Sreenivasa Instityte of Technology and Mangement studies,

(Autonomous)

CHITTOOR. AP.

Engineering Physics(18SAH112)

Lecture Notes PREPARED BY

P V Ramana Moorthy

ASSOCIATE PROFESSOR

DEPARTMENT OF SCIENCE AND HUMANITIES

SREENIVASA INSTITUTE OF TECHNOLOGY AND MANAGEMENT STUDIES

BANGALORE-TIRUPATHI HIGHWAY, MURAKAMBATTU-517127 PHONE: 08572-246298,

246299 FAX: 08572-246297 EMAIL: principal@sitams.org Website: www.sitams.org

UNIT-I

OPTICS, LASERS

AND

FIBER OPTICS

OPTICS

Introduction

Basically optics is the branch of science which deals with the study of light.

It is also known as the branch of physics, which deals with the study of properties

and nature of light. Optics is mainly divided into two parts.

i) Geometrical optics which deals with the image formation by optical systems.

That is the Geometrical optics concerns with the formation of images, when light rays

passes through an optical system, such as a lens and a prism.

ii) Physical optics which deals with the nature of light.

That is the physical optics deals with the nature of light, such as Interference, Diffraction

and polarization.

Interference

Interference is that phenomena in which two wave trains, when superposed at a

point, produce collinear oscillations such that the resultant intensity at the point of

superposition not only depends on the amplitudes of the component waves but also on

their phase difference at the point of interference.

The interfered effect at any point can be observed by the eye, only if the effect is

steady over sufficiently long intervals of observation.

The effect is steady only if the phase relations between the interfering waves remain

constant over that time interval.

The phase emission of a wave train from a source, change at random. This random

change in the emission phase changes the phase of waves train at the given point.

The phase difference between two wave trains at a point of their superposition will

vary with time, if their frequencies are not equal.

Thus constant phase relations between the interfering waves requires sources of

i) Same and single frequency and

ii) Constant emission phase difference.

The condition (i) is fulfilled if the sources are monochromatic and of the same frequency.

The condition (ii) requires coherent sources.

Coherent source

Coherent sources are those sources, which maintain their emission phase

difference constant for al time although each one may change its emission phase abruptly

and at random.

Constructive interference If two wave trains at the point of superposition produced collinear vibrations

interfere in the same phase, then the interference is said to be constructive. This is

possible when the phase difference of the two wave trains at the point of superposition is

2nπ, where n is an integer.

In that case the resultant amplitude is the sum of the individual amplitudes and the

intensity is maximum. The corresponding path difference between the two interfering

wave trains is an integral multiple of the wavelength, provided the sources are

equiphased.

Path difference = n , n = 1,2,3,……

Destructive Interference

If the two wave trains interfere in the opposite phase, then the interference is said

to be destructive. This is possible when the phase difference of the two wave trains at the

point of super position is (2n+1), Where n = an integer.

In this case the resultant amplitude is the difference of the individual amplitudes and the

intensity is minimum.

The corresponding path difference between the interfering waves should be an odd

multiple of half the wavelength, if the sources are equally phased.

Path difference = 2 1 , 1,2,3,.....2

n n

Interference in thin films

The colours of thin films, soap bubbles and oil slicks can be explained as due to

the phenomena of interference.

Let a plane wave front be allowed to incident normally on a thin film of uniform

thickness t.

The plane wave front is obtained with the help of a partially reflecting a glass plate G

inclined at an angle 450

with the parallel monochromatic beam of light.

The plane wave front is partly reflected at the upper surface of the film and partly

transmitted into the film. This is shown in figure (1).

The transmitted wave front is reflected again from the bottom surface of the film and

emerges through the first surface.

The wavefront reflected from the upper surface and the lower surface interfere with each

other. The resultant interference pattern can be observed with eye without obstructing the

incident wave front.

Here the following two points are observed.

i) The wavelength reflected from the lower surface of the film, traverses an

additional path 2 t.

