MODERN PHYSICS

Lecture note 1

Dr.Narasimha H Ayachit1 and Dr.G.Neeraja Rani2

1. Prof. and Head of Department of Physics, Dean (examination), SDM College of Engineering and Technology, Dharwad, India-580002 and Special Officer VTU

Belgaum-590014.2. Asst. Prof., Department of Physics, SDM College of Engineering and Technology,

Dharwad, India-580002.

What we observe is not nature itself, but nature exposed to our method of questioning -

---- Heisenberg

Classical mechanics and classical physics which are studied to understand the law of nature in terms of what we see and feel about our surroundings is basically depends upon our common sense. The overall performance of all these classical phenomena depends upon the nature of basic constituents of these surroundings that is on atomic and molecular behavior. The phenomena occur at these microstructures do not seems to work on common sense. This is because of the concept of quantization of energy which led to the branch of Quantum Mechanics further which led to the new beginning of Modern Physics.

In this lecture note 1, a brief introduction to this concept has been given trying to drive home the ‘duality of nature’ as given by de-Broglie hypothesis. This note also includes some of the references to the work carried out earlier to de-Broglie’s hypothesis, which are the basis of this hypothesis, like interference, photo-electric effect etc.

The study of any science and in particular Engineering deals with the acquiring

knowledge about the energy transfer from one place to another place in its original

form or in a form in which it is needed by converting the original energy in the most

efficient manner. The technology we say is better if the efficiency increases i.e. the

loss of energy is minimized in its transformation. To manage and manipulate the

energy for efficient transfer, the mode of its propagation and knowledge about it are

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essential features and hence it is always the rate equations of propagation of energy

becomes important, which has the knowledge of energy and medium in which it is

happening.

The initial study was on light energy and its propagation. Initially it was

assumed that light travels in a straight line (rectilinear propagation of light) which

was explained through Pin-Hole camera (Appendix1.1) , which could only explain

the formation of shadow and images (due to lenses and mirrors) through reflections,

refractions and total internal reflections. The Young’s double slit experiment

(Appendix1.2) lead to the wave theory of light through Hygen’s Wave theory and

superposition principle of waves. Further, the diffraction of light confirmed the

wave theory. The polarization ( Appendix1. 3) of light lead to show that light is a

transverse Electro Magnetic radiation with electrical vector and magnetic vector

perpendicular to each other. The wave theory could also explain all those, which

rectilinear propagation of light could explain and the rectilinear propagation of

light was totally gone in to the dust.

During 19th century, Hertz experiment of photoelectric effect (Appendix1.4)

and the explanations of the experimental results of photoelectric effect by Einstein

through Plank’s radiation concepts lead to show that, the light radiation when

interacts with some alkali metals rather behave with particle nature i.e. energy in

the form of quanta (photons) ( Appendix1.5), i.e. when more than one come together

also show that they act individually, contrary to the wave nature ( the wave nature

as per the principle of superposition act collectively when more than one wave act at

point and also its energy acts on surroundings). This was evident from the fact that

2

no electrons are liberated even when enormous number of photons are made to fall

on alkali metals, the frequency of whose are less than the threshold frequency and

only increase in the kinetic energy but no increase in the number of electrons when

the frequency of the photons is increased by keeping the density of photons same.

The above facts were explained by Einstein’s equation,

hν = W + (1/2)mv2

Where h is Planck’s constant, ν is the frequency, W is the work function, m mass of

the electron and v is the velocity of the ejected electrons. Thus, one can conclude

that the photoelectric effect can be explained by considering that, the incident

energy behaves as small energy packets leading to one to one interaction between

photons and electrons. This lead to show that light some times behaving like waves

(under interference, diffraction and polarization) and some times as particle (Photo

electric effect), leading to the duality of light.

The other experiment which showed the behavior of duality is Compton

effect. In this X-rays are made to collide with the electrons of a system and during

collision one can see that the energy exchange between two happens as if it is a

particle-particle collision. Here although X-rays are waves as shown by diffraction,

in this particular experiment, they behave like a particle which is inferred through

the shift in wavelength observed. Further, the explanation of Planck’s black body

radiation using Planck’s equation, that energy is in quanta, namely E = hν show that

the energy behave like a particle.

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WAVE – PARTICLE DUALISM (de-Broglie’s hypothesis)

1. The experiments on intereference, diffraction and polarization in light infer

that light behaves as waves. While the Photoelectric effect infers that the

light at the time of interaction show that they behave as quanta( particle) .

2. The X- rays through the process of diffraction show that they are waves,

while the same X-rays show that they are scattered as if they are particles

following the law of conservation of energy.

3. The above two examples clearly talk of duality of electromagnetic radiation.

de-Broglie in his study, assuming that what is true with energy ( X-

ray/light ) is also true with matter, as they are interchangeable according to

Einstein’s theory 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.

de-Broglie went on to prove that λ = h/p, where h is Planck’s constant

and p is momentum of the particle/ wave. λ is the wave length, known as de-

Broglie wavelength.

4

de-Broglie hypothesis opened up a new thinking in almost all the

fields of Physics. In fact it can be treated as the new beginning of the Modern

Physics. To quote an example, is of the motion of an electron in an orbit

around the nucleus of an atom. The de-Broglie hypothesis compel one to

think that the electron just do not follow the path of the circle but has to go

whizzing around this path in the form of a wave. But this is not sufficient.

Everyone knows as long as an electron exists in an orbit no dissipation of

energy takes place and if this has to happen there is no other way of thinking

that the number of waves around the orbit should be an integer? Of course

this has been proved later leading to the selection rules of transitions.

5

Appendix I

1.1 PINHOLE CAMERA

A pinhole camera, also known as camera obscura, or “dark

chamber”. Although the concept of pinhole camera was visualized in 1545, it

was put in to practice to take a photograph during 1850 by Sir David

Brewster.

A typical Schematic diagram (Fig.1) and explanation of its working is

presented below.

Figure1. Pinhole camera

When light falls on an object, each point on its surface reflects light

rays in all the directions. If these rays are made to pass through a hole they

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produce a inverted image of the object on the projection plane and this image

being at all parts proportional to the object clearly indicates the receiving of

energy at a point through a straight ray coming out of a hole.

In fact technically the best camera in the world is a pinhole camera,

because it does not involve aberrations and the one which works perfectly as

per the principles of rectilinear propagation of light.

1.2 YOUNG’S DOUBLE EXPERIMENT

In the year 1802 Thomas Young demonstrated experiment on

interference of light. Superposition of two or more waves which are coherent

(same frequency, nearly the same amplitude and are always in phase with

each other) results a modified wave pattern (Fig.2).

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Figure 2. Young’s double slit experiment

The coherent waves spreading from the two pin holes S1 and S2

superimpose on each other and produces alternatively bright and dark bands

on the screen CD. When crest falls on crest or trough falls on trough

produces a bright band (maximum energy) and when crest falls on trough or

vice versa results a dark band(minimum energy).

The interference experiment explained above indicates division of

amplitude and hence the division of energy from the same source S and

combination of these energies from S1 & S2 to have a modified energy in the

form of variation of intensity on the screen strongly advocate towards the

wave nature of light.

1.3 POLARIZATION

Any thing which is normal as per the nature’s requirement is

unpolarized. Polarization is the state of a system which is under a force

because of an external agency which looks to be more systematic and

conditioned compared to the natural state. Is it not right to wonder under the

state of polarization the system is under strain!!!? The greatest inference the

polarization of light has given is not only that light is a transverse wave, it is

an electromagnetic wave.

Different types of polarizations like Circular polarization, Elliptical

polarization etc are possible through the combination of plane polarized light

which can be achieved through different methods like use of Nicole prism,

polaroiders etc.,

8

The process of plane polarization is shown in the figure3 below.

Figure3. Plane polarization

Let the unpolarized light waves vibrating in all the planes perpendicular to

the direction of propagation are passed through the polarizers 1 and 2, when

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both the slits are parallel to each other and are perpendicular to the direction

of propagation the plane polarization is observed. In any other direction the

sine waves of reduced amplitude are observed.

1.4 PHOTOELECTIC EFFECT

When a light radiation of certain minimum frequency known as threshold

frequency falls on a certain alkali metal surfaces it has been observed that the

electrons get ejected (example working of photovoltaic cell). This phenomenon is

shown in the figure4.

Figure 4. Photoelectric effect

Hertz working systematically on the photoelectric effect by placing alkali

metals in a vaccum tube and noting the current created in the vaccum tube to avoid

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the influence of the surrounding on the experiment achieved the following results.

Here in are also presented the explanation how these results lead to the concept of

quantization of the energy associated with waves, here after called as photon and

their behavior while interacting with the matter as if they are particles but not

waves.

a) Results obtained by varying the frequency of the incident light.

As the frequency increased from a minimum value gradually it was found

that below a certain frequency not a single electron was liberated and above this

frequency the electrons were liberated. This frequency was named as threshold

frequency. This threshold frequency is different for different metals. It is also found

that increase in the frequency of the incident light increases the energy of the ejected

electrons while their number remained the same. This indicates the transfer of light

energy to the electrons in one to one corresponding manner leading to think that the

energy associated with each light wave as quanta of energy referred earlier as

photon. This one to one interaction is obviously similar to the transfer of momentum

from particle to particle involving elastic collision. No release of electron below the

threshold frequency indicates the quanta of energy associated with photon is not

sufficient to liberate the electron.

b) Variation of intensity of the light below the threshold frequency.

The experiment was also conducted by varying the intensity of light below

the threshold frequency. Under, no intensity of light it is found that the electrons are

liberated showing that they are not collectively been able to create an energy which

is sufficient to knock out a single electron from the metal surface. This non collective

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effect of light waves clearly indicates the energy transfer is not in favor of wave

theory but is in favor of particle theory.

c) Variation of intensity of the light above the threshold frequency.

Above the threshold frequency it has been found that the number of

electrons liberated is proportional to the intensity of light irrespective of the

frequency used. This nature of being proportional is in it self an indication of one to

one interaction between photon and electrons which have interacted with each other

and in turn pointing towards particle nature.

The above inferences drawn form the results made Einstein to propose the

Einstein’s photoelectric theory borrowing the concept of Planck’s radiation theory,

which mathematically can be presented as below.

hν = W+ (½)mv max2

Where ν is the frequency associated with the incident light, h being the

Planck’s constant and hence hν is the energy associated with the photon. W is

the work function given by hν0 with ν0 being threshold frequency. The last term

in the above equation is associated kinetic energy with the liberated electrons.

The general misunderstanding at lower level and some of the

statements made in the literature stating that the energy hν is partly used to

liberate an electron and the remaining energy is converted into kinetic

energy, which is very much against the statement of quantization of energy.

Rather it should be understood that the whole of the energy hν is transferred

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to the electron through a perfect elastic collision and this electron which gets

excited converts this energy into the mechanical energy which is used for the

ejection of electrons and its kinetic energy.

1.5 Quanta (photons)

Whenever an electron transits from a higher energy level to a lower energy

level giving an electromagnetic radiation having an energy E = hν in the form of a

pocket which can not be broken in to parts and hence this energy associated

with each transition is quantized and this energy is known as a quanta. These

quanta of energy which gets transferred to the material after interaction in full,

under the propagation of this quanta of energy from a source to material or

otherwise where the momentum transfer does not take place is physically

realized in the form of a photon and hence a photon is a visualization of

quantized energy with no mass.

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MODERN PHYSICS

Lecture note 2

Dr.Narasimha H Ayachit1 and Dr.G.Neeraja Rani2

1. Prof. and Head of Department of Physics, Dean (examination), SDM College of Engineering and Technology, Dharwad, India-580002 and Special Officer VTU

Belgaum-590014.2. Asst. Prof., Department of Physics, SDM College of Engineering and Technology,

Dharwad, India-580002.

Implications of de-Broglie hypothesis

Dilemma about radiation leading to dilemma of matter by de-Broglie triggered for

opening of scientific gold mine consisting of great research work. This is evident from

the scientific out put that led in between 1925-1928, for example;

a) Pauli’s exclusion principle, which later became the theoretical basis of

Periodic table,

b) Heisenberg, Born Jordan’s matrix mechanics which led to the

understanding of the motion of an electron within an atom leading to

the explanation of spectral lines.

c) Schrödinger’s wave mechanics associating any thing in motion with a

wave function led to the revolution in Solid State Physics.

d) In continuation of Schrödinger Fermi-Dirac and Bose-Einstein

proposed the statistical nature of the universe by giving the different

statistical laws. The statistical nature of the universe and the principle of

superposition associated with wave mechanics led to the

conceptualization of uncertainty principle, which led later to show that

a free electron can not exist with in a nucleus. This inference drawn

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explained the nuclear structure and later made us to stay in this world

with threat of nuclear weapon!!!

e) Dirac’s relativistic wave mechanics led to the explanation of the

electrons spin. Another greatest concept of the relativistic wave

equation led to the prediction of antimatter.

f) Dirac’s another fundamental work of quantum field theory provided in

sight in to the understanding of the electromagnetic field.

g) The Photoelectric effect which is basis of de-broglie’s hypothesis which

involves a statement of symmetry of nature made one to think of the

production of radiation after a particle interacts with matter, that is,

nothing but the understanding of production of X-rays, which were just

till then were only X (an alphabet as used in case of Algebra).

Thousands of pages of spectral data with accurate values of wavelengths of

elements were lying in the dust and nobody knew how they resulted from the atoms or

molecules and what information they convey i.e., this whole data was as if coming from a

black box. The understanding of this and its impact in the later developments through

quantum physics formed on the basis of de-broglie hypothesis stand to the testimony of

greatness of de-Broglie’s hypothesis.

There is no end to make the list of implications. We can add a few if you want to

this non ending list like invention like LASER, concepts of symmetry and identity,

probability concept of wave function and very recent development of Nanoscience and

Nanotechnology.

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Davisson Germer Experiment

The de-Broglie’s hypothesis and in fact the whole of the quantum theory which

hypothesis the energy of electromagnetic wave of a frequency f which is quantized in

units of hf and the concept that only this unit can interact in toto with matter was not well

accepted by all probably because of not able to visualize the association of a wave with a

particle. The two important experiments which removed all the doubts regarding this are

one Compton effect and the other most important one is Davisson Germer experiment.

In line with Bragg’s law one may infer that if diffraction has to happen there

should be a variation of intensity up and down as the angle of scattering increases. This

indicates the wave nature, while gradual decrease in the intensity with the scattering

angle indicates Rayleigh’s scattering in consistent with the particle nature. As the

electrons are well known as particles it is expected them to show the Rayleigh’s

scattering but as per the de-Broglie’s hypothesis if these electrons are made to move fast

they should behave like waves after scattering. As per the de-Broglie wave length

expression for electrons having momentum generated by applying a voltage of 54V the

wave length will be 1.66 X 10-10m as shown below.

1/λ = p/h (momentum p= = )

1/λ = p/h = =

Or λ = 1.66 X 10-10m.

Thus, as per condition at which diffraction has to take place we should have a

material having set of planes with interplanar spacing approximately around 1.66A0.

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Requirement of this value will slightly vary depending upon the applied voltage used to

accelerate electrons.

Thus, the major requirement to observe diffraction is to find a diffracting target of

the dimensions mentioned above and to carry out the experiment in such a way that the

motion of electrons is not affected before scattering and after scattering. Davisson

Germer were successful in achieving this with nickel metal as a target by carrying out the

experiment in vaccum. This experiment commonly known as DG experiment which was

carried out in vaccum has been discussed in detailed along with the inferences drawn

below.

Davisson Germer experimental set up is as shown in the Figure5 below,

Figure 5. Davisson and Germer’s experimental arrangement

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Experimental set up

It consists of a tungsten filament (F), variable voltage source, Nickel metal (N)

and an electron detector (D). When the filament is heated the electrons are liberated from

it by thermionic emission. 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 a support by the applied voltage. When the electrons hit the

target metal they get scattered around the metal. The electron detector was mounted on an

arc so that it could be rotated to different angles (ф) to observe the scattered electrons

from the metal surface. Keeping the accelerating potential constant the intensity of

electrons was measured by varying the angle ф. Polar graphs (figure6) below show the

dependence of electron intensity on ф.

Figure 6. Polar Graphs

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It was a great surprise to find that an intense peak at certain angle and increase in

the intensity with the applied accelerating potentials up to 54V and then decreases. The

maximum intensity for this peak was observed for 54V and ф=500.

If the electrons are considered to behave like particles then inconsistent with

Rayleigh’s scattering, in the above experiment the graph should not contain any peak

with intensity gradually increasing. Thus, the peak indicates the occurrence the

phenomena of diffraction and hence strongly indicate the wave nature of electrons, which

could be interpreted by the Bragg’s diffraction law as below (figure 7).

