)
WAVE PARTICLE DUALITY
86

Wave nature duality
He suggested that any particle, not only photons, travelling with a linear
momentum p should have a wavelength.
Given by the de Broglie relation:
French physicist
Louis de Broglie
(1892-1987)
wave–particle duality is the concept that all matter and energy exhibits
both wave-like and particle-like properties.
87

Cont.
• The wave is associated with a particle (shortly this wave will be seen to be
the wave function of the particle).
• A particle with high momentum has a wavef unction with a short wavelength,
and vice versa.
88

Cont.
• Relate energy to frequency and momentum to wavelength.
• To explore the consequences of these relations, Standing de Broglie waves
• i.e. electron in motion in confined 1D box and standing wave.
,
,
,
89

Electron Microscope
The Photoelectric Effect
The Compton Effect
The Davisson-Germer Electron Diffraction Experiment
X-ray diffraction
Born based on the concepts
90

Cont.
Lyman
series
Balmer
series
Pashen
series
UV
Visible
IR
-13.60 eV
-3.40 eV
-1.51 eV
-0.85 eV
-0.54 eV
-0.37 eV
-0.28 eV
0.00 eV
1
∞
7
6
5
4
3
2
n
Energy
Atomic Spectroscopy Absorption or Emission
91

n1 → n2 name Converges to
(nm)
1 → ∞ Lyman 91
2 → ∞ Balmer 365
3 → ∞ Pashen 821
4 → ∞ Brackett 1459
5 → ∞ Pfund 2280
6 → ∞ Humphreys 3283
Where , Rydberg Constant =
Bracket
series
Pashen
series
Lyman
series
The wave lengths corresponding to the emission lines correlated
92
Cont.

Instruments Magnification Resolution
( )
Sample
environment
Sample
Thickness
Eye 1 200 Air Thick
Magnifying
glass
2-10 15-100 Air Thick
Optical
Microscope
200-1000 0.2 Air, Oil Thick
SEM 0.001 vacuum Thick
TEM 0.00014 Vacuum Thin
(~50nm)
Lecture Note on Advance Surface Characterization (2016) by Prof. Jinn P. Chu, MSE,NTUST, Taiwan. 93
• An electron microscope is a type of microscope that uses electrons to
illuminate a specimen and create an image.
• The greater resolution and magnification of the electron microscope is
due to the wavelength of an electron.
Cont.

Resolution
The smallest distance between two points that can be resolved by
• Human eye 0.1-0.2 mm
• Light microscope-0.2
• Scanning Electron Microscope (SEM)-~1.2nm
• Transmission Electron Microscope (TEM)
• TEM uses high energy electron beam
Lecture Note on Advance Surface Characterization (2016) by Prof. Jinn P. Chu, MSE,NTUST, Taiwan.
94
Cont.

Scanning Electron Microscope (SEM)
95

97
Topography
The surface and texture of a matter.
Morphology
The size, shape and order of the particles that matter
consist of.
Composition
The elements and compounds that the object is composed
of, and the relative amounts of them .
Crystallographic Information
The arrangement of atoms in the object .
Important character of SEM

99
Transmission Electron Microscope (TEM)
 TEM, is a microscopy technique in which a beam of electrons is transmitted
through a specimen to form an image.
 The specimen is ultrathin section less than 100 nm thick or a suspension on
a grid.
How is the image formed?
 An image is formed from the interaction of the electrons with the sample
as the beam is transmitted through the specimen.
Important character of TEM
 Due higher resolution, TEM enables to observe:
 Individual atoms of a surface,
 Helpful in studying small metal crystallites .
 Due to the Careful and systematic measurements using TEM also allow:
 Estimation of the particle size distributions occurring in a particular
sample.

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