(t from upper surface to lower surface and t from lower surface to upper surface).

Where is the refractive index of the film.

ii) When the film is placed in air, the wave front reflected from the upper surface

undergoes an additional phase change of (Because the reflection takes place

at the surface of a denser medium). Here it should be noted that no phase

change takes place at lower surface because the reflection takes place at the

surface of rarer medium.

Now when the path difference, 2 t = n, Constructive interference takes place and the

film appears bright.

Here n = 1,2,3,……

When the path difference, 22 (2 1)t n ,destructive interference takes place and the

film appears dark.

Here n = 0,1,2,3……

Note : t is the optical thickness of the film.

Constructive interference

Destructive interference

The constructive and destructive interferences are shown

Above.

B

t

D

A

C

Eye

G

Glass Plate

Figure (1) Interference in thin films

Types of Interference

Interference takes place in two ways.

i) Due to divisions of wave forms or wave front.

ii) Due to division of Amplitude.

i) Division of wave front

The phenomenon such as reflection, refraction or diffraction aids in splitting the

incident wave front into two parts. This is division of wave front.

These two parts of the same wave front transverse equal distances and combines at

some angle to produce interference.

Fresnel Biprism, Lloyd‟s mirror etc are examples of this class.

ii) Division of Amplitude.

The amplitude of the incident beam is split into two parts either by parallel

reflection or refraction.

These divided parts combine after travelling different paths and produce

interference.

Unlike the phenomenon of division of wavefront where a point or a narrow line

source is used, broad light source may be used to produce bright bands.

Newton‟s Rings, Michelson Interferometer, interferences in thin films by

Reflection etc are examples of this class.

Analytical Treatment of Interference

Division of wave front

Let A and B be the

two coherent sources

separated by a distance „d‟

and „D‟ is the distance

between source and screen.

Consider a point P (where

interference is taking place)

at a distance y, from centre of

screen „C‟ and 2y from E.

This is shown in figure (2).

Fig.2: Interference – Analytical treatment

d

D

s

A

B

y1

C

E

y2

AC= BC

S is the source of light.

Now 1 siny a t ------- (1)

And 2 siny a t -------- (2)

But according to the principle of superposition

1 2y y y

sin siny a t a t

sin sin cos cos siny a t a t t

sin sin cos cos siny a t a t a t

sin 1 cos cos siny a t a t ------ (3)

Now let a 1 cos cosA ------ (4)

sin sina A ------ (5)

Now sin cos sin cosy A t A t

siny A t ------- (6)

Equation (6) represent the equation of two superposed waves.

Now squaring and Adding equations (4) and (5), we get

22 2 2 2 2 2 2sin cos sin 1 cosA A a a

2 2 2 2 2 2 2sin cos sin 1 cos 2cosA a a

2 2 2 2 2 2 2

2 2 2 2

2 2 2 2

2 2

2 2 2

2 2

sin cos 2 cos

sin 1 cos 2cos

sin cos 1 2cos

1 1 2cos

2 2cos

2 1 cos

A a a a a

A a

A a

A a

A a

A a

Now 2cos 2cos / 2 1

2 22 1A a 22cos / 2 1

2 2 22 2cos / 2A a

2 2 24 cos / 2A a ------ (7)

Where A = Amplitude of the Result and superposed waves.

But Intensity 2I A , Square of the Amplitude

2 24 cos / 2I a

Now the Intensity of the Resultant wave depends on the term 2cos / 2 and the value

/ 2 .

Case (i) : When 0,2 ,2 2 ,...... 2n Where n=1,2,3….

(OR) path difference , 0, , 2 ,......,x n Then 24I a

Case (ii) : when

,3 ,...... 2 1 : 0,1,2,3,n n

(OR) path difference

3

, ,......., 2 12 2 2

x n

Energy distribution

curve of the resultant

wave is shown in the

figure (3).

Interference in the films by Reflection: Let us consider a plane parallel film, as shown in

figure (4) below.