The interplanar spacing value is 0.91A0 for a set of planes under study in nickel

metal. With this interplanar spacing and with applied voltage of 54V Davisson Germer

found the diffraction observed at an angle of 650. With these values and using the Bragg’s

law (2dsinӨ = nλ) with n = 1, λ turns out to be 1.65X10 -10m. Comparing this value with

the one calculated using de-Broglie equation infer that electrons do behave like waves. It

also infers the duality of matter. As acceleration of electrons before the scattering stand

with the particle nature while diffraction observed after scattering stands for wave

nature.

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Concept of Group velocity and Phase velocity.

Another important consequence is how do I see the energy under motion?

Undoubtedly in the form of a wave which is a resultant of many waves almost having

nearly the same amplitude determined through the de-Broglie wave equation λ = h/p, the

value of which is being very small but having slightly different frequencies. That is what

one is trying to understand through an experiment which involves the motion of energy

through a wave, which is resultant of group of waves. Thus, any concept to study

involving de-Broglie hypothesis has to have its basis on group wave concept. However,

when momentum transfers take place, these individual waves are looked as quanta of

energy having a concept of particle. However, for an individual wave it is the change in

phase that becomes important. In view of these concepts and the fact that any energy

associated with motion of the system is visualized through its velocity, the present study

demands defining of three types of velocities, namely Group velocity, Particle velocity

and Phase velocity.

Take some amount of water in a non rigid container in which each molecule is

under motion with respect to other molecules. When this container is moving with certain

velocity v the individual molecules will have velocity of motion very close to the velocity

of the container. Thus the individual velocities of the water molecules can be associated

with phase velocity and that of a container which is the mean of all the phase velocities to

be forming a group velocity. When this container transfers its momentum after impact to

some other body the body will be feeling the velocity of the container as if it is one

particle. This shows that the particle velocity and the group velocities are one and the

same. This can be further visualized in better manner assuming that this type of

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containers are received continuously at equal intervals of time by the body which gets

impacted will only be able to sense the velocities of the individual containers rather than

the velocities of individual molecules in a container. This can be taken as an analogy to

feel concept of particle velocity equivalent to group velocity. Thus the group velocity of

the wave of different cycles can only be felt as one pocket.

Phase velocity

Describes how a peak (crest and troughs) of a wave moves along the direction of

propagation and the velocity with which it moves is known as phase velocity.

To understand the movement of the peak of a wave, let the weights are suspended

on the ends of the springs and all them bounce with same frequency but all of them start

slightly at different times. Then these weights will form a traveling sine wave as shown in

the figure 8 below.

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Consider a point on the sine wave at a particular instant of time the wave equation

can be written as Y = Asin ωt

After certain time say t0 if the point is displaced by a distance x the equation is

given by

Y = Asin[ω(t-t0)].

Velocity of the point v = x/to, or to = x/v

Therefore Y = A sin [(ω (t-x/v)] = Asin[ωt- (ω/v)x] = Asin(ωt-kx), where k= ω/v.

Where k is defined as the propagation constant or the wave number which is also

defined as k= 2Π/ λ.

‘v’ mentioned above as stated earlier is associated with the individual wave and

hence it is nothing but phase velocity. Thus phase velocity vphase = ω/k. ---------1.1

Group Velocity

When a group of two or more waves of nearly same frequencies are superimposed

on each other the resultant wave will have different properties from those of the

individual waves. Variation in the amplitude is found in the resultant wave which

represents the wave group or wave packet.

To obtain an expression for group velocity one has to consider the superposition of

constituent waves, according to the principle of superposition of waves. Below is

discussed the superposition of two waves which are a part of a group wave that is the

waves which have slightly different angular frequency which are superimposed along a

line. For simplicity only two waves are taken, whose amplitudes(say A) have been

considered as same. Any wave can be represented by a sinosoidal equation.

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If Y1 is the displacement of one wave at a particular instant of time(t) and

position(x) whose frequency is ω with wave number k is then,

Y1 = Asin(ωt-kx) -------- (1.2)

Considering another wave with a displacement Y2 at the same instant (t) and

position (x) at that of Y1 but having slightly different frequency (ω+Δω) and hence also

the wave number (k+ Δk), then

Y2 = A sin [(ω+ Δω) t – (k+ Δk) x], -------- (1.3)

The resultant displacement due to the superposition of the above two waves is,

Y = Y1 + Y2 (as per the principle of superposition)

= Asin(ωt-kx) + A sin[(ω+ Δω) t – (k+ Δk)x] ------(1.4)

Since, sinA + sinB = 2cos ( ) sin ( ),

Equation (1.4) is written as

Y = 2Acos {( ) t ( ) x} sin {( ) t – ( ) x}

As the difference in frequency of the two waves is very small because Y1 and Y2

form part of a group Δω and Δk terms in the sin part can be neglected as they appear in

combination with 2ω and 2k which are quite large compared to these quantities. The

same can not be done in cosine part as these terms do not appear in combination with 2ω

and 2k. Hence one can write the above equation as

Y = 2Acos {( ) t ( ) x} sin (ωt-kx) ------- (1.5)

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From a look at the equations (1.2) and (1.5) one can conclude that under the

approximation mentioned above, the wave component of the original reference wave and

the resultant wave remains the same while the amplitude changes from

‘A’ to ‘2Acos {( ) t ( ) x}’

The demonstration of the superposition of two waves having slightly different

frequencies is illustrated in the following figure 9.

Figure 9.

From the figure it is evident that the amplitude of the resulting wave is as per the

cosine component of equation (1.5) and it reaches its maximum when cosine component

value becomes 1. That is

cos {( ) t ( ) x} = 1 i.e. {( ) t ( ) x} = 0

From the above equation one can write

x/t = ( ) , x being the position at a given instant t of the resultant

wave i.e. group wave, obviously x/t represents the group velocity (Vg). Under the

limiting case Δk tending to zero the group velocity Vg can be written as

Vg = dω/dk ------- (1.6)

MODERN PHYSICS

24

Lecture note 3

Dr.Narasimha H Ayachit1 and Dr.G.Neeraja Rani2

1. Prof. and Head of Department of Physics, Dean (examination), SDM College of Engineering and Technology, Dharwad, India-580002 and Special Officer VTU

Belgaum-590014.2. Asst. Prof., Department of Physics, SDM College of Engineering and Technology,

Dharwad, India-580002.

De-Broglie wavelength (Derivation)

de-Broglie in consistent with his hypothesis had given an expression for the

wavelength associated with the matter wave namely, λ = h/p where h is a plank’s

constant and p is the momentum associated with the matter wave (in some of the books

the above equation has been derived by equating, Plank’s quantized energy E = hν and

mass energy relation E = mc2, obtained through Einstein’s theory of relativity. That is by

putting, E = hν = hc/ λ = mc2 i.e. λ = h/ (mc). Treating m as the mass and c as the

velocity and hence mc as momentum, one may write λ = h/p. But the question here is m

is of what? and c is of what? Certainly both do not belong to the same quantity and hence

although the approach seems to give the right equation but it is not the right way to do it).

It has been proved that the de-Broglie wave length is in consistent with the

wavelength associated with matter waves, undoubtedly arising out of the duality and

hence out of group wave and particle concepts. Below is presented the derivation of

expression for de-Broglie wavelength with this concept. As per the Plank’s quantization

and photo electric effect, we know

E = (1/2)mv2 + V -------(1.10)

And E = hν ------- (1.11)

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By equating the above two equations,

hν = (1/2)mv2 + V --------(1.12)

The left hand side is synonymous with duality of energy as per the photo-electric

effect and the right hand side of the above equation stands for particle nature as the two

terms are associated with the motion of electrons liberated in the photoelectric effect.

We know, vg = dω/dk, where ω = 2Πν and k= (2Π)/ λ

dω = 2Πdν and dk = 2Πd(1/ λ)

Thus dω/dk = dν/d(1/λ) --------(1.13)

This leads to d(1/λ) = dν/vg, writing vg = v as is being commonly written

Differentiating the equation for photo electric effect i.e. (1.12) we get

hdν = mvdv or dν = (m/h)vdv.

(dν/v) = (m/h)dv

d(1/λ) = (m/h)dv --------(1.14)

Integrating we get,

1/λ = (m/h) v + constant. -------(1.15)

For a particle having a velocity zero the wavelength can be assumed to be infinite

i.e. λ = for v = 0. Substituting this in the above equation we obtain the constant of

integration as zero. Thus,

1/λ = p/h

Or, λ = h/p = h/mv --------(1.16)

Please note v is the velocity of matter waves which are associated with those

particles having mass m only.

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Relation between group velocity and particle velocity

Energy of a photon E = hν or ν = E/h ------ (1.17)

We know ω = 2Πν or ω = (2ΠE)/h

dω = (2Π/h)dE ------(1.18)

further, k = 2Π/ λ = (2Πp)/h

dk= (2Π/h)dp ------(1.19)

dividing (1.18) by (1.19)

dω/dk = dE/dp ------(1.20)

by definition group velocity vg = dω/dk

vg = dω/dk = dE/dp ------ (1.21)

If a particle of mass m is moving with a velocity vparticle

Then the Non relativistic energy

E = (1/2)mv2particle = p2/2m -------(1.22)

Differentiate with respect to p

We get dE = (2p/2m)dp = (p/m)dp

dE/dp = p/m = (mvparticle)/m = vparticle

Hence vg = vparticle ------ (1.23)

By relativistic approach we can also obtain the same results as shown below.

The relativistic energy is given by the relation

E2 = p2c2 + m02c4 , where m0 is the rest mass ------- (1.24)

Differentiate the above equation with respect to p

dE/dp = pc2/ ------(1.25)

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by substituting, p= m0vparticle/ in (1.25) we get

vg = vparticle ----- (1.26)

It can be seen that this result is in consistent with our argument put forth earlier

with an example.

Characteristics of matter waves

Characteristics of matter waves are,

(i) They represent the resultant of group of waves

(ii) They represent, the wave associated with a particle if the constituents are

particles and represent the waves associated with photons if it is an

electromagnetic radiation.

(iii) The wavelength associated with the particles λ = h/p where h is Plank’s

constant, p is the momentum associated with the particle. If the de-Broglie

wavelength associated with electromagnetic radiation λ is given by hc/E where c

is the velocity of light and E is the energy associated with the photon.

(iv) The amplitude of these matter waves being very small and being the resultant

of group of waves the amplitude determined of a de-Broglie wave only gives us

the probability of occurrence of the constituent particle at a given position and at a

given time.

(v) Under the limiting case the relation between group velocity and the phase

velocity is given by vgvphase = c2, which leads to an interesting observation that

matter waves have the velocity more than that of light. This observation compel

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one to think that matter waves are not the waves which can be physically felt but

indicate only the probabilistic nature of these waves.

(vi) The velocity associated with the electromagnetic radiation remains constant

for all the wave lengths while the velocity of matter wave differs under different

conditions associated with it.

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In the sharp formulation of the law of causality-- "if we know the present exactly, we can calculate the future"-it is not the conclusion that is wrong

but the premise.--Heisenberg, in uncertainty principle paper, 1927

We are treating motion of a particle in the form of group. Thus any thing we want to measure physically, what we see is because of the parameters connected with a group. When we want to measure we try to individualize them and hence the deviation in the measurement which exist between that of an individual parameter and a group parameters. Thus to this extent we have always an uncertainty built up.

The effect seen even in an individual wave is because of the interaction of matter waves with the system under study. As per the Planck’s quantum theory the energy hν cannot be broken i.e. when an interaction takes place what part of the wave has come in interaction is uncertain. Apart from this the proof can be done in Relativistic and Non-Relativistic.

According to “classical physics” It is easy to measure both the position and velocity of a macroscopic system i.e. the future motion of a particle could be exactly predicted, or "determined," from knowledge of its present position and momentum and all of the forces acting upon it. But when it comes to microscopic system (subatomic particles) this is not true as explained by the Heisenberg's uncertainty principle. According to Heisenberg, because one cannot know the precise position and momentum of a particle at a given instant, so its future cannot be determined. One cannot calculate the precise future motion of a particle, but only a range of possibilities for the future motion of the particle. Because a moving particle is associated with a group wave hence according to Born’s probability interpretation the particle may be present any where with in the group wave. If we consider the group wave is to be narrow, then the position of the particle can be found accurately where as the inaccuracy in calculation of its velocity or momentum increases. On the other hand if we consider the group wave to be wide, then its velocity or momentum can be found accurately whereas there is lot of inaccuracy in locating the particle. That is

The Uncertainty Principle The more precisely the position is determined, the less precisely the

momentum is known in this instant, and vice versa. --Heisenberg, uncertainty paper, 1927

Simultaneous measurement of both the position and momentum (two conjugate variables) of a particle cannot be done with arbitrarily high precision. If one tries to measure the position of a particle precisely there is an imprecision (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 of things.

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Following 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 properties of the particle. Four of these measured properties which are important for the uncertainty principle, are the position, its momentum, its energy, and the time. These properties appear as "parameters" in equations that describe the particles motion.

The uncertainty relations may be expressed as.

xp > h/2Et > h/2LӨ> h/2

Where,

1 x is the uncertainty or imprecision (standard deviation) of the position measurement.

2 p is the uncertainty of the momentum measurement in the x direction at the same time as the x measurement.

3 E is the uncertainty in the energy measurement.

4 t is the uncertainty in the time measurement at the same time as the energy is measured.

5 L is the uncertainty in the angular momentum measurement

6 Ө is the uncertainty in the angle measurement at the same time as the angular momentum is measured

7 h is a constant from quantum theory known as Planck's constant, a very tiny number.

8 Π is pi from the geometry of circles.

If we measure the position of a moving atomic particle with a great accuracy that is x is very small almost tending to zero then, what happens to the measurement of the momentum p, which we measure at the same instant? From the equation

p > h / 2 x

we can see that the uncertainty in the momentum measurement, p is very large, that is it approaches infinity since x in the denominator is very small.

Proof of Heisenberg’s Uncertainty Principle

When two or more waves superimpose on each other forming a group wave the resultant we get is

Y = 2Acos {( ) t ( ) x} sin (ωt-kx)

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Of the above equation the term 2Acos {( ) t ( ) x} determines the amplitude of the

group wave. Thus the certainty lies in between two nodes of the wave shown in the Fig.10.

The distance between these two consecutive nodes(x1,x2) can be obtained as follows. First

node is found at ( ) t ( ) x1 = (2n+1)2 and the second node at ( ) t ( ) x2 =

{(2n+1)2}+. Therefore by finding the difference between these two one can write (

) (x1-x2) = . By taking (x1-x2) = Δx, as the uncertainty in position one can write Δx =

2/Δk. But k = 2π/λ = 2πp/h.

Therefore Δk = (2πΔp)/h.

Using this expression for Δk in the expression for Δx we get

Δx Δp = h.

Extending this argument to two and three dimensions respectively one can prove,

Δx Δp = h/2π, Δx Δp = h/4π

Gamma-Ray Microscope

The one of the best experiment which can illustrate the uncertainty in the measurement of two conjugate quantities simultaneously is the gamma ray microscope through resolving power of a very high resolution microscope (Fig.11).

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Fig.11

According to the definition of the resolving power Δx = (λ)/(2sinθ). To make Δx smaller one has to select a radiation whose wavelength is very small. So the best choice of it would be either X-ray or γ-ray. When this radiation falls on a surface it is scattered by an electron which transfers momentum of (hν)/c as per the Compton Effect and which leads to a disturbance in the central position of the electron. If the radiation gets scattered at an angle ‘θ’ then the error in the measurement of momentum can be maximum of

Δp=(2hν sinθ)/(c).

Therefore Δx Δp={(λ)/(2sinθ)}{(2hν sinθ)/(c)}

= h

This is along a point, if it is taken over an area in 2 dimensions it becomes h/2and over the space h/4Applications of Uncertainty Principle1. The non-existence of free electron in the nucleus.

The diameter of nucleus of any atom is of the order of 10-14m. If any electron is confined within the nucleus then the uncertainty in its position (x) must not be greater than 10-14m.According to Heisenberg’s uncertainty principle, equation (1.27)

x p > h / 2

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The uncertainty in momentum is

p > h / 2x , where x = 10-14m

p > (6.63X10-34) / (2X3.14X10-14)

p > 1.055X10-20 kg.m /s --------------(1.30)

This is the uncertainty in the momentum of electron and then the momentum of the electron must be in the same order of magnitude. The energy of the electron can be found in two ways one is by non relativistic method and the other is by relativistic method.