Let PA be a ray of light incidenting on the upper

surface as shown in the figure (4).

PA light ray makes an angle of incidence i.

Now part of the light is reflected into the film in

the direction AB and the other part is refracted

into film

In the direction AC.

The light AC which is refracted, is reflected at C

and emerges at D. The emerged light DF is

parallel to AB.

At the Normal incidence, the path difference

between rays AB and DF is the two times the

optical thickness of the film 2 t .

The two parallel rays of light AB and DF will

interfere in the field of Eye and produce

interference pattern.

Now the path difference between the rays AB

and DF, for Normal Incidence is given by

2 t ------ (1)

At oblique incidence the path difference is given

by

AC CD AB ----- (2)

Now from the figure (4), triangle AEC is a right angled triangle.

900

P

B

H

E i

r r

D

F

A

C

t

No phase

change Fig. 4: Interference in thin films

(thin parallel films)

Phase change

of π

i

1

2

4a2

-4 -3 -2 - 0 2 3 4

| | | | | | | | |

X

Figure 3 : Energy distribution curve.

cosEC

rAC

=> cos

ECAC

r ----- (3)

Triangle CED and Triangle AEC are similar and are right angled triangles.

cosEC

rCD

=>cos

ECCD

r ----------(4)

Now cos cos

EC ECAC CD

r r

2

cos

ECAC CD

r

But EC =t, thickness of the film. 2

cos

tAC CD

r ------ (5)

Also from the right angled triangle ABD,

sinAB

iAD

=> sinAB AD i

sin ,

2 sin

AB AE ED i AE ED AE ED AD

AB AE i

Also Tan r = AE

EC from the right anlged triangle AEC

nAE EC Ta r

2AB t TanrSini ------- (6)

From equations (2), (5) and (6), we get

But we know that (Snell‟s Law) sin

sin

i

r ,µ=Refractive index of material of the Film.

sin sini r

22 an sin

cos

tt T r r

r

22 an sin

cos

tt T r r

r

12 an sin

cost T r r

r

2 cost r ----- (7)

Where is the refractive index of the

medium between the surfaces of the film.

For the reflected ray AB, the reflection is

occurring in the denser medium, a phase

change of occurs. This phase change

is equivalent to path difference of 2

.

21 sin2

cos

rt

r

2cos2

cos

rt

r

C ^ ^ ^

^ ^

^^ ^ ^

^ ^

^

^ ^

^ ^ ^ ^

L

C

S

G2

450

M

Figure (5) Experimental

setup for Newton’s Rings

^

^ ^

^^ ^ ^ ^

L

S

G2

450

M

Figure (5) Experimental

setup for Newton’s Rings

G1

^

^^ ^ ^

^ ^

^

^ ^

^ ^ ^ ^

L

C

S

G2

450

M

Figure (5) Experimental

setup for Newton’s Rings

^

The condition for maxima for the air film to appear bright is

2 cos2

2 cos2

t r n

t r n

2 cos 2 12

t r n

------ (8)

For the reflected ray CD and transmitted ray of light DF, No phase change occurs.

Because, the reflection of light CD takes place at a surface of lower refractive index.

The film appear dark in the reflected light

When

2 cos 2 12 2

2 cos 2 12 2

2 cos 2 1 12

2 cos

t r n

t r n

t r n

t r n

Where 0,1,2,3,....n

2

H F

L

G

D

B

C

G E

1 A

Newton’s Rings

When a Plano convex lens with its convex surface is placed on a plane glass plate, an air film of

gradually increasing thickness is formed between the two. The thickness of the film at the point

of contact is zero. If a monochromatic light is allowed to fall normally and viewed as shown in

figure (5), then alternative dark and bright circular fringes are observed.

The fringes are circular because the air film has a circular symmetry.

Newton‟s Rings are formed because of the interference between the waves reflected from the top

and bottom surfaces of the air film between the curved surface and the glass plate as shown in

figure (5).

figure (5) shows the experimental setup for Newton‟s Rings. In the setup

G, is the plane glass plate. L is a Plano convex lens. S is a monochromatic

source of light. G2 is the glass plate inclined at an angle 450 with the

incident parallel light from the source S. C is a double convex lens. M is

the microscope, through which we can observe interference fringes.