Non-Relativistic method:

The kinetic energy of the electron is given by,

E = p2/ 2m

p is the momentum of the electron = 1.055X10-20 kg.m /s

m is the mass of the electron = 9.11X10-31kg

E = (1.055X10-20)2/ (2X9.11X10-31) J

= 0.0610X10-9J

= 3.8X108eV

The above value for the kinetic energy indicates that an electron with a momentum of 1.055X10-20 kg.m /s and mass of 9.11X10-31kg to exist with in the nucleus, it must have energy equal to or greater than this value. But the experimental results on β decay show that the maximum kinetic an electron can have when it is confined with in the nucleus is of the order of 3 – 4 Mev. Therefore the free electrons cannot exist within the nucleus.Relativistic method:

According to the theory of relativity the energy of a particle is given by

E = mc2 = (m0c2)/(1-v2/c2)1/2 ------- (1.31)

Where m0 is the particle’s rest mass and m is the mass of the particle with velocity v.Squaring the above equation we get,

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E2 = (m02c4)/ (1-v2/c2)= (m0

2c6)/ (c2-v2) ------- (1.32)

Momentum of the particle is given by p = mv = (m0v)/ (1-v2/c2)1/2

And p2 =(m02v2)/ (1-v2/c2)

= (m02v2c2)/ (c2-v2)

then p2c2 = (m02v2c4)/ (c2-v2) ------------(1.33)

Subtract eqation (1.33) from (1.32)

E2 - p2c2 = {(m02c4) (c2-v2)}/ (c2-v2)

= (m02c4)

or E2 = p2c2 + m02c4 (1.34)

= c2(p2 + m02c2)

Substituting the value of momentum from eq(1.30) and the rest mass as = 9.11X10-31kg we get the kinetic energy of the electron as

E2 > (3X108)2 (0.25 X 10-40 + 7.469 X 10-44)

The second term in the above equation being very small and may be neglected then we get

E > 1.5X10-12 J

Or E > 9.4 MeV

The above value for the kinetic energy indicates that an electron with a momentum of 1.055X10-20 kg.m /s and mass of 9.11X10-31kg to exist with in the nucleus it must have energy equal to or greater than this value. But the experimental results on β decay show that the maximum kinetic an electron can have when it is confined with in the nucleus is of the order of 3 - 4 Mev. Therefore the electrons cannot exist within the nucleus.

2. Width of spectral lines (Natural Broadening)

Whenever the energy interacts with the matter the atoms get excited and the excited atom gives up its excess energy by emitting a photon of certain frequency which leads to the spectrum. Whatever may be the resolving power of the spectrometer is used to record these spectral lines no spectral line is observed perfectly sharp. The broadening in the spectral lines is observed due to the

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indeterminateness in the atomic energies. To measure the energies accurately the time (t) required is more that is t tends to infinity or otherwise. According to Heisenberg’s uncertainty relation

E = h / 2t

Where E is the uncertainty in the measurement of energies and t is the mean life time of the level is finite and of the order of 10 -8s which is a finite value. Therefore E must have a finite energy spread that means the energy levels are not sharp and hence the broadening of the spectral lines. Thus broadening of spectral line which cannot be reduced further for any reason is known as natural broadening.

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How does the uncertainty measured? It is through the probability that means de-Broglie wave which is basically a group wave defined by group velocity, a corresponding frequency wave number and de-Broglie wavelength has to be represented by probability equation. Since it is being a wave Therefore any mathematical model representing this shall be a combination of sine and cosine terms which are basically complex in nature. This model shall represent the probabilistic value of finding a particle at a given instant at a given point. Such a representation is what is known as a wave function. This wave function does physically means a representation of probabilistic value and since it involves sine and cosine terms it has to be complex in its nature. This type of a wave function is written as

ψ = Ae-i(ωt-kx) ------(1.35)

This wave function as mentioned earlier is complex and to obtain a real part which gives a physical meaning is only through multiplying with by its complex conjugate. This shows that the probability is measured through (ψψ*)1/2 where ψ* complex conjugate of ψ.The value of amplitude of ψ is determined through the following equations. For an existing particle the probability of finding it is under the limits of boundary conditions imposed

1. Integral of ψ ψ* dψ is equal to one.

2. For a non existing particle ψ ψ* dψ is equal to zero..The above two equations strongly advocate towards quantized energy states for an existing particle leading to its physical significance. From the above discussion the characteristics of a waveform representing a de-Broglie wave can be listed as below1. Wave function is complex2. It is Probabilistic in nature 3. Its value is such that ψ ψ* dψ = 14. Since it is to find a probability through integration it means the function has to

be differentiable.5. Since it is differentiable in the existing space it should be continuous.6. It is not only continuous it should be single valued over the space.

There can be two types of wave function

1. Time dependent →These are the wave equations associated with de-Broglie waves in which amplitude decreases with time. For example the motion of the electron/hole across the pn junction of a semiconductor diode. The equation of motion for this type of de-Broglie waves are known as Time Dependent Schrödinger Wave equation

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2. Time independent → these are the wave equations associated with de-Broglie waves in which amplitude remains constant at all times. This is associated with those systems where the energy is conserved in which the de-Broglie wave forms a stationary wave. Example being the motion of an electron inside a metal (topic discussed later in the form of a quantum model of a free electron) in which the electron get confined to the metal as the potential in side the metal is less than the potential outside the metal. However these electrons move from one place to another place and form de-Broglie waves of stationary type. The equation of motion written for such particles is known as Time Independent Schrödinger wave equation.

Time Independent Schrödinger wave equation.

Let us consider a particle having a mass ‘m’ associated with a de-Broglie wave of wave length ‘λ’ and moving with a group velocity ‘v’ then λ = h/mv. Let this wave equation be a stationary wave that is time independent type then the wave function associated with this wave ψ can be written as ψ = Ae-i(ωt-kx) Since it forms a stationary wave the value of ψ changes with position rather than time and hence to obtain the realistic differential equation the above wave function is only differentiable with ‘x’ and ‘t’ is taken as constant,

dψ/dx = -ikAe-i(ωt-kx) and

d2ψ/dx2 = -i2k2Ae-i(ωt-kx)

= -k2Ae-i(ωt-kx)

= -k2 ψ

Therefore (d2ψ/dx2) + k2 ψ = 0 ---------(1.36)

But k = 2 λ then k2 = 4 λ2

= 4 (h2/p2)

= 4mv h2

= (8m h2)(m v

8mKE)h2Always energy is the measure of a system and hence it has to be represented in terms of energy).

If E is the total energy of the particle and V is the potential energy associated with the particle then KE = E –V

Therefore k2 = 8m(E-V)}h2

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Substituting this in the equation (1.36) we get

d2ψ/dx2 + [8m(E-V)}h2] ψ = 0 (1.37)

The above equation is known as Time independent Schrödinger wave equation.

Eigen Values and Eigen functions

The solution of a equation of motion written for a de-Broglie wave has seen earlier is trigonometrical in nature(complex). Thus any trigonometrical function will have a most general solution and hence they are not unique. All these possible solutions written in the form of wave functions are known as Eigen functions. This can be seen from the fact that a particle which forms a stationary wave in a given potential can have more than one wavelength and hence frequency.

In view of the fact the particle can exist with different frequencies corresponding to different Eigen functions naturally corresponds to different energies that is a particle can have more than one energy at different timings satisfying the Schrödinger wave equation are known as Eigen values.

Process of Normalization

In obtaining the Eigen functions from equation of motion associated with a de_Broglie wave involves the process of integration such that the

ψ ψ* dψ = 1 for an existing particle that is the constants have to adjusted in such a manner. This process is known as process of Normalization of a wave function. Application of Schrödinger wave equation to a particle inside a potential well of infinite height

Let us consider a particle of mass ‘m’ inside a potential well moving in a direction say along the breadth of the potential well. Let us assume that this as X-axis and let the breadth of the potential well be ‘L’ (Fig.12). Let this particle move along X-axis with out doing any work in other words at any energy of the particle it is moving in an equipotential space. Since always potential difference is a measure of the difference between the potentials the potential difference inside the box can be considered as zero. Thus the Schrödinger wave equation of time independent type for this particle can be written as

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Fig.12

d2ψ/dx2 + [8mE)h2] ψ = 0.

This is taken as time independent type because the particle is confined to the potential well and hence can be taken as a case where the energy dissipation does not take place. This can only happen when formation of stationary waves takes place.

As already known the solution of this type of Schrödinger wave equation will have a solution in its complex form. That is it contains sine and cosine terms. Thus, the most general solution can be for the wave function ψ associated with particle can be written as

Ψ = Asinkx + Bcoskx ------- (1.38)

Note that here‘t’ is considered as zero because the solution is for the time independent type.

ψ = 0 at x≤ 0 and also at x≥ L.

Using these boundary conditions in the solution of the wave function can be written as

1. at x = 0, ψ = 0

i.e. 0 = Asin0 + Bcos0

Then B = 0

Therefore ψ = Asinkx ----------(1.39)

2. at x = L, ψ = 0

i.e.0 = AsinkL (Since B=0)

Therefore kL = n

Or k = (nL

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Therefore ψ = Asin(nxL ------ (1.40)

This wave function ‘ψ’ has to satisfy the condition that ψ ψ* dψ = 1. Thus using the boundary conditions we can write L

∫ A2 sin2(nxL dx = 10

Then A2/2 { ∫ 1dx} - A2/2 { ∫ Acos2(nxLdx} = 1 {sin2θ = (1-cos2θ)/2}A2L/2 = 1Or A = (2/L)1/2

ψn = (2/L)1/2sin(nxL ------------ (1.41)

Since ‘n’ can take all possible integral values that means there is more than one wave function. These all values of sin are called Eigen functions. For each Eigen function we associate an Eigen value i.e. the allowed energies associated with that particle.We know that k = nL

Then k2 = n2L2

Or 4 λ 2 = n2L2 ( k = 2 λ )

4 (h2/p2) = n2L2 (λ = h/p)

4mv h2 = n2L2

(8m h2)(m v= n2L2

Since the potential energy is equal to zero the kinetic energy KE = (m v is the total energy E.

Then 8mEn)h2= n2L2

Therefore En = n2 h2mL2 ---------- (1.42)

En are called Eigen values. ‘n’ can take any value i.e. energy can be any value since it is an infinite potential well that is the energy gap between the surroundings and the well is infinite. Below is represented a schematic diagram of wave function associated with particle having n = 1,2,3…. (the position of a particle shown at any point should not misunderstood as its exact position as it only indicates the most probabilistic value). Infact, dealing with quantum mechanics one has to discourage the interpretation of anything through a diagram. However for understanding the concept and visualizing the things in the probabilistic manner these pictures are represented.

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For n = 1; E1 = h2mL2

And ψ1 = (2/L)1/2sin(xL with ψ1 = 0 at x = 0ψ1 = (2/L)1/2 at x = L/2ψ1 = 0 at x = Lψ1 = - (2/L)1/2 at x = 3L/2

For n = 2; E2 = 4h2mL2

And Ψ2 = (2/L)1/2sin(2xLwith ψ2 = 0 at x = 0ψ2 = (2/L)1/2 at x = L/4ψ2 = 0 at x = L/2ψ2 = - (2/L)1/2 at x = 3L/4ψ2 = 0 at x = L

Similarly for n = 3; E3 = 9h2mL2 and so on

The wave can be represented as in the following diagram(Fig.13).

Fig.13It may please be noted that1. The wave formed are stationary waves2. Energies are discrete3. The gap between two energy levels is not uniform.

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Traditionally light is defined as a radiation which when falls on objects and comes to our eyes (reflection) gives a sense of vision. The modulation in the reflected light gives a lot of information. However it will not be able to acquire the full knowledge as this light is most unsystematic in nature. More scientifically light is defined as an electromagnetic radiation which carries an information after reflection through the process of modulation and which can be sensed by the use of proper detectors. Thus, basically light is the energy in the form of electromagnetic radiation. This can happen only through the process of emission, that is, a system coming from a higher energy state to a lower energy state. Of course, all emissions need not give us light. The one which gives us light is said to perform radiative transition and while the others are called non-radiative transitions (example, cooling of water). The process involved in which one is interested is the emission of radiation in the form of light. This light although have enormous good characteristics has the following negative characteristics.

1. Highly disordered in nature

2. Not coherent

3. Highly scattered in all possible angles

4. Energy per unit cross sectional area is very less

5. Highly divergent in nature

All these characteristics make it inefficient as a source of energy. What matters is not the total energy but the energy per unit cross sectional area which increases the efficiency of the system and hence its power. Thus, there is a need to amplify light such that the energy per unit cross sectional area increases which does not have much of the negative characteristics mentioned earlier.

Amplification of any parameter of a system is defined as achieving the out put more than the normal out put that is an efficient out put compare to its in put in a parameter in which we are interested at a given instant of time. This can be achieved by the following processes.

(i) Reducing the loss

(ii) Negative feed back

(iii) Systematizing out put

(iv) Supplying an external energy of different type and converting it in to desired type of energy

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In case of light, for example, a small converging lens can produce a large amount of heating effect. This is because of an amplification achieved through the process of reduction of loss. The working of a table lamp with a reflector increases the intensity per unit area on the table is an example of negative feed back. Increasing the intensity of light in Phosphorescence is an example of external energy being fed. Thus, for any amplification to happen the process is to be controlled from out side. This is known as process of stimulation. If there is no control from out side the system behaves on its own as per its spontaneous nature and hence a process that happens without control is spontaneous process and the performance will be more erratic. In case of an ordinary emission of light the properties as listed above exist because the process involved is a spontaneous emission. Some how if this light is amplified that is the process is controlled externally one would get an amplified light. This light expected to give us other than the characteristics mentioned above through the process of amplification achieved through stimulation as explained by Einstein and this light is called LASER an acronym for Light Amplification by Stimulated Emission of Radiation.

The following properties are expected from the LASERS

1. Highly concentrated energy that is the energy per unit cross sectional area should be very high.

2. Highly unidirectional because the divergence has to be less. If the divergence is less the action of the beam at any point away from the source will remain constant.

3. Highly monochromatic. If the light contains more than one frequency naturally they can not be in phase and hence the reduction of energy takes place. Thus it is expected that to bring about coherence the source should be highly monochromatic

4. Phase coherence, it is not just sufficient that light is monochromatic but every ray in a beam should be in phase with respect to each other so that intensity do not reduce. This is known as phase coherence

5. Temporal coherence, the above parameters of phase coherence and monochromatic makes the light to have the intensity at any point across the cross section of the beam to be same. This is known as temporal coherence.

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THEORY OF LASERS

To achieve Laser the basic requirement is to achieve amplification so as to get the characteristics mentioned earlier. Thus one needs to achieve controlled emission against spontaneous emission. Before achieving emission the system has to be excited to the higher state that is in a typical two energy level system the atoms or molecules have to be put to a higher energy level through the process of absorption of radiation. Let us say at any given instant of time N1 be the number of atoms/molecules in the ground state and N2

be the number of atoms/molecules in the excited state corresponding to energy of energy levels E1 and E2. The number of atoms/molecules are determined through the equations E1

– E2 = hν and according to Boltzman’s equation N2 = N1Exp (-h ν /kT), where h is the Planck’s constant, ν is the frequency of the radiation, k is the Boltzman’s constant and T is the temperature at which absorption is taking place.

The rate of absorption (stimulated absorption shown in Fig.1) is directly proportional to the number of atoms in the ground state and also the radiation density ‘ρ’ incident on the system.

Fig.1

Therefore dN1/dt α - ρN1

The negative sign indicates that the N1 decreases as the absorption takes place. This is also written as

(dN1/dt)ab = - B12ρN1 (1.1)

B12 is known as Einstein’s coefficient of absorption. Once a system gets excited the emission starts to takes place. The most probable is the one which happens on its own that is spontaneous emission (Fig.2). Thus the rate of spontaneous emission (dN2/dt)sp is only proportional to the number of atoms in the excited state i.e, N2.

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Fig.2

Therefore dN2/dt α - N2

The negative sign indicates that the N2 decreases as the emission takes place. This is also written as

(dN2/dt)sp. = -A21 N2 (1.2)

A21 is known as Einstein’s coefficient of spontaneous emission.

The interest is in controlled emission which is more systematic and is known as Stimulated emission (Fig.3). The efficiency of the stimulated emission do depend upon the number of atoms/molecules in the excited state and the parameter which is controlling from out side which is nothing but the radiation involved in the process of absorption i.e. energy density ‘ρ’.

Fig.3

Therefore the rate of stimulated emission is proportional to the number atoms/molecules in the excited state and the radiation density ’ρ’

(dN2/dt)st α - ρN2

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The negative sign indicates that the N2 decreases as the emission takes place. This is also written as

(dN2/dt)st = - B21ρN2 (1.3)

B21 is known as Einstein’s coefficient of Stimulated emission.