Theory: Newton‟s rings are formed due to interference

between the waves reflected from the top and

bottom surfaces of the air film formed between the

glass plate and curved surface of the plano convex

lens. The formation of Newton‟s Rings can be

explained by using the Figure (6).

L is the Plano Convex lens. G is a plane glass,

plate. AB is the monochromatic Ray of light,

which is incidenting on the system.

A part of the light is reflected at C (glass air boundary), which goes out in the form of

rays (1). Without any phase reversal.

This is because at the point „C‟ a light ray is reflected from a rarer medium.

The other part is refracted along CD, at the point D it is again reflected and goes out in the form

of ray (2). (DEF Ray of light).

The ray (2) suffers a phase reversal of . This is because at the point D, the light ray is reflected

from the denser medium glass.

The reflected rays (1) and (2) [GH and EF] are in a position to produce interference fringes as

they have been derived from the same ray AB. Hence they fulfill the condition of interference.

As the rings are formed in the reflected light, the path difference between them is

2 cos2

r

---- (1)

Since the interference is taking place because of the air film, for air film 1 .

Figure (6): Interference in Newton’s rings setup.

And for Normal incidence, r=0.

Now the path difference 2 1 cos 02

t

22

t

---- (2)

Where t= thickness of the air film.

At the point of contact, t=0, and the path difference 2

.

This is the condition of minimum intensity. Hence the central spot is dark.

Now the condition for bright fringe is

22

t n

22

t n

22

2

nt

2 2 12

t n

, ----- (3)

Where n= 1,2,3,……

The condition for dark fringe is

2 2 12 2

t n

2 2 12 2

t n

2 2 1 12

t n

2t n ---- (4) Here n= 0,1,2,….

Calculation of the Diameters of the

Fringes

Let LOL‟ be the plano convex lens

placed on the glass plate AB. Here the

curved surface is touching the plane

surface of the glass plate.

The curved surface LOL1 is part of the

spherical surface with the centre at C.

Let R be the Radius of curvature of the

curved surface.

Let r be Radius of Newton‟s Rings

corresponding to constant film thickness

t. From the property of the circle.

PN NQ ON ND

r r ON OD QN

2 2r t R t

2 22r Rt t

As t is very small, 2t is also very very small.

Hence 2t can be neglected.

2 2r Rt 2

2

rt

R ------- (5)

For a

Brighter Fringe

2 2 12

t n

the condition is

Now substituting the value of t, we get

22

2

r 2 1

2n

R

22 1

2

n Rr

----- (6)

Here r Radius of the Ring.

If D = diameter of the Brighter Ring, then

2

Dr

22 1

2 2

n RD

2

4 2 1

2

n RD

2 2 2 1D n R

t

O M

Q

A B

C

R

P

L1

L

D

r

PM=t

PN= r

QN = r

N

Figure (7) calculation of the Diameter of the Ring

2 2 1D R n ---- (7)

From Equation (7) 2 1D n

The diameter of the Bright Ring is proportional to the Square root of odd natural number.

For mth

Bright Ring (m is a higher order fringe).

2 2 1mD R m

For nth

the Bright Ring (n is a lower order fringe).

2 2 1nD R n

Similarly 2 2 2 1mD R m

2 2 2 1nD R n

2 2 2 2 1 2 2 1m nD D R m R n

2 2 2 2 1 2 1m nD D R m n

2 2 2 2 1m nD D R m 2 1n

2 2 4m nD D R m n

2

2

4

m nD D

Rm n

----- (8)

Also for a dark fringe, the condition is 2t n ------ (9)

But 2

2

rt

R

22

2

rn

R

2r n R

But 2

Dr

Diameter of the Ring is given by 2

2

Dn R

2

4

Dn R

2 4D n R

2D n R --- (10)

Thus the diameter of the rings are proportional to the square root of the Natural Numbers.