RELATION BETWEEN EINSTEIN’S COEFFICIENTS AND EXPRESSION FOR ENERGY

DENSITY

Under equilibrium conditions the rate of absorption is equal to the rate of emission.

Therefore (dN1/dt) ab = (dN2/dt)sp. + (dN2/dt)st. (1.4)

That is B12ρN1 = A21 N2 + B21ρN2 (1.5)

Rearranging the terms we get

ρ = A21 N / (B12N1 - B21ρN2) = (A21/ B21){( N2) / [(B12/ B21)(N1/ N2)-1]}

ρ = {(A21/ B21)/ [(B12/ B21)(N1/ N2)-1]} (1.6)

According to Boltzman’s equation

N2 = N1Exp(-h ν /kT) or N1/N2 = exp((h ν /kT) substituting this in the above equation

ρ = {(A21/ B21)/ [(B12/ B21) exp((h ν /kT)-1]} (1.7)

Since the above equation is for an equilibrium state the radiation density ‘ρ’ is the same as given for a black body by Planck.

According to Planck’s radiation law

ρ =[ (8 Пhν3)/(c3)] {1/[exp((h ν /kT)-1]} (1.8)

Where ‘ν’ is the frequency of the radiation, ‘c’ velocity of light, ‘k’ Boltzman’s constant and T is the temperature of the system

Comparing the equations (1.7) and (1.8) one can write

A21/ B21 = (8 Пhν3)/(c3) and B12/ B21 = 1 (1.9)

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Equation (1.9) gives the relation between Einstein’s coefficients. It may be here noted that a system which has good absorption power that is higher value of Einstein’s coefficient of absorption has also good ability to give stimulated emission.

Therefore

ρ = {(A21/ B21)/ [(N1/ N2)-1]} (1.10)

The equations (1.7) and (1.10) can be taken as the expression for energy density.

POPULATION INVERSION AND ITS NEED

The energy density is given by the equation (1.10). In this expression the nature of ‘ρ’ determines whether the system is under amplification or not. If ‘ρ’ is positive then there is no gain and hence, no amplification. Thus if the system has to act as an amplifier then ‘ρ’ has to be negative. According to Boltzman's equation N2 = N1Exp(-h ν /kT). As per this equation at T = ∞, N1 can be made equal to N2. It means at ordinary working temperatures it is impossible to have N2 greater than N1, that is, in the expression for ‘ρ’ the denominator is always positive and hence ‘ρ’ is positive i.e. the system does not act as amplifier. If ‘ρ’ has to be negative, the denominator has to be negative in the equation (1.10). This is only possible when N2 is greater than N1. That is, to say population of atoms/molecules in the higher energy has to be greater than the population in its ground state, which is much contradictory to the existing condition as per Boltzman. Between two states this requirement for laser is known as Population inversion.

The direct implication of requirement of population inversion is that the laser can not be achieved through a system which has two energy levels as it is a relation between only those two energy levels. (However in the recent times people have achieved laser through two energy level systems, the basic principle being that the rate of absorption is made faster than the rate of emission. This can be achieved only when one type of laser is used as the source to achieve another type of laser).

METASTABLE STATE AND ITS NEED

The energy states are classified in three categories according to their life time (it is the time spent by an atom/molecule in that energy state).

(i). Stable (Ground) state – It is that state in which if an atom/molecule exists for ever unless forced upon to leave, that is, its life time is infinite.

(ii). Unstable (Excited) state – Whenever an atom/molecule excited to certain states it spends a time of about nano seconds to pico seconds. Such states are referred to unstable states. Emission from such states are always spontaneous in nature as the life time is small and here they can not be made to interact with the external controls.

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(iii). Metastable state – The atom/molecule excited to this state spends time of about milli second, that is, nearly 1010 times more than the unstable state, such states are known as Metastable state. The emission from such states is known as delayed emissions or Phosphorescence. Since life time is higher these states can be utilized to bring about population inversion and made to interact with the external control so as to get a stimulated emission.

A TYPICAL LASER ACTION

THREE ENERGY LEVEL SYSTEMS

Let an energy ‘hν/’ be made to fall on a system at energy state E1 (Fig.4) so as to get excited to E2. E2 being unstable state part of the atoms/molecules comes back to E1 i.e. some of the atoms /molecules come directly to E1 by emitting radiation while few gets in to the state E3. The excitation from E1 to E2 is continuous and hence with time the population in E3 goes on increasing as E3 is metastable state. A continuous absorption between E1 and E2 and transitions between E2 and E3 the population in E3 goes on building up till a stage reaches where population in E3 level becomes more than that in E1. After a milli second when transition happens from metastable state to stable state i.e. E3 to E1 that will be the amplified signal as the population in E3 is greater than the population in E1. Note that the Boltzman equation has not been violated because absorption is between E 1

and E2 while the population inversion is achieved between E3 and E1. This amplified signal (Phosphorescence) which comes due to the transition from E3 to E1 say with a frequency ‘ν’ is made to fall back on the system. This signal interacts with the atoms/molecules in the metastable state resulting stimulated emission of radiation. These emitted photons will have the same direction, same frequency and same phase with respect to each other. This process is continuously repeated and hence an enough amount of photons are emitted. A part of this is taken out as a source and the remaining is sent back into the system for further continuous process. The part of the light that is coming out is highly monochromatic, highly directional and in phase and hence it is coherence in all respects and hence it is a Laser.

Apart from the above theoretical phenomena, the laser action is supported through experimental setups and construction. The laser active material is kept in a cavity which is long in length and smaller in breadth with two mirrors along its cross section with a small hole at one side through which part of the laser beam emerges. A typical diagram is presented as below (Fig.5). The mirrors used are dielectric mirrors with 99.9% reflection. The energy to the system is given along its length of the cavity. The distance between the two mirrors is so adjusted that it is exactly equal to integral multiples of amplified frequency. This creates a resonance and helps in absorption of the signal leading to the better stimulated emission. Since the length of the tube is long and two parallel mirrors with a small hole in one of the mirrors are used, the light that is emerging through this small hole becomes unidirectional.

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Fig.4

Fig.5

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Some of the terminologies used are as below

(i) Active material. The material which is responsible for laser action that is a material in which amplified signal is attained through population inversion.

(ii) Resonator. The cavity in which the active material is placed and in which a resonance is brought about by adjusting the two mirrors such that the distance between them is integral multiples of wavelength of the amplified signal.

(iii) Pumping technique. The method of exciting the system to a higher state by supplying external energy in the above example from E1 to E2 is what is known as pumping. There are different methods of pumping. If the light energy is used to excite the system then the pumping is called as optical pumping, when the electrical energy is used it is called as electrical pumping, chemical energy brings the chemical pumping and so on.

As can be understood from the above discussion the material to act as a laser has to meet lot of conditions like B21 to be large, existence of metastable state etc. and hence the choice of material becomes very limited.

In a similar manner the discussion can be extended to more than three energy level systems. However, in any system transition from a metastable state to lower energy state is essential. In case of four energy level system the transition is from metastable state to another state which is mostly unstable from which nonradiative transition takes place to the lower state. In view of this building of population inversion becomes much faster and hence are found to be more efficient than three energy level systems.

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Lecture notes 8

RUBY LASER

Ruby laser was fabricated by Maiman in 1960 and was the first laser to be operated successfully. It consists of abundant amount of aluminum oxide with some of the alumina ions, replaced by chromium ions. The concentration of chromium ions is approximately 0.05% by weight. The aluminum oxide is known as host material which absorbs the incident energy and transfers it to chromium ions, which take part in lasing action. The energy level diagram is as shown in Figure 6. From the figure, it can be observed that ruby laser is a three level laser.

Figure 7 shows a typical setup with a flash lamp (e.g., Xenon or Krypton) pumped pulsed laser. As the absorption bands of the ruby laser and the emission spectra of available flash lamps matches very well with each other makes the pumping of atoms to higher energy levels more efficient. The helical flash lamp is surrounded by a cylindrical reflector to direct the pump light onto the ruby efficiently. A typical ruby was in the form of a rod of length about 2 to 20cm and diameter about 0.1 to 2cm. The ends of the rod was grounded and polished to make them parallel and flat. One end of the rod was completely silvered and the other end was partially silvered through which the laser beam emerges.

When the light from the flash lamp falls on the ruby rod the chromium ions get excited to E1 and E2 levels from the ground level. These excited chromium ions relax rapidly through a non radiative transition to the pair of levels M, which is the upper laser level. The life time of the level M which is metastable state is of the order of 3ms from where the lasing action takes place by emitting a photon of wavelength 6943Ao.

One of the interesting things about ruby laser is that it is a pulsed laser. This is because of the flash lamp used which in itself is pulsating type so in a given pulse the population inversion gets built up as it reaches the threshold value till the next pulse is absorbed. The population depletion takes place below the threshold value of inversion and hence lasing action stops for a very small time. This leads to pulsating laser to come out. A very close look at the spectrum of the out put, thus, gives a spike like structure.

In spite of ruby being an inefficient laser because it is a 3 energy level system still it finds a large amount of applications because of the laser characteristics.

1. The absorption band of ruby laser matches very well with the emission spectra of flash lamp in it self can be used for pumping.

2. It has long life time.3. It has narrow line width.

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4. Because it falls in the visible region it is used in photographic emulsion, photodetectors and in pulsed holography.

Figure 6

Figure 7

He-Ne LASER

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He-Ne laser was the first gas laser to be operated successfully and is the most widely used with continuous power out put in the range of a fraction of mW to tens of mW. In gases the atoms are characterized by sharp energy levels compared to those in solids. Therefore an electrical discharge is used to pump the atoms instead of a flash lamp or continuous high power lamp, which is used in a system having broad absorption band.

Figure 8 shows the schematic of a He-Ne laser. It consists of a long narrow discharge tube of diameter about 2 to 8mm and length about 10 to 100cm, which is filled with the mixture of He – Ne gas molecules in 10:1 ratio. The laser transitions take place in Ne atom and the He atoms are used for selective pumping of Ne atoms so as to achieve population inversion. He-Ne laser may consist of either internal mirrors or external mirrors Figure 9. Usually the small lasers consist of internal mirrors. The space between mirrors constitutes a cavity or a resonator.

In case of external mirror type, to avoid the reflection from the ends of discharge tube, the Brewster windows are attached at the two ends. These windows have almost 100% transmittance excluding very small absorption and scattering losses. The laser beam gets polarized in the plane of incidence and transmits through the windows and this polarized laser will be perpendicular to the plane of incidence gets reflected back. This leads to linearly polarized beam as an out put. The orientation of the laser beam can be varied either by rotating the tube or the laser itself.

Figure 8

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Figure 9

When an electrical discharge is passed through the gas, the electrons, which are accelerated collides with the He atoms in the discharge tube and excites them to a higher energy levels i.e. to F2 and F3 (Figure 10). As the life time of these atoms being more (10-4

&5X10-6 s respectively) many He atoms get collected there and these atoms can not go to directly to the ground energy level by emitting a photon as this transition is not optically allowed. The energy levels F2 and F3 act as metastable levels and the energies of these levels almost coincide with the E4 and E6 energy levels of Ne. The excited He atoms collide (perfectly elastic collision) ground Ne atoms and transfers whole of energy to Ne atom, excites them to E4 and E6 energy levels. As the gas mixture contains He and Ne in the ratio 10:1, the Ne atoms are selectively populated to E4 and E6 levels.

The Ne atoms in E6 level take a transition to E3 level by emitting the very popular radiation of He-Ne laser of wavelength 6328A0. In Ne atom the life time of E6 energy level is very much greater than the life time of E3 level and hence the population inversion is achieved between E6 and E3 levels. From E3 level the atom goes to E2

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spontaneously as the life time of E3 is too small (~10-8s). The E2 energy level is a metastable energy level and thus the Ne atoms get collected there. These atoms instead of going to ground state by colliding with the walls of the tube may absorb the spontaneously emitted radiation and there by get excited back to E3 level which effects the achievement of population inversion and hence the efficiency of the laser. To avoid the absorption of the radiation in E2 level the diameter of the tube is decreased to as narrow as possible.

Two more possible transitions E4 to E3 and E6 to E5 can be observed in the energy level diagram, which emits the radiations of wavelength 1.15µm and 3.39µm respectively.

Figure 10

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Lecture notes 9

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. The major property of laser is its high power, high directionality. The character of high directionality means the energy is concentrated with in the beam and hence the precession in space is highly uniform. The major problem of a laser is its high energy a major part of which goes as reflected light. So to find any application of laser the remaining energy after reflection should be capable of interacting with the material. In this part also a small amount of energy also gets wasted as transmission energy. In view of all these one need to select a laser source that has very high energy and such lasers are known as industrial laser. The examples of the industrial laser are CO2 laser and Nd-YAG lasers. The combination of these lasers with robotics and optical fibers has made these mechanical applications more diverse in nature.

LASER DRILLING

In case of a conventional drilling a drill bit is held to the specimen with large amount of force between them apart from the force used to hold the drilling machine and the specimen. This brings about large amount of mechanical friction and hence most of the energy gets wasted in producing of heat, sound and other energies. The heat energy produced creates an uneven distribution 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 in to deformation of drill bit. This leads to non-uniform drills created by the same drill bit. Apart from this if the drill is needed to be very small in diameter then having a drill bit with high mechanical strength of that size becomes impossible but in the today’s world of miniaturization thus becomes essential. These practical problems can be overcome by using high power lasers as there is no mechanical contact between the laser and the specimen and the highly directional laser can also be concentrated on a narrow space. Drilling can be done at any angle and through very hard and brittle materials. However the drilling through lasers needs a high precession both in 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 heat generated 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-metallic materials. The pulsed laser exposure is controlled through a time controller, as any variation in this will result in not only the variation in depth but also in width. Further, to avoid accumulation of molten metal in the drill a high-pressure inert gas preferably argon gas is passed. A typical schematic diagram is shown in figure 11. The drills created through Lasers are highly uniform. Since lasers can be concentrated on a small space very small drills can be created.

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Figure 11

LASER WELDING

Laser welding plays very important role to reduce the roughness of the melted surface, eliminate mechanical effects and to limit the heat effected area in the material. Laser welding are generally used for welding multilayer in which the thermal properties differ at interfaces. Compared to the other types of welding such as arc welding and electron beam welding the laser welding is more efficient because it is contact less process, does not destruct the material and the structure remains homogeneous. The laser welding process is as explained below.

The laser beam is focused on to the material as shown in the figure 12 at a particular point at which the junction is to be formed between the two layers. Because of high intensity of the beam falling over a smaller region generates lot of heat energy which melts the material with in that small region. When the temperature of the material is reduced the molten part of the material solidifies and makes a stronger joint.

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Figure 12

LASER CUTTING

Laser beam is made to concentrate at a point and the specimen is moved with a uniform speed. This creates molten state in the metal. Repeating this movement to and fro for number of times the depth is made to go on increasing and ultimately the whole of the specimen can be cut. Since laser can be concentrated on a very small space and the surface energy is very smooth the cutting becomes uniform. This cutting through laser light is facilitated using high pressure oxygen so that the efficiency becomes better. This also helps in high uniform cutting. The combination of this system with the programmed robot will give a better control over the whole process. Coupling the laser with NC machine will make the three dimensional cutting possible. Surface after cutting will be very smooth and hence no polishing is required. A typical schematic laser cutting diagram is presented in figure13.

Figure 13

MEASUREMENT OF POLLUTANTS IN THE ATMOSPHERE

Of all the pollutions the air pollution in the atmosphere is considered to be the most hazardous which basically results from the creation of dust, smoke, flash etc. These pollutants are in the size of nano meters. The challenge here is not only the size of the particle but also the very less quantity of pollutant. The conventional method of Chemical analysis fails because of the less quantity and the practical limitations in the measurement of quantity accurately. The accurate measurement becomes essential at lesser quantities because even small variation of air pollutants can create drastic effect on a life cycle.

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The other method of quantitative analysis is through the spectroscopic analysis which depends on Kirchhoff’s law which states that a material which has a capability to emit certain wavelengths at higher temperatures also has the capacity to absorb the same wavelength at lower temperatures. These wavelengths are the characteristic of that particular material and the amount of absorption 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 as Absorption technique. By measuring the absorption and the wavelength the quantitative and qualitative 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 a source of light because of their important properties like monochromatic, high radiation density and highly concentrated energy so that it can even interact with the smallest percentage of element present in the atmosphere. For such type of experiments Dye laser or tunable lasers which gives continuous source of wavelengths, of course, one wavelength at a given instant time are used. Further, to increase the concentration of the pollutant the sample is collected at atmospheric pressure is compressed to high pressure, so that the concentration of the pollutant per unit volume is increased so that the detection becomes easier.

A laser light is split into two parts one is made to fall on the sample and reflected back to the detector with the help of a concave dielectric mirror. The other beam is used as reference beam and made to fall on the detector such that it is out of phase with the one which comes through the sample. By comparing these two the variation in the intensity at different wavelengths of Dye laser can be recorded. By knowing the wavelengths the elements can be identified and by knowing the variation in the intensity with reference to Nitrogen or Carbon dioxide or Oxygen, the concentration of the pollutants can be measured. With these values the corresponding concentration of the pollutants at atmospheric pressure can be calculated.

Pollutants can also be identified using Raman Back-scattering method. In this method the laser light is made to pass through the sample and the spectrum is recorded for the transmitted light. As the laser is a monochromatic source the spectrum is expected to give single intense line corresponding to the frequency of laser light. But also it consists of several lines very close to the intense line on either side of it. It is because of Raman scattering. These spectral lines are called side bands. The side bands in the Raman spectrum observed at certain frequencies correspond to oscillating frequency of the molecule plus or minus the 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 can find the type of pollutant present in the sample.

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Lecture notes 10HOLOGRAPHY

When we see around us we can feel that the objects are in three dimensions that is, we have the ability to analyze the intensity distribution and the depth of the different points of an object. In other words we have the ability to analyze the amplitude variation and path difference between the two points. When a photograph of an object is taken all that we see is the amplitude variation but no path difference. This leads to a two dimensional view. In view of this, there is a necessity that a technique has to be developed, such that when photograph is taken it should have the information about amplitude variation and path difference between the two points. This is what is known as Holography. The information about the path difference between the two points is nothing but the information regarding the phase difference between the two rays coming from two different points, 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 is an interference experiment and hence the techniques of holography has to essentially depend upon the interference phenomena.

Some times some sketches gives us a three dimensional feel and also with the help of specially prepared spectacle, one can get three dimensions feeling. This should not be understood as holography, they are just illusions in reality.

In principle holography can be had using multiple wave lengths of source but a single color will give a better information regarding the phase difference and hence use of monochromatic source of light is preferable. Further, whenever a difference is measured if the reference is uniform, the measurement becomes clearer. Hence, a laser as a source becomes very important in holography because not only it is monochromatic but also it has both the phase and temporal coherence.

PRINCIPLE

The idea of recording the whole information (Holography) of a three dimensional objects on 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 wave called reference wave. These two waves produce the interference pattern on the recording medium which is the characteristic of the object. This is known as recording process. The interference pattern which was recorded consists of amplitude distribution and the phase of the waves scattered from the object hence it is called hologram. Construction of a hologram and reconstruction of the image are as explained below.

Consider two coherent waves P and Q are made to fall on the photographic plate with different angles of incidence as shown in the Figure14. Superposition of these two waves results in interference fringes on the photographic plate which can be observed after developing the film. These fringes carry the information regarding phase and

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intensity of the beam. If once again the coherent waves P and Q are made to fall on the film at the same positions then these waves pass through fringe pattern and the waves P /

and Q/ can be observed from the transmission side these waves appear as if they are the continuation of the waves P and Q respectively (Figure 15). If one of the incident waves say Q is blocked and only one wave P is made to fall on the film. This wave gets diffracted by the fringes that are present on the film and the emergent wave gets split in to two parts. One of the part gets deviated from its path and emerges as if it is the continuation wave Q (Q/) which is not present there and the other part emerges as P/

Figure 16. In this way in holography, the arrangement is done to get one of the coherent waves reflected from the object called object beam on to the photographic plate 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 produces secondary wavelets. These secondary wavelets superimpose on each other and produce maxima in certain directions and generate a real and virtual images of the object. On the transmission side one can view the real image and on the other side virtual image of a three dimensional object (Figure 17).

Figure 14

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Figure 15

Figure 16

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Figure 17

Recording of the image of an object can be done in two ways

(i) wave front division technique (figure 18) and (ii) Amplitude division technique (figure 19)

In both the techniques part of the incident laser beam is made to fall on the object and the other part on to the photographic plate which has been placed in front of the object.

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Figure 18

Figure 19

Applications

1. Holographic Interferometry.

The one of the important testing process is non destructive testing of specimen which basically depends upon interferometric methods. The nature of holographic process to give objective analysis of a specimen when illuminated with a reconstruction wave allows us to have interference leading to a method called double exposure holographic interferometry. This uses the photographic process between the object wave and a reference wave of a specimen which is under stress alternatively to create a hologram. The holograms so created give us two object waves when illuminated with reconstructive 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 the information about the qualitative and quantitative stress distribution. Since interference is involved in the whole process the slightest variation in the stress distribution can be analyzed.

There is another type of holography known as variant holography which gives us what is known as real time interferometry in which a hologram is formed of an

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unstrained object and the image produced by the hologram is superimposed on the reconstructed object of a strained specimen. If the object undergoes any strain, fringes will be formed and this can be studied as function of time. Such type of studies is used in study of vibrating objects like musical instruments etc.

2. Microscopy

The very principle of holography which involves the measurement of path variation gives us a tool in which depth measurements are involved of microscopic level which are constant or transient in nature. The ordinary microscopic studies involving interference can be 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 handy which is recorded over a time and event and can be studied later through a hologram as it is frozen in this dynamic holography. Such studies can be utilized in the analysis of cloud chamber, rocket engine and aerosols etc.

3. Character recognition

This is used in optical image in which the recognition of a character is essential. This involves 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 will be plane polarized and hence alternatively bright and dark lines of uniform thickness will be formed. Using this, holograms are produced which are nothing but diffraction gratings. Such a uniform formation of gratings is impossible as all other methods involve mechanical approach which needs a very high control.

5. Information coding

In the present era of quantum computing which involves optical coding requires the information 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 be reconstructed and acquired through this hologram.

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OPTICAL FIBERS

Two of the greatest developments of recent past are in communication and

Miniaturization. The contribution to this has come through different developments in

Science and Engineering. Polymer Science is one of such field, which has resulted into

the realization of an optical fiber.

Optical fiber, basically, does a job of efficient transfer of light signal from one

point to another. In past also the light was used for communication system so also at

present, for example traffic lights. The basic problem with light is that it has the property

of scattering and hence the distance through which it can be communicated. In any solid

medium if this light is made to confine, then also it is impossible to make the light to

travel from one point to another point with out much scattering. This is basically because

of the large number of scattering points (scattering points should not be considered as

some other particles in side the material but it is the change in the refractive index of the

material because of change in density at various points). Polymers are able to overcome

this, of course, in very low dimensions.

Optical fiber with such polymers works on the principle of Total Internal

Reflection and hence the material has to have more than one density, which is to be well

controlled. Although Total Internal Reflection was known long back, development in

transmission of light in a confined space could not be achieved because of non-

availability of technology described above. With the development of optical fiber

practically we find its applications in every field of Communication, Medicine, Industry

and sensors. The combination of optical fibers with lasers has revolutionized the Science

and Technology. In the following paragraphs the development in communication as one

of the examples is discussed.

Our conventional communication system was earlier through cable and later

through microwaves although they gave us a large advantage, have their own limitations.

For example through the cables, the communication is achieved by the modulation of

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electrical signals, which are governed by the flow of electrons. As every one knows

electrons have mass and hence inertia. This in itself creates a problem in the speed of

communication. This speed not only depends upon the electrons but also on the material

of the cable, which results in to resistance. Apart from this, speed gets affected because

the electrons have charge hence they interact with the surroundings leading to loss of

energy. To overcome this, regular amplification is required. Because electrons have

inertia, whenever a cable bends the electrons are likely to penetrate from one cable to

another and hence cross hearing is possible. This is a severe problem in rainy season.

In a bundle of a wire through which electrons are moving, a part of that can be cut

and the information can be tapped and hence affects the security. For the reasons

mentioned above we have to use a copper wire or an alloy of it, which comes as a natural

limited resource. With increasing demand and decrease in natural resources it has put a

lot of problem in communication. There are several other drawbacks like weight of the

cable, durability, corrosion etc. But the greatest drawback of this cable communication

system is its limited bandwidth because each can transmit a very limited number signal

through it simultaneously.

The microwave signal, which belongs to wire less communication, has a problem

in its production basically because it is an electronic gadget and hence the speed of

production of such signals is slow. Earlier these signals were reflected through

ionosphere but at present through the satellite from one point to another. This distance

itself creates a lot of delay in the signal processing. More than any thing these signals are

left in space and whoever is capable of collecting it will be able to do so and hence there

is no security. Especially in an era in which we are talking of SKUD missiles and HIGH

END machines in defense the delay in signal transmission can create havoc and also the

insecurity. There are many more problems associated with the microwave technology one

of them again being low band width. Signal distortion is also very high in this system.

Replacing the above communication system by combination of light and optical

fibers where light acts as a signal and the optical fiber acts as a waveguide using the

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principle of Total Internal Reflection all the disadvantages observed in cable

communication and microwave communication system can be overcome. For example

light being a photon has no inertia and its speed is enormous. Since they are

electromagnetic radiation they do not interact with each other and hence minimum losses.

You have an infinite number of visible wavelengths and hence bandwidth is enormously

high. Since there is only one fiber for one modulated signal tapping is impossible as the

communication gets cut down. The optical fiber, which is used to send a signal, can also

be used to receive the signal. Since the material of an optical fiber is a polymer it can be

synthesized in abundance and it can have high mechanical strength to weight ratio etc.

Total Internal Reflection

Consider a ray S1 of light made to travel from a denser medium of refractive index

n1 to rarer medium of refractive index n2 (n1 > n2 ) which fall on the interface between two

mediums at an angle of incidence Ө1 with the normal. Due to the change in medium at

point ‘P’ the ray refracts at an angle r1. The angle of refraction will be more than the

angle of incidence as the ray passing from a denser to a rarer medium. If the angle of

incidence is increased from Ө1 to Ө2 the angle of refraction also increases from r1 to r2. As

the angle of incidence increased the angle of refraction also increases and at a particular

angle of incidence the angle of refraction becomes 900, that is, the refracted ray moves

along the interface between the two mediums. This angle of incidence is known as

critical angle and represented by Өc. Further increase in the angle of incidence increases

the angle of refraction greater than 900, which makes the ray to get back to the same

medium, which is known as total internal reflection (fig. 1a).

Figure 1a

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According to Snell’s law n1 sinӨ1 = n2 sinӨ2 and When Ө1 = Өc , Ө2 = 900

Then n1 sinӨc = n2 sin900

sinӨc = n2/ n1 (since, sin900 = 1)

Or Өc = sin-1 (n2/ n1) -------------- (1)

Figure 1b below gives the schematic representation of an optical fiber which is

used as a wave guide for the propagation of light on the principle of total internal

reflection. Using very high-end technology the glass fibers with low refractive indices are

drawn in micron thicknesses in the cylindrical form. This is known as a core. In fact the

bare core can be used as wave-guide as this is surrounded by air of low refractive indices,

which can easily facilitate total internal reflection. However around the core a material

with lower refractive indices is drawn which is again a glass fiber and is known as clad.

This gives the uniformity of variation in refractive indices between core and clad and to

give strength to the fiber. Another cover which gives mechanical strength avoids effect

from the environment and stops the stray light to enter in to the fiber surrounds the clad.

A typical fiber will be of the order of few microns. This smaller thickness ensures the

total internal reflection in an efficient manner.

Figure 1b

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Angle of acceptance and Numerical ApertureThe angle of acceptance is defined as the maximum angle with which a ray can be

sent in to the optical fiber such that it suffers total internal reflection and reaches the other

end of the fiber.

Consider a ray SO is made to enter in to the optical fiber with an angle ‘Ө0’ with

the fiber axis. Let angle of refraction at ‘O’ is Ө1 = 0. After refraction the ray falls on the

interface between core and cladding mediums at point ‘A’ at an angle Ө2 [( 90- Ө1)]. If Ө2

is equal to the critical angle Өc then the ray travels along the interface between core and

the cladding mediums as shown in figure 2. Consider another ray PO is made to enter the

optical fiber with an angle of incidence greater than Ө0 with the fiber axis. The angle of

refraction at ‘O’ will be greater than Ө1 and further the ray falls at point ‘B’ on the

interface with an angle of incidence less than Ө2 and hence the ray escapes through the

cladding in to the surrounding medium. Now, Consider another ray QO is made to enter

the optical fiber with an angle of incidence less than Ө0 with the fiber axis. The angle of

refraction at O will be lesser than Ө1 and further the ray falls at point ‘C’ on the interface

with an angle of incidence greater than Ө2 that is greater than Өc and hence the ray suffers

total internal reflection and propagates through the core medium and reaches the other

end. Thus any ray that enters the optical fiber with an angle either equal to Ө0 or lesser

than Ө0 at ‘O’ propagates through the optical fiber and the other rays escapes through the

cladding. The angle Ө0 is called the angle of acceptance. If SO is rotated around the fiber

axis keeping Ө0 it produces a conical surface. Any ray sent in to the fiber with in this

conical surface is accepted by the fiber and hence Ө0 is also known as the acceptance

cone half angle.

Figure 2

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The Numerical Aperture (NA) is defined as the light gathering capacity of an

optical fiber which is expressed as sin of angle of acceptance that is,

NA = sin Ө0

Condition for propagationConsider the figure 2 and assume that the refractive indices of the surrounding

medium as n0, core as n1 and cladding as n2.

By applying the Snell’s law at ‘O’ one can write

n0 sin Ө0 = n1 sin Ө1 ---------- (2) At point ‘A’

n1 sin (90-Ө1) = n2 sin 90n1 cosӨ1 = n2 ,

or cosӨ1 = n2 / n1 ----------- (3)

Equation one can be written as

sin Ө0 = (n1/ n0) sin Ө1

or,sin Ө0 = (n1/ n0) (1-cos2 Ө1)1/2 ------- (4)

Substituting Equation (2) in (4) we get,

sin Ө0 = (n1/ n0) (1- n22 / n1

2)1/2 ,that is sin Ө0 = (n1/ n0) [(n12- n2

2) / n12]1/2

sin Ө0 = (n12- n2

2) 1/2/ n0 ----- (5)

If the fiber is surrounded by air then n0 = 1, then

sin Ө0 = (n12- n2

2) 1/2 ------ (6)

orN.A. = (n1

2- n22)

1/2 -------- (7)

The ray will be able to propagate through the fiber only when

sin Ө0 ≤ N.A.

This is the condition for propagation of light.

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Fractional Index Change (∆)

Is defined as the ratio of the refractive index difference between the core and

cladding to the refractive index of the core.

∆ = (n1- n2) / n1 ------ (8)

Numerical aperture can be expressed in terms of ‘∆’ as

N.A. = (n12- n2

2) 1/2

= [(n1- n2)( n1+ n2)]1/2

= [(n1∆)( n1+ n2)]1/2 from equation (8)

Since the difference between the refractive indices of core and the cladding is very small

one can assume that n1 = n2, n1+ n2 = 2 n1

Then N.A. = (2n12∆)1/2

Or N.A. = n1(2∆)1/2 ------ (9)

From the above equation it can be observed that increasing the value of ‘∆’, one

can increase the light gathering capacity of an optical fiber. But a large value of ‘∆’ leads

to intermodal dispersion, which causes signal distortion.

Types of optical fibers and Modes of propagation

1. Step index single mode fiber

This optical fiber is made up of core of constant refractive index n1 with lesser

diameter and cladding of slightly lower refractive index n2. The name step index is

given because the refractive index profile (Fig.3) for this fiber makes a step change at

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the interface between the core and the cladding medium. Single mode is because of

smaller diameter because of which the optical fiber can allow for propagation of only

few modes through it at a time. As the signal transmitted is less, the signal broadening

is not practically found which makes these fibers highly advantages. The signal

propagates almost along the axis of the core as shown in the figure. The source used

for coupling the signal in to the fiber can be only lasers because of its diameter. These

are widely used in sub marine cable systems.

Refractive index profile Mode of transmission

Figure 3

2. Step index multimode fiber

The refractive index profile remains same as the step index single mode fiber

except for the core diameter which is almost five times greater than the single mode

fiber. Because of the larger core diameter this fiber allows many modes to propagate

through it. The refractive index profile and propagation of modes is as illustrated in

the figure 4. Each mode is made to enter in to the optical fiber with different angles of

incidents which makes these modes to propagate different distances through the fiber,

due to which broadening of the signal (intermodal dispersion) takes place at the out

put end. This in turn restricts the bandwidth of the optical fiber. These fibers can use

the spatially along with incoherent optical sources such as LED’s. In addition to

lasers can be easily coupled because of larger core diameter and larger numerical

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apertures. These are basically used in DATA links, which don’t require very large

bandwidths.