Now Diameter of the mth

Dark Ring is given by 2 4mD m R ---- (11)

Diameter of the nth

Dark ring is given by 2 4nD n R ----- (12)

By measuring the diameters of the dark rings.

We can calculate the Radius of curvature of the Plano convex lens.

From Equations (11) and (12), we have 2 2 4 4m nD D m R n R

2 2 4m nD D R m n

Radius of curvature of the Plano convex lens

2 2

4

m nD DR

m n

---- (13)

Here m n .

If R is known, the wavelength of the source can be calculated as follows.

2 2

4

m nD D

R m n

----- (14)

Note : 1. To show that PN NQ ON ND .

Now consider 2 2 2PN PC CN

2 2 2PN R CN (from the right angled triangle PNC)

Also from the Right angle triangle, QNC 2 2 2QN QC CN 2 2 2QN R CN

2 2PN R CN and 2 2QN R CN

2 2 2 2PN QN R CN R CN 2 2PN QN R CN ------ (1)

Now consider ND CN CD

ND CD CN

ND R CN

Also ON OC CN

Q

C

P

D

N

O

Q

C

P

D

N

O

R R

Fig 8 Fig 9

But OC R

ON R CN

Now ( )ON ND R CN R CN

2 2ON ND R CN ------ (2)

from (1) & (2), we have

PN QN ON ND

Note: 2. Determination of wave length of a

light source

Let R be the Radius of curvature of a Plano

convex lens. Let be the wavelength of

Monochromatic light used.

Let mD and nD are the diameters of thm

and thn dark Rings respectively.

Then 2 4mD m R

and 2 4nD n R

Now 2 2 4m nD D m n R

2 2

4

m nD Dm n

R m n

Newton‟s Rings are formed with Newton‟s

Rings setup. By using a traveling

microscope, the readings of the different

orders of dark rings were noted from one

edge of the Rings to other edge. The

diameters of different orders of the Rings

are calculated.

A graph between 2D and the order of the

Rings in drawn. A straight line graph is

obtained as shown in figure (10).

From the graph 2 2

m nAB D D

From the graph, the values of m n and 2 2

m nD D are calculated.

The radius of curvature R of the Plano Convex lens can be obtained with the help of the

spherometer. Substituting these values in the formulae.

2 2

,4

m nD D

R m n

can be calculated.

CD m n

Fig 10: Graph between D 2 and order of ring

Dm2 B

Dm2-Dn2

(m-n)

Dn2

D2

O n m Y

X

Order of the Rings

A

D C

Note 3: Determine of Refractive Index of a Liquid.

Now the Newton‟s Rings system is placed into a containers containing a liquid of

refractive index . Now we have to find the value of refractive index of the liquid.

Now the air film is replaced by the liquid film.

Now again the experiment is repeated. The diameters of thm and

thn dark Rings are now

obtained.

Then we have

2 21 1

4m n

m n RD D

--- (1)

Also for air film, we have

2 2 4m nD D m n R ---- (2)

From equations (1) and (2), we get

Using this formulae, we can calculate .

2 2

2 21 1

m n

m n

D D

D D

DIFFRACTION

Introduction

Diffraction confirm the wave nature of light. Usually waves bend round the corner of the

obstacles their path. For example, water waves coming from a small hole spread out in all

directions as if they have originated at the hole. Similarly sound waves pass round obstacles of

moderate dimensions. Similarly light waves bends round the corners of an obstacle is called

diffraction.

Diffraction – Explanation

Figure (1) Diffraction at a straight edge.

Light from a monochromatic source „s‟ is allowed to fall on a lens L. Now the light is rendered

parallel. 1S is a slit. AB is a straight edge. The parallel beam of light passes through slit

1S . The

light from the slit 1S falls on the straight edge. Now a geometrical shadow is observed on the

screen. The shadow is not a sharp one. Above the shadow, parallel to the edge A, several bright

and dark bands are seen due to diffraction. Thus the bending of light waves round the edges of

opaque obstacle or narrow slits and spreading of light into geometrical shadow region is known

as diffraction of light.