Refractive index profile Mode of transmission

Figure 4

3. Graded index multimode fiber

In this fiber the refractive index of the core is varied as a function of radial

distance from the center of the fiber and the refractive index of the cladding is

maintained constant. The refractive index profile and the propagation of modes is

as shown in the figure 5. Though the signals take different distances through the

fiber the signal distortion is not seen as they travel through the medium with

different refractive indices. The one that covers more distance travels through low

refractive index than the one that covers less distance. These are used for

illumination purposes and where the total variation in the light matters as in the

case of sensors.

Refractive index profile Mode of transmission

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Figure 5

Attenuation

In any communication system the major concern is the loss of information, which

basically comes because of attenuation. That is attenuation is the measure of loss in the

energy in its transformation from one point to another point. The effort of technology is

to always see that this attenuation is minimum. To achieve this one has to have the

knowledge of different parameters leading to attenuation that is, attenuation basically

comes because of the interaction of the energy or radiation with the material of the wave-

guide. Naturally the characteristic of attenuation will thus be dependent on both the

power loss and the length of the wave-guide. In view of the radiative properties always

the loss of power is logarithmic in nature and this loss being proportional to the length,

the coefficient of attenuation is defined as

= {[10/L) (log (Pin /Pout)]}

The factor 10 comes because of the unit used is decibels. In an ideal wave guide P in =

Pout is the one. Here Pin, Pout and L are in put power, out put power and the length of the

fiber respectively. The attenuation or the fiber loss (the loss of signal or power) because

of different parameters are as listed below,

1. Absorption loss

2. Scattering loss

3. Bending loss

4. Core and cladding loss

5. Dispersion loss etc.

1. Absorption loss:

Since optical fibers are made up of glass polymers and hence the absorption is

resulted due to

1. Defects in the composition

2. Impurities in the material (extrinsic absorption)

3. Characteristic absorption of the material of the fiber (intrinsic absorption)

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Apart from the above factors the amount of attenuation happens because of OH ions

present and the existence of transition metals. The presence of OH ions cannot be

avoided because the hydrolysis process involved in the preparation of optical fibers.

Thus, in the transmission of signal through the optical fibers, care has to be taken not

to use the wavelength of light where absorption occurs due to OH ions and the

characteristic peaks of the material of the fiber appear.

2. Scattering loss

However best the material may be, it is impossible to have its density uniform

through out. The variation in the density leads to the variation in the refractive index,

of course to a very little extent; even this little variation causes refraction resulting

into attenuation. The complexity increases because of the variation in the refractive

index with wavelength also. Thus, the scattering induced attenuation, which is a

Rayleigh’s type of scattering of light, is very complex in nature.

Structural inhomogenities and defects created in the fiber fabrication can also

cause scattering. This scattering being dependent on value of wavelength and it

decreases with increase in wavelength. Thus, using higher wavelengths one can

reduce this type of losses.

4. Bending losses

Because of the bending of the fiber arising out of its installation from one point to

another which are known as macroscopic bending, that is bending because of the

curvatures in the fiber also leads to attenuation. Another type of bending arising out

of uneven forces on the interface leading to change in the refractive indices of core

and clad resulting in to microscopic losses. These can be reduced by having high

radius of curvatures at the time of cabling and by using a high ordered clad which has

a good resistance for small variation in external forces.

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5. Core and cladding losses:

Since core and cladding have to have different refractive indices and their

multiplicity being high in step index multimode and graded index fibers, where

more than one material is used, the attenuation coefficients are going to be

different. This in itself creates scattering and hence attenuation when ray of light

tries to travel from one medium to another.

6. Dispersion loss:

The dispersion in the light is obvious because of the refractive indices of the

different wavelengths being different in the transmission. This arises out of the

characteristic of the material. Because of this the angle of incidence for different

wavelengths will be different when a ray tries to travel towards the clad resulting

in to uneven total internal reflection and some times transmission of light from

core to clad. This type of attenuation thus arises because of the wavelength itself.

The different attenuation discussed above is because of the some important

parameters of the material of fiber and its construction; however the attenuation

also happens because of group delay pulse broadening etc which are of not much

importance for ordinary applications. However, they play a very important role in

case of signal processing. A major loss of signal occurs because of the coupling

between two optical fibers, although it cannot be considered as an attenuation

problem for an individual fiber but it becomes a major problem of attenuation

when whole of the system is considered.

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Application of optical fibers for communication

The optical fibers are widely used in various fields like communications, Medical,

Domestic etc. One of its applications in point to point communication is discussed in

detail here (Figure 6).

In optical fiber communication the signal can be transmitted through the fiber only in the

form of optical signal. For which first the audio signal is converted into electrical analog

signals with the help of transmitter in the telephone and these analog signal are fed in to a

coder where they get converted in to binary form. This binary data is made to enter in to a

optical transmitter where these get converted in to optical signals by modulating the light

signals. These optical signals are fed in to the optical fiber with in the angle of

acceptance. At the out put end the optical signals are fed in to a photo detector which

converts them in to Binary form and then in to a decoder where the signal converted in to

analog form and finally gets converted in to voice at the receiver end. As the signal

propagates through the optical fiber there may be loss of signal due to various factors like

as discussed in attenuation and also the signal may be distorted due to the spreading of

pulses with time which is mainly because of the variation in the velocity of the different

spectral components through the optical fiber. With distance the loss of the signal

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becomes more. Hence before the entire gets lost it is necessary to have repeaters which

amplifies the signal but there is no such device which can directly amplify the optical

signal and therefore at each repeater the optical signal needs to be converted in to

electrical signal then amplify and again convert it back in to optical signal and then fed in

to the fiber. This process has to be followed at every repeater, which restricts the speed of

the signal transmission.

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3. Dielectrics

IntroductionOne way of studying a material is to characterize it by study of the influence of the

applied electrical field. One of the way in which, the materials are classified is by looking

into their electrical conductivity under the influence of external electrical field. The

classification has lead into naming of materials on this effect, namely as conductors, semi

conductor and insulators.

Insulators are those materials, which do not conduct electricity. With this statement one

may conclude that these are the materials which do not get influenced by electric field,

which is however not true. The insulators, which are also known as Dielectrics have been

found to have an influence of external electrical field. Faraday first observed this, through

the working of capacitors. An insulator is defined as that material which does not allow

the conduction of electrons when an electrical field is applied due to the absence of free

electrons. The dielectric being a non-conductor has the ability to hold the electrostatic

field and hence can store the electrical energy. This is possible due to the phenomenon of

polarization of what are known as electric dipoles (permanent and induced).

3.1 Dielectrics & Dielectric constant.

The electrical flux density D in a material is related to electrical field strength E in any

point in space, is related to D = o r E, as per the Gauss theorem, with o =8.854 x 10-

12 farad /m represents the permittivity of vacuum & r the relative permittivity of the

material. Further, = o r which is known as absolute permittivity of the material.

r which is a constant & dimensionless is known as dielectric constant of the material as

this is the measure of influenced electrical field within a dielectric material.

The other way of looking dielectric & dielectric constant through the experimental

observations in a parallel plate capacitor experiment when a voltage V is applied to

parallel plate capacitor, a change (Q) which is proportional to applied voltage (V)

accumulates on the two plates, with the relation Q=CV, Here, C is called capacitance of

parallel plate capacitor. The capacitance of the capacitor has been found to be directly

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proportional to area of the plates & inversely proportional to the distance between the

plates. It is found that the insertion of a dielectric material between the plates increases

the capacitance of the capacitor. Further the value of C has been found to be proportional

to the thickness of the dielectrics. This increase in capacitance can be only explained by

considering that when a dielectric material is inserted between the plates should decrease

the electrical field inside the dielectrics. The capacitance of a capacitor with a dielectric

and without a dielectric i.e. vacuum are compared, it will give dielectric constant i.e.

relative permittivity of the dielectric material as explained above.

, the permittivity of vacuum has no much of physical meaning other than that it gives

the fundamental conversion factor. On the other hand the dielectric constant , gives the

insight into atomic structure of the material as is equal to 1 for vacuum & for all other

materials it is greater than 1.

3.2 Dipole moments, polarization, dielectric constant and electric

susceptibility.

A dipole consists of two equal & opposite charges, which are separated by a distance. If

Q is the charges separated by distance‘d’ then, the dipole – moment P of the dipole is

given by d. The moment acts towards the positive charge.

In a system, the centers of negative charges & positive charges act at the same point and

then dipole moment does not exist. However, under the influence of an external field, if

the centers are displaced, then a dipole will come into existence and the dipole is said to

be an induced dipole. This is the case as shown in the figure (1). The dipole moment of

such a system is said to be an induced dipole –moment and will exist as long as external

field exists.

In certain materials like ionic crystals the charges are displaced and hence have

permanent dipoles and are said to have permanent dipole moment. Under the influence of

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external electric field, these will have additional effects along with the permanent effect

i.e. both induced dipole moment and permanent dipole moments.

In case of molecules depending upon, whether they are polar or non-polar in nature, their

exist a permanent dipole moment or not respectively. However, under influence of

electric field the non-polar molecules show induced dipole moments because of change in

electrical field distribution within a molecule. This also depends upon symmetry of the

molecules. The examples of polar dielectric materials are CH3Cl, H2O while that of non-

polar are H2, CO2, CH4, etc.

When no electrical field is applied, for these systems where permanent dipole moments

exist, will have their resultant equal to zero. They arrange themselves in such a way that

the net effects because of individual dipoles are nullified as shown in the figure (2).

When an external electrical field is applied, these dipoles get partially aligned along

applied electric field. The extent to which they are polarized depends upon the applied

electric field & the permittivity of the material. This process of re-orientation is known as

polarization.

In case of system where no permanent dipole moments exist, under the influence of

external electric field, the dipoles are induced & these dipole moments so induced will

align themselves in a direction opposite to the direction of the applied field in accordance

with Lenz’s law. This process in which a net induced dipole moment is results is also a

process of polarization.

The amount of polarization depends upon the resultant dipole –moment. The resultant

dipole moment creates an electrical field inside the dielectrics, so as to reduce the effect

of external electrical field. This explains why the capacitance of a capacitor increases

with the increases with the introduction of dielectric material inside a capacitor. The

amount of polarization is measured by a quantity known as polarization density P. The

polarization density P is defined as resultant dipole moment induced per unit volume of a

material. The polarization is thus naturally is a measure of change in capacitance with

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vacuum and a dielectric material within the parallel plates of a capacitor. It is further, can

be shown that E = - P(in SI units) where Ep is the average macroscopic electric field

within a dielectric material due to induced polarization. At atomic level or molecular

level, the dipole moment induced is also proportional to applied electric field. If p is the

dipole moment induced per atom, p E i.e. p = E, where is called polarizability

(defined as electric dipole moment induced in the atom by an electric field of unit

strength). i.e. P = Np= N E. Thus under polarization state D is given by D = E + P (in SI

units), D is also known as displacement field vector. But D = εE i.e. D= o r E = E + P.

As, the polarization density is a measure of the change in the capacitance of the capacitor,

we have

P = Cm - Co

Where, Co and Cm are the capacitances of the capacitor with vacuum and dielectric

material as medium respectively.

.: P= [Evac- Em] = [(Evac/ Em) - 1] Em = [ε-1] Em

P/E = ( – 1)

From the above equation we get

=1 +P/E

Here, P/E=ψ is called as electrical susceptibility

=1+ ψ

3.3 Different Methods involved in polarizations:

The polarization is basically brought about because of three aspects, which contribute

towards polarization density. They are

i) The Electronic polarization

ii) The anomic polarization

iii) Orientational polarization

iv) Space Charge polarization

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1) The electronic polarization:

In some systems of dielectrics where the permanent dipole moment of an atom/molecule

involved is zero, i.e. in which the centers of negative and positive charges meet in such a

way to give net dipole moment zero, an induced dipole-moment comes into

effect under the influence of external electrical field due to the separation of charge

centers. The polarization brought about by such cases is known as electronic polarization

(figure 1) & the polarizability is known as electronic polarizability and denoted by αe.

If the polarization is considered as displacement of two spheres of positive and negative

of radius R & If the displacement is r then

e E = (Ze/4Пε0). (1/r2).(Zer3/R3)

This shows that r = .E

But dipole moment defined as pe = Ze.r = (Ze. 4Пε0 R3E)/Ze

= 4Пε0 R3E

and hence, the pliability

In the above equation Z is the atomic number.

The electric polarizability of He is 1.78 x 10-42 f m2 Ne 3.49 x 10-42 f m2 and Xe is 35.33 x

10-42 showing that polarizability increases with increase in atomic number of an atom.

2) Ionic Polarization:

Ionic crystalline materials like KCl, NaCl, etc. show ionic polarization. The external

electric field shifts the positive ions relative to negative ones (figure3a &b). This shift

produces polarization. The electronic polarizability measures the shift in the centre of

electric clouds while the ionic polarizability measures the shift of the ions relative to each

other. The ionic polarization is demanded by .

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3) Orientation polarization :

When an external field is applied to systems of molecules like water, phenol etc carrying

permanent dipole moment Po, they tend to align themselves along the field i.e. the

orientation of these dipoles are changed leading to polarization effect (figure 4). This

polarization is known as orientation polarization and the corresponding polarizability as

induced polarizability. The polarizability of this type is found to be proportional to square

of the dipole moment ‘p’ and inversely proportional to T (temperature). This orientation

polarizability is the only polarizability, which depends upon temperature.

4) Space charge polarization:

In the process moment of charges from one place to another place within a dielectric, the

charges are trapped because of different materials within the system. This creates a space

charge leading to polarization known as Space charge polarization & the corresponding

polarization is known as space–charge polarizability. This effect is very negligible and

most of the cases it is neglected.

Thus if N is number of dipoles created per unit volume, then polarization density is given

by

P=Np = N ( )

3.4 Internal fields in liquids & solids (One –dimensional approach)

Let us consider a dielectric system in case of liquids and solids in particular in this

section. In earlier discussions the effect of external field was discussed on the system.

The effect produced by this external field at atomic level depending upon the nature and

the surrounding is going to be different. One of the major effect is the formation of

induced dipoles at each atomic centers of the system. These induced dipoles create their

own electric field around that and creates local fields much different from the external

electrical fields. This effect can be neglected in case of gaseous dielectrics in view of

87

large distance between molecules however, which is not the case in solids and liquids. In

view of this the local field in liquids and solids differ from the microscopic field as

considered in earlier discussion.

Thus the study of effect at a particular location where atom is situated has to be studied

because

a) Influence of external electric field.

b) Field arising from the manual interactions of other, atoms, known as induced

field.

The equation for induced dipole moment is

Pi =

Where Ei is total induced field called local field. Thus the potential at any point in

space, at a distance r and angle θ from the mid-point of a dipole is given by

V (r, θ) =

Thus, the field around the dipole will have two component with respect r &

.: Er =

=

and E θ = -(1/r).[dv(r, θ)]/d θ = (1/4Пε0).(pisin θ/r3)

If one restricts to a one dimensional array as has been purposed by Epstein, E is equal to

zero and only that component Er exists with =0. Thus, for a one dimensional array the

problem reduces to Er = here cos θ = 1 & sin θ =0

The model with dimensional array is schematically is pictured as below. If Ao is a

molecule under study,

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The effect due to the adjust atom A+1, A+2,------------------, and A-1, A-2, have to be taken

into effect. Assuming systematic arrangement of atoms, at a distance ‘a’ from each other,

the effect of A+1& A-1, on Ao are respectively, (1/4Пε0).(2pi/a3) because of A+1, it is

(1/4Пε0).[2(-pi)/(-a)3] because of A-1. The total effect because of both is then (1/Пε0).

( pi/a3) similarly, because of atoms A+2 and A-2 it will be (1/Пε0).[pi/(2a)3] and so on.

Because of nth atom on both sides it will be (1/Пε0).[pi/(na)3]. Thus, the electrical field,

due to the other atoms in general will be,

(1/Пε0).[pi/(a)3] + (1/Пε0).[pi/(2a)3] + (1/Пε0).[pi/(3a)3] + ……+ (1/Пε0).[pi/(na)3]

= n=0Σα (1/Пε0).[pi/(na)3] = (1/Пε0).[pi/(a)3]n=0Σα 1/n3

for infinite array of atoms

Therefore, the local electrical field at an atom is given by

Ei=E + pi/ Пε0a3

Where E is the external field,

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.: Ei=E + 1.2 pi/ Пε0a3 since =1+

3.5 Static Dielectric constant – determination & temperature

dependency.

The dielectric constant is already discussed in the earlier section 3.1. This dielectric

constant under the influence of a constant electrical field i.e. in static electric field is

called static dielectric constant. (The effect of variable electric field will be discussed

later).

It can be shown from the equation mentioned in section (3.3) of this chapter for

polarization during “P’ for poly atomic system

ε 0[εr-1] = a +b/T With a = N (

This equation is a straight line. If one plots the graph of ( ) against , the

interception Y axis will give N ( while the slope . Thus, holds good only in

gaseous state. From the above it is clear that, the effect of temperature on those materials,

which do not have permanent dipole moment is equal to zero. i.e. only those gaseous

systems whose molecules contain permanent dipole moment show a temperature effect.

In consistent with the above observations the groups of CH4, CCl3, CH3Cl2 and CH3Cl the

value of slope goes on increasing in order of their dipole moment values.

A further, general equation has been given, Clausius – Mosotti, which is

Where, as explained in section (3.3.)

If MA is the molecular weight of the molecule involved then , then the above

equation can be written as

90

Therefore

The plot of against with gives straight by with slope as . These

results also go along with discussion as done in case of gases system above. The above

equation is known as Debye’s equation, and it helps in the determination of permanent

dipole moment of a molecule. To measure the dipole moment; the dielectric constants at

different temperatures will have to be measured using different methods.

One of the methods of measuring dielectric constant is to determine the capacitance of

the capacitor with a vacuum or air and with the dielectric medium and to compare them.

Take a parallel plate capacitor. The capacitance of the capacitor is measured using air as

the medium. If C1 is the capacitance of the capacitor in air is equal to where A and

d are the area of the plates and d is the distance between them. The capacitance, is

determined with a material whose dielectric measurement is to be carried out. If Cm is the

capacitance of the capacitor with dielectric medium, with dielectric constant then

Cm =

.:

The variation of dielectric constant depends upon the temperature in general on the state

of the material and nature of molecules like its shape, size and mass etc. In general, in

liquid state the dipoles are comparatively free to rotate in an external electrical field. At

freezing point these dipoles also freeze & at this stage only contributions from ionic &

91

electronic part come into effect. Thus, the variation of with temperature changes

depending upon which type of polarizability is more predominant.

3.6 The complex dielectric constant & Dielectric Loss.

If a dielectric is subjected to an alternating external field, say, a sinosoidal electrical field,

E=Eocosωt, then the polarizability will be forced to change. However, it cannot be

expected that this polarizability to fall in line with the applied electric field in view of the

constraints. Thus, this will be a phase lag between polarizability and hence with

displacement vector D in comparison with the applied electric field. If ‘δ’ is the phase lag

at any time between D & E then,

D=Do cos(ωt- δ)

=Do cosωt cosδ + Do sinωtsinδ.

The phase lag being constant for a given system, as can be fairly assumed that D is

proportional to E, it can be written as

D1 = Do cosδ & D2= cosδ

Therefore, D under the influence of applied electrical field will have two components,

one being cosine & other being sine i.e. D=D1 Cosωt +D2 sin ωt

Since D is having the components dielectric will also have two components one

corresponding to cosine component namely cosδ and sine component as,

sinδ . These values of dielectric

constants have been found to be frequency dependent. The two dielectric constants in a

complex notation can be, thus written as ε/(ω) – i ε//(ω) and is usually denoted by ε*(ω)

i.e. ε*(ω) = ε/(ω) – i ε//(ω). ε(ω ) being a cosine component has been taken as real part and

ε//(ω) being a sine component as imaginary part. Then, the displacement vector can be

written in its complex notation as

D =

From above, it is clear that

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tan is known as loss tangent or dielectric loss. In fact is a measure of energy loss

during the application of the electrical field, which is used up in the form of the work

carried out in changing polarization & hence appearing as Joule heating. This heating has

to be thus appeared as loss and hence appears as an imaginary part of the dielectric

constant.

The energy loss i.e. dissipation of energy per unit time is given by

W=

Where J is the current density given by J = dE/dt.

Thus J = ω(D1sin ωt + D2cos ωt).

Substituting the value for J and E, the expression for W will be

W=

=

=

since

ω = =

.: ω = and also

ω =

93

The equation for ‘ω’ thus, indicates that the dielectric loss is directly proportional to sinδ.

At low values sin ~ tanδ is known as loss tangent. is known as loss angle.

3.7 Frequency dependence of Polarizability and hence Dielectric

Constant

The dielectric constant has two parts and so also the polarization. As noted in earlier

sections the polarizability is of three types, namely, electronic, ionic & dipolar i,e,

orientation. Depending upon the intensity of vibration, involved under the influence of

varying electrical field the absorption of electromagnetic radiation takes place and hence

depending upon the nature of polarization the contribution towards dielectric varies. The

electrical polarizability is responsible for dielectric constant at optical frequencies. The

dielectric associated with ionic polarizations fall in the region of infrared radiation. For

orientation polarization or permanent dipole moments, the range of frequency under

study will be radio frequency and microwave ranges. The figure 5 indicates a schematic

diagram of variation of with frequency. The variation in the dielectric constant will be

of the same type except for the numerical value along Y-axis as the variation in dielectric

constant is directly proportional to polarizability.

Figure 5

Dipoles tend to align itself along the direction of the applied electrical field and in

varying field it tries to follow the field. In the process the dipoles try to interact with each

94

other and this results into dielectric loss and appears as heat. This energy loss is asociated

with imaginary dielectric constant. This is also evident in case of an LCR circuit, with

power is being given by

Power = VR IR Cos

With tan = . Thus, it may be concluded that the dielectric losses are because of

ionization, leakage current, polarization and structural inhomogeneity.

3.7 Ferro Electrics:

These are the materials which are similar to ferromagnetic materials under the influence

of electric field i.e. the materials in which the polarization is not linear with applied

electrical fields. Naturally, such materials exhibit hysterisis(Figure 6). After applying the

electrical fields, the dielectrics get polarized and even after the removal of the electrical

field they process the spontaneous polarization Ps. The value of Ps is found to be the

function of temperature. Every ferroelectric has a critical temperature above which they

behave as Para-electric as per the Curie –Weiss-law, namely is in case

ferromagnetic material ψ is the electrical susceptibility C Tc are constants. Tc is known

as Curie temperature. It is found that, while going from ferroelectric to Para –electrics the

dielectrics involved go into transitions, transition being occurring from order to disorder

in crystal structure or displacement of ions. Some of the examples of ferroelectric are

BaTio3 (curie temp2-3800k, Ps 0.26 C/m2), Rochelle salt KNbO3(Curie temp2-7080k; Ps

0.026667 C/m2), etc. Another important feature of ferroelectric crystals is they do not

have centre of symmetry.

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Figure 6

3.8 Piezoelectricity:

Polarization can also be changed in dielectrics by application of stress or temperature. If

polarization changes because of the application of stress such materials are as known as

Piezoelectric. This type of change in polarization is seen in all the crystals where no

centre of symmetry exists. Nearly twenty classes of crystals have been found to exhibit

the change in polarization under stress. BaTiO3 has been found to be piezoelectric. The

change in symmetry an account of change in temper is known piezoelectricity BaTiO3

has found to be pyroelectric also(Tourmaline is an example of pyroelectric). However,

every material, which is piezoelectric, is not essentially a pyroelectric. The piezoelectric

materials are characterized by piezoelectric strain co-efficient and electromechanical

coupling K. Quartz is most widely used piezoelectric.

Variation of polarized in all these materials have produced large application as

transducer.

96

Figure 1

Figure 2

+

-+

-

-

Electric Field Electric Field

97

Figure3a E=0

98

Electric field

Figure3b

99

Figure 4

-e

+e

Electric field

-e

+e

Electric field

100

Eng Physics Unit VIICRYSTAL STRUCTURES

Professor B V Sadashivmurty, Sri Jayachamarajendra College of Engineering, Mysore

Narasimha Ayachit and G Neeraja Rani, SDM College of Engineering and Technology, Dharwad

IntroductionOne has to understand the crystal structure (arrangement of atoms) thoroughly because solids have specific properties which depend on the structure. The materials are generally classified into (i) solids, (ii) liquids and (iii) gases. Solids are further classified into Crystalline and Non-Crystalline (Amorphous) solids depending on the arrangement of atoms. If the atoms are arranged periodically through out the solid then it is said to be crystalline other wise it is said to be non crystalline solid.

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)

There appears a 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

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r0 = the equilibrium distance between atoms or moleculesFigure 1

Att

ract

ive

forc

eR

epul

sive

for

ce

r0

Inter atomic/inter molecular distance

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

m

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+ =

Lattice + Basis = Crystal StructureFigure 3

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 4(a) to 4(e) representing both Bravais lattice and non- Bravais lattices.

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Fig.4(c)Examples of Bravais Lattice & Non-Bravais Structure (Two Dimensional)Fig4(d)

+ =

Fig 4(a) Bravais Lattice

/ + =

Fig. 4(b)Non-Bravais Lattice

Fig. 4(e)

Bravais Lattices and 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 System No. Unit Cell Coordinate Description

1 Triclinic 1 Primitivea ≠ b ≠ cα ≠ β ≠ γ

2 Monoclinic2 Primitive a ≠ b ≠ c

α = β = 90° ≠ γ 3 Body   Centered

3 Orthorhombic

4 Primitive

a ≠ b ≠ cα = β = γ = 90°

5 Base   Centered

6 Body   Centered

7 Face   Centered

4 Tetragonal8 Primitive a = b ≠ c

α = β = γ = 90° 9 Body   Centered

5 Trigonal 10 Primitivea = b = c

α = β = γ < 120°, ≠ 90°

6 Hexagonal 11 Primitivea = b ≠ c

α = β = 90°, γ = 120°

7 Cubic 12 Primitive a = b ≠ cα = β = γ = 90°

13 Body   Centered

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14 Face   Centered

Crystal Systems

Symbols used P - Primitive - simple unit cell F - Face-centered - additional point in the center of each face I - Body-centered - additional point in the center of the cell C - Centered - additional point in the center of each end R - Rhombohedral - Hexagonal class only

Table2

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 vector .

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Figure 5Let and be two vectors having equal magnitudes and oriented along AB and AC respectively as shown in the figure 5. With and as coordinate vectors, the position vector of any lattice point can be written as,

= n1 + n2

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

For a three dimensional representation of position vector is written as:=n1 + n2 +n3

Where, the terms have their usual meanings.

CRYSTAL LATTICE AND UNIT CELL:

A crystal lattice is a space lattice in which the lattice sites are occupied by atoms or clusters of atoms. Each lattice point is associated with atom or group atoms, are called the basis.

Figure 6

The Basis must be identified in composition, arrangement and orientation such that the crystal appears exactly the same at one point as it does at other equivalent points. The figure 6 shows the basis consisting of a group of two atoms. When the basis is repeated with correct periodicity in all direction the crystal structure is obtained. Thus,

Space lattice + Basis ------------ Crystal Structure

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UNIT CELL

The entire lattice structure of a crystal can be generated by identical blocks known as unit cell. The unit cell may be a group of ions, atoms or molecules. The unit cell is the smallest building block or geometric figure from which the crystal is built up by repeating it in three dimensions figure 7.

Figure 7

PRIMITIVE AND NON PRIMITIVECELLS:

A unit cell can be chosen in a number of ways. The figure8 shows two ways of choosing a unit cell in two dimensions. In the first way the unit cell contains lattice points only at the corners. This type of unit cells is called as Primitive Cells. In the second way the unit cell contains lattice points in addition to the corners. These are known as Non Primitive Cells.

Figure8Note: The unit cells differ from the primitive cell in that it is not restricted to being the equivalent of one lattice point. Thus, unit cells may also be primitive cells, but all the primitive cells need not be Unit Cells.

LATTICE PARAMETERS

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

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

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(2) The angles ( , β 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 β, 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.

Directions and Planes in a Crystal Lattice

While dealing with a 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.

Figure9A resultant vector which joins Lattice points A and B (figure9) can be represented by an Equation.

= n1 +n2 +n3 , If n1= n2 = n3 =1, then = + +

CRYSTAL PLANES AND MILLER INDICES:

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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-m y- and z co-ordinates of the lattice points of the plane lying on the x-, y- and the z- directions of a reference frame

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

(3) Convert these fractions into whole numbers by multiplying all the numbers by a common multiplier.

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

These simplified numbers, derived from the x-, y- and 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 index 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 index a set of parallel planes is written as <hkl).Example: Given that,

X= Y= Z=

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

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Taking reciprocals of the three fractions

Multiplying throughout by least common multiple ‘4’ 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 crystallographic point of view. In such cases, Miller indices are enclosed in braces {} instead of parenthesis or brackets, which represents all the equivalent planes. For example in the case of cubic lattice, the 6 planes referring to 6 faces of a unit cell are represented by the Miller indices as (1 0 0) (0 0 1) (010) (0 0) (0 0 1) collectively designated as {1 0 0 }

EXPRESSION FOR INTERPLANAR SPACING IN TERMS OF MILLER INDICES:

Figure11

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To get the interplanar distance, consider a plane ABC with Miller indices (hkl). In the reference frame, draw a normal to the plane from the origin. Let OP is the normal to the plane ABC. Let angle POA = ’, and POB = β’ and POC =’ be the angles made by the normal to the plane with the x, y and the z directions. OP = d is the interplanar distance

Then, from the figure, Cos = ON/OA = d/x

Cos β = ON/OB = d/y

Cos = ON/OC = d/zBut from the definition of Millers incise derived in the earlier section,

h = a/x therefore, x = a/h

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,

Cos

Cos

Cos

From solid geometry, Cos2 + cos2 β + Cos2 =1

Substituting the values of the trigonometric relations we get

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d2hkl =

Then, the interplanar distance‘d’ is given by

dhkl =

For cubic lattice a=b=c then,

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 =

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

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

From equations ( ii ) and ( iii ),

From this equation, the density of the crystal is

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

Therefore

The lattice constant ‘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 can be taken as the first nearest neighbors of an atom in the structure.

Atomic packing factor is the ratio of the volume of the unit cell occupied by atoms to the net volume of the unit cell

The following are the possible structures found in a cubic system of crystals, resulting due to the kinds of packing of atoms,

(a) Simple Cubic (SC),(b) Body Centered Cubic ( BCC ) and(c) Face Centered Cubic (FCC)

(a) Simple Cubic Structure (SC )

In a simple cubic structure, the space lattice is cubic. Atoms are placed at all lattice locations. Therefore, an atom at a lattice location will be in contact with all its six nearest neighbors (figure12). Thus the co-ordination number for a simple cubic structure is N=6. In a simple cubic structure, the lattice constant is the cube edge of the conventional unit cell. Thus the lattice constant is the distance between the centers of two neighboring atoms. That is the lattice constant a = 2r, where r is the atomic radius. The atoms at the lattice locations are shared by eight unit cells. Each atom at the lattice location of the unit cell contributes (1/8) to the unit cell.

The number of atoms contained in a unit cell in the structure is given by the number of lattice locations in the unit cell multiplied by the contribution of the atom at each location. That is the number of atoms in a unit cell of the simple cubic structure is given by

n = (no. lattice locations X contribution of atoms at these location)n = [8 X (1 /8) ] =1

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Figure12

The parameters describing the structure of a simple cubic structure are:

1 Co-ordination Number N = 62 Number of atoms in a unit

celln = 1

3 Atomic radius R4 Lattice constant r = 2r5 Volume of a unit cell V = a3

6 Volume of an atom (4/3) r3

Volume of the unit cell occupied by the atom is given by

v = (number of atoms in a unit cell) X [volume of an atom]v = (n) X [4/3 r3]

Atomic packing fraction in a simple cubic crystal structure is

APF =

(b) Body Centered Cubic Structure (BCC )

In body centered cubic structure, the space lattice is cubic. Atoms are placed at all lattice locations and there will be an additional atom at the body center of the cubic unit cell. In this structure, the atoms stacked along the body diagonal of the cubic unit cell are in contact. That is, the atom at the body center of the unit cell will be in contact with all other atoms at the cubic lattice locations of the unit cell. Each atom in the structure will have eight nearest neighbors (figure 13). Thus, the co-ordination

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number for a body centered cubic structure is N = 8. In a body centered cubic structure, the lattice constant (a), is the cube edge of the conventional unit cell.

Figure13Calculation of lattice constant in terms of atomic radius

Let d be diagonal distance between the atoms in a plane of the unit cell. Then,

If D is the diagonal of the cubic unit cell, then

But, the atoms along the body diagonal, are in contact, Therefore D = 4rAnd D2 = (4r)2 - - - - - - - - (iii)

From equations (i) and ( iii) , D2 = (4r)2 = 3a2,

Thus, the relation between the lattice constant (a ) and atomic radius is,

The atoms at the lattice locations are shared by eight unit cell. Therefore, each atom at the lattice location of the unit cell contributes ( 1/8 ) to the unit cell. The number of atoms contained in the unit cell of the structure is given by the number of lattice locations in the unit cell multiplied by the contribution of the atom at each location plus the atom at the body center of the unit cell. That is the number of atoms in a unit cell the simple cubic structure is given by

n = (no. of lattice locations X contribution of atoms at these location ) + ( atom at the body center)

n = [ 8 X ( 1 X 8 ) ] + 1 =2

The parameters describing the structure of a body centered cubic structure are :

1 Co-ordination Number N = 82 Number of atoms in a unit

celln = 2

3 Atomic radius r4 Lattice constant a =(4 / ) r5 Volume of a unit cell V = a3

6 Volume of an atom a = (4/3)

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r3

Volume of the unit cell occupied by the atom is given byv = (number of atoms in a unit cell ) X [ volume of an atom ]v = (n ) X [ 4/3 r3]

Atomic packing fraction in a simple cubic crystal structure is

APF =

(C) Face Centered Cubic Structure (FCC)

In a face centered cubic structure, the space lattice is cubic. The conventional unit cell consists of eight small cubelets. Atoms are placed at alternate lattice locations. The atoms stacked along the face diagonal of the cubic unit cell are in contact. The atom at any lattice location of the unit cell will have twelve nearest neighbors (figure14)

Figure14

Calculation of lattice constant in terms of atomic radius

Let d be the diagonal distance between the atoms in a plane of the conventional unit cell. Then,

d2= (a2 + a2 ) = 2a2 - - - - - - - ( i )But, the atoms along the face diagonal of the conventional unit cell are in contact.

Therefore, d = 4r - - - - - - - ( ii )

And d2 = ( a2 + a2 ) = 2a2 - - - - ( iii )From equations ( i ) and ( iii ) ,

d2 = ( 4r)2 = 2a2 ,

Thus, the relation between the lattice constant ( a ) and atomic radius is,

A =

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The atoms at the lattice locations of the conventional unit cell are shared by eight such cells. Therefore each atom at the lattice location of the unit cell contributes ( 1 /8 ) to the unit cell. The number of atoms belonging to the unit cell in the structure is given by the number of lattice locations in the unit cell multiplied by the contribution of the atom at each location plus the contributions of the atoms at the face centered locations. The number of atoms in a unit cell of the face centered cubic structure is given by

n = ( No. of lattice locations X contribution of atoms at these location ) ( No. of atoms at the face centers X their contribution to the unit cell )

n = [ 8 X (1 /8 )] + [ 6 X ( 1 / 2)] = ( 1 +3 ) = 4

The parameters describing the structure of a face centered cubic structure are :

1 Co-ordination Number N =12 2 Number of atoms in a unit

celln = 4

3 Atomic radius r4 Lattice constant a =(4 / ) r5 Volume of a unit cell V = a3

6 Volume of an atom a = (4/3) r3

Atomic packing factor is the ratio of the volume of the unit cell occupied by atoms to the net volume of the unit cell

Volume of the unit cell occupied by the atom is given byv = (number of atoms in a unit cell ) X [volume of an atom]v = (n ) X [ 4 / 3 ]

Atomic packing fraction in a face centered cubic crystal structure is

APF =

CRYSTAL STRUCTURE OF SODIUM CHLORIDE:

Sodium chloride is an ionic crystal. It is a compound of the alkali halide family. Due to the proximity of the sodium atom with a chlorine atom, the valence electron from

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the sodium atom is transferred to the chlorine atom. With this transfer of an electron, the sodium atom becomes a cation and the chlorine is converted to an anion. Then, these ions are held by the ionic force and the bond that holds the atoms together is ionic.

A conventional unit cell of the structure consists of eight cubic primitive cells (figure 18). In the conventional unit cell of the crystal, sodium atoms occupy the FCC positions of a cubic structure. Chlorine atoms are placed at the intermediate positions between the sodium atoms. Chlorine atoms independently from an FCC structure similarly sodium atoms also from an independent FCC structure. The lattice constant of both the FCC structures is a = 5.63Au. A unit cell of a sodium chloride crystal can be regarded as the structure formed by the inter-penetration of sodium FCC lattice with the Chlorine FCC lattice through one half of the lattice constant. This can be treated as the graphical representation of the crystal structure.

The crystal structure can also be illustrated in terms of the co-ordinates of the atoms in the structure. For this, an orthogonal co-ordinate system is considered. The origins of the co-ordinate system can either be at the location of a chlorine atom or at the location of a sodium atom. With the origin at the location of the sodium atom, the co-ordinates of the sodium and chlorine atoms in the crystal structure are shown below. Sodium chloride is an ionic crystal formed by the transfer of the valence electron from sodium atom to the chlorine atom. Due to this transfer the sodium atom is converted to a cation and the chlorine is converted to an anion. These ions are held by the ionic force and the bond that holds the atoms together is ionic bond. In the unit cell of the compound, chlorine atoms occupy the FCC positions of a cubic structure. Sodium atoms occupy edge center positions.

Figure18

Atoms Co-ordinates

Sodium

000. 100, 101 , 001 , ½ 0 ½

½ ½ 0 , 1 ½ ½ , ½ ½ 1 , 0 ½ ½ ,

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110 , 111 , 011 , 010 , ½ 1 ½

Chlorine

½ 0, 10 ½, ½ 01 , 00 ½ ,

1 ½ 0, 1 ½ 1, 0 ½ 1, 0 ½ 0, ½ ½ ½

½ 1 0, 1 ½ ½ , ½ 1 1, 0 1 ½

Density of sodium Chloride

Density of a crystal in terms of its lattice parameter is given by

Here n is the number of sodium chloride molecules in a conventional unit cell. In place of A the molecular weight of sodium chloride molecule is to be considered. V is the volume of the conventional unit cell with a lattice constant a = 5.63 Au. This is the distance between any two neighboring sodium atoms or the neighboring chlorine atoms. The number of molecules in a unit cell is equal to the sum of sodium atoms [n (Na)] contained in the cell and the chlorine atoms [n(CI)]. The number of sodium chloride molecules in a unit cell are :

N = [ n (Na) ] + [ n ( CI ) ]

The number of sodium atoms in the cell are :

[ n (Na)] = 8 ( 1/8 ) + 6 (1/2) = 1 + 3 = 4

The number of chlorine atoms in the cell are :

[ n ( CI ) ] = 4 ( 1/4 )+ 4 ( 1/4 )+ 1 = 4

Therefore, there will be four sodium chloride molecules in a conventional cell

Atomic weight of sodium is A (Na) = 22.98 X 10-3 Kg

And that of Chlorine is A ( CI ) = 35.45 X 10-3 Kg

Therefore, the molecular weight of sodium chloride molecule is

A = A (Na ) + A ( CI ) = ( 23 X 10-3 ) + ( 33.45 X 10-3 ) kgA = ( 58.43 X 10-3 ) Kg

Volume of the unit cell is V = a3 = ( 5.63 X 10-10 ) 3 m

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And the Avogadro number is NA = ( 6.023 X 1023 )

The density of sodium chloride crystal is

CRYSTAL STRUCTURE DIAMOND

Diamond is the crystalline form of carbon. The structure consists of the periodic stacking of carbon atoms in its crystalline state.

A conventional unit cell of structure consists of eight cubic primitive cells ( figure 16 ). On the basis of the positions of the carbon atoms in the unit cell they are classified as first kind and second kind only for the sake of convenience. Otherwise they are the same carbon atoms. The carbon atoms placed at the FCC positions of the conventional unit cell are the first kind of carbon atoms. Another set of carbon atoms occupy the body centered positions in the primitive cubic cells in the structure. The second type of carbon atoms are placed at the body centers of the alternate cubic primitive cells. These carbon atoms independently form another FCC structure. Therefore, diamond crystal can be regarded as a structure produced by the penetration of one type of carbon atoms forming FCC structure in to the other FCC structure, formed by the second kind of carbon atoms, through a distance equal to three fourth the body diagonal (figure 19).

If the locations of the atoms in a diamond structure is seen in the primitive part of the unit cell (one of the eight cubelets), they look like a tetrahedra. In these tetrahedra, one atom at the center is bonded to three others in the primitive cell. Periodic stacking of the tetrahedral structure also produces the diamond crystal.

The distance between the two carbon atoms place on the cube edge of the conventional unit cell is the lattice constant and it is, a = 3.567 Au for diamond crystal. This can be treated as the graphical representation of the crystal structure.

The crystal structure can also be illustrated in terms of the co-ordinates of the atoms in the structure. To write the co-ordinates of the carbon atoms in the structure, an orthogonal co-ordinate system is considered. The origin of the co-ordinate system is fixed at one of the carbon atoms. With this co-ordinate system the co-ordinates of the atoms in the structure are shown below.

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Figure19

Atoms Co-ordinates

Carbon atoms of first kind

000. 100, 101 , 001 , ½ 0 ½ ½ ½ 0 , 1 ½ ½ , ½ ½ 1 , 0 ½ ½ ,

110 , 111 , 011 , 010 , ½ 1 ½

Carbon atoms of second kind

¼ ¼ ¼, 3/4 3/4 1/4,

1/4 3/4, 3/4 ,3/4 1/4 3/4

Density of Diamond Density of a crystal in terms of its lattice parameter is given by

In diamond structure, the number of carbon atoms in the conventional unit cell is

Contributions from the atoms at the corners of the cube = 8 ( 1/8 ) = 1 +

Contributions from the atoms at the centers of the faces = 6 (1/2 ) = 3 +

Contributions from the atoms at the body centers = 4 ( 1 ) = 4

n = 8

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n =

Atomic weight of carbon is A = 12.01 X 10-3 kg.Lattice constant a = 3.567 AuVolume of the vunit cell V= a3 = ( 3.567 X 10-10)3

And the Avogadro number is NA = ( 6.023 X 1023)

The density of diamond is

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Notes by Prof. B.N.Meera, Bangalore University

Introduction: Classical Free electron theory of metals

Metals are excellent conductors of heat and electricity. First successful attempt to

explain the electrical behavior of metals was put forth by Drude in the year 1900. (Note 1). To

understand the mechanism of electrical conduction (in metals), one needs to look at the

process at the atomic level. In 1900, three years after Thompson's discovery of the electron,

Drude constructed his theory of electrical and heat conduction by applying the highly

successful kinetic theory of gases to a metal, considered as a gas of independent (non-

interacting or 'free') electrons (Note 2). According to this theory the atoms condensed into a

solid give up their loosely bound valence electrons, which are free to wander through the

metal and lead to conduction of charge and energy/heat. The simple and the elegant model of

this conduction process put forward by Drude are based on the following assumptions.

Basic assumptions of free electron theory

1. The valence electron of the atom is loosely bound to their respective atom. In metals they become the free electrons. These electrons are free to move throughout the metal and are hence termed as free electrons. Since electrical conductivity is due to these free electrons, these are also termed as conduction electrons.

2. These free electrons move in the metal according to Kinetic theory of gases.

3. Like the molecules of gas in a container, free electrons in these metals move randomly (in the absence of electric field) with a velocity, which is the average velocity or rms (root mean square) velocity. In the absence of externally applied potential difference there are on an average as many electrons wandering through a given cross section of the conductor in a given direction as there are in the opposite direction. Hence the net current us zero.

4. However on application of electrical field, the random motion gets slightly affected and the electrons experience a drift velocity in the direction of the applied field. This drift velocity is less than the rms velocity by several orders of magnitude.

These simplistic assumptions give a considerably accurate explanation of conduction mechanism.

Expression for electrical conductivity:

Let us now derive an expression for electrical conductivity of a metal according to the classical free electron theory.

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Consider a conductor of uniform cross section. If a voltage V is applied to a metal of resistance R, and I is the current in the conductor, we know by ohm’s law,

1)

The law can be equally well expressed in terms of current density J as

2)

where σ is the electrical conductivity and A is the cross sectional area of the conductor. (Note 3).

Let us find an expression for J. Let vd be the drift velocity of the electrons. Hence in one second the electrons travel a distance of vd. Let n be the number density of the electrons in the metal. Consider unit volume of the metal. This contains n free electrons, each of charge e. The total charge is en. The total charge crossing unit area per unit time is nevd. This is the definition of current density.

.

The applied voltage generates an electric field E. The force F acting on the electrons due to the field is -eE. It is important to note that free electrons obey Newton’s Laws. For example, electrons moving in the opposite direction to the field will be speeded up and those moving in the same direction as E will be slowed down. Those moving perpendicular to E will acquire a sideways component of velocity and so on. The equation of motion for an electron is

where m is the electron mass, a is its acceleration . This equation suggest that the electron is accelerated indefinitely and their velocity should grow continuously as a result of electric field. However, this is not correct. The electrons undergo collisions during their transit. Drude's original view was that collisions are due to electrons bumping into heavy ions. Hence according to Drude’s model, resistance to the motion of electrons comes because of collisions of electrons.

Integrating we get

where is the time for which the electron moves before undergoing collision. This average time between collisions is known as the scattering time or relaxation time

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Hence conductivity is given by

Note electrical conductivity depends on n – free electron number density. If the carrier density is however not known, then the conductive properties of the materials cannot be fully determined, for and only the product of the two quantities can be obtained.

Mean time between collisions and the mean free path

An important concept that is of importance in Drudes theory (which is a important concept in Kinetic theory of gases) is the concept of mean time between collision and mean free path. As discussed above, the electrons travels in the presence of electric field for a time (mean free time) before it undergoes a collision. The electron travels with the rms velocity in the absence of electric field. In the presence of applied electric field, the drift velocity imparted to the electron is very small compared to the rms velocity. For ex, in copper metal the rms velocity is equal to where as the drift velocity is . The time taken by the electrons in traversing during time is hence decided not by the drift velocity but by the much greater velocity – the rms velcity (average velocity).

where is the average distance traveled by the electron between collisions. Here is the average electron speed, which Drude estimated from the classical partition of energy:

i.e.

It is also important to note that drift velocity is a concept which is of relevance in the Drude free electron theory but not in Kinetic theory of gases (Note 4).

Note:

1. Drude put forth this model in the year 1900. J.J. Thomson discovered electron in the year 1897. However, a complete understanding of atom was established (in the year 1914) by Rutherford. He used the results of scattering of alpha particles by gold foil to arrive at the model. It is interesting to note that the details of the

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nature of the atom (as understood today) was not available to Drude at the time of formulating of his theory- a fine example of insight and intuition.

2. Certain processes can be understood and explained by experiences that are accessible to human sensory perceptions. For example, motion due to application of force can be “seen” (via any of our sensory perceptions). However atomic processes cannot be “seen”. For example, we cannot “see” atoms. Understanding of physical processes at the atomic level is possible only by modeling the atom. The model presupposes certain assumptions. The resulting effects are validated by comparison with experimental measurements. The model is said to be a true representation of the underlying process when there exists an agreement with the theory and the experiment. If not the model is changed iteratively so as to arrive at an agreement.

3. Ohms law expressed by involves readily measurable macroscopic physical quantities like current and voltage. Let us now look at the alternative representation of Ohm’s law, which involves parameters representing microscopic description of the system.

The current in conductor can be expressed in terms of current density as , where A is the cross sectional area of the conductor and J is the current per unit area of cross section. Similarly the voltage V can be expressed as where E is the electric field applied to the length of the conductor l. We can write above equation as

.

For a conductor of resistance R, the resistance is directly proportional to the length of the conductor, and is inversely proportional to the area of the conductor. Hence R is written as where ρ is the coefficient of resistance or the resistivity of the material. Hence above equation becomes . We can write . The equation can also be written as

where σ = 1/ρ is called the conductivity of the material.

6. It is very instructive to calculate the values of vd and for a typical case.

Substituting we get,

= 1.154 x 105 m/s

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Note that the rms velocity is determined by temperature alone.

Let us now calculate vd.

We get 3.67 x 10-4 m/s. Note the vast difference in the magnitudes of drift velocity and rms velocity.

Note 4: Considering the extremely simplistic assumptions made in modeling the atomic

processes in the Drude free electron theory, it is natural to expect that the model cannot give

a good agreement with experimental results. However, the model quite successfully explains

the electrical behavior of metals and is even at present considered as a workable theory as a

starting point in understanding electrical processes in metals.

The main objections to Drude theory are the following. In the model proposed by Drude,

electrons are treated as “non-interacting” gas molecules. Unlike gas molecules electrons are

charged. There is a strong electrostatic interaction between these electrons and this

interaction is assumed to be non-existent in the Drude theory. Drude also did not dwell in any

detail the exact cause and mechanism of collision in his theory (remember the theory was

proposed in 1900 when there existed no clear picture of an atom!!). Also the expression

derived for conductivity of metal does not explicitly show any temperature dependence of

conductivity. Electrical conductivity was known to depend on temperature.

More and more contradictions surfaced as experimental studies on electrical conductivity of

different materials were reported. Sommerfeld and others developed more refined theories

and the failures of Drude model were addressed.

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