Types of diffraction

Fresnel Diffraction

In this class of diffraction, the source of light and the screen are at finite distance from the

aperture or obstacle having sharp edge. The incident wave front on the aperture or obstacle is

either spherical or cylindrical. For the study of this diffraction lenses are not required.

Fraunhofer Diffraction: In this class of diffraction the source of light and the screen are at

infinite distance from the diffraction aperture or obstacle. Due to this for focusing the light, we

need a lens. This diffraction can be studied in any direction. Here the incident wavefront is a

plane wave front.

Fresnel Diffraction Fraunhofer Diffraction

1. Point source of light or an illuminated

narrow slit is used as light source

1. Extended source of light at infinite distance is

used as light source.

2. Light incident on the obstacle or

aperture is a spherical wave front.

2. Light incident an the obstacle or aperture is a

plane wave front.

3. The source and screen are at finite

distance from the aperture or obstacle

producing diffraction.

3. The source and screen are at infinite distance

from the aperture or obstacle.

4. Lenses are not used to focus the light

rays.

4. Converging lens is used to focus the light rays.

^ ^ ^

L

S1 A

B

Straight edge

Screen

Geometrical shadow

S

Fraunhofer Diffraction at a Single site

Figure (2) Fraunhofer diffraction at a single

Consider a slit AB of width „e‟. 'ww is a plane wavefront of monochromatic light of wavelength

is incidenting normally on the slit. The diffracted light through the slit is focused by using a

convex lens on to a screen placed in the focal plane of the lens. According to Huygens – Fresnel

every point on the wavefront in the plane of the slits a source of secondary wavelet. These

secondary wavelets spread out in all directions to the right.

The secondary wavelets traveling normal to the slit, along the direction 0OP are brought

to focus at 0P by the convex lens L. Thus 0P is a central bright image.

The central bright image is formed because there is no path difference for the Ray traveling

normal to the slit.

The secondary wavelets traveling at an angle with the normal are brought to focus at a point

1p on the screen.

The intensity of point 1p depends upon the path difference between the secondary waves

originating from the corresponding points of the wavefront.

To find intensity at 1p , draw a normal AC from A to the light ray at B.

Now the path difference between the secondary wavelets from A and B in the direction is

given by

Path difference = BC.

From the figure (2) triangle ABC is a right angled triangle.

sinBC

AB

sinBC AB But AB = e

sinBC e ------ (1)

Now the phase difference 2

path difference.

A

B

e

Lens

L

P1

P0

W

W1

WW1=Plane wave front

AB= Rectangular slit

L=Lens

o C

2

sine

---- (2)

Now let the width of the slit is divided into „n‟ equal parts. The amplitude of the wave from each

part is „a‟.

The phase difference between any two successive waves from these parts will be given by

1 1 2

total phase sine dn n

----- (3)

By the method of vector addition of amplitudes, the Resultant amplitude R is given by

sin2

sin2

nda

Rd

---- (4)

From equations (3) and (4)

sina n

R

1

n

2

2sin

2sin

e

sin

2

e

n

sinsin

sinsin

ea

Re

n

Now let sine

----- (5)

sin

sin

aR

n

In the above expression n

is very small

Hence sinn

n

.

sinaR

n

sinnaR

sinAR

, Here A na ---- (6)

We know that intensity of light is proportional to square of the amplitude.

Intensity 2I R

2

2 sinI A

---- (7)

1

Analysis of Intensity Distribution Princial Maximum

The resultant amplitude is given by

sin

R A

3 5 7

.........3! 5! 7!

AR

2 4 6

1 .........3! 5! 7!

R A

If the negative terms vanish, the values of R will be maximum i.e. 0

sin

0e

sin 0

0 ------ (8)

Now the maximum value of R is A, R=A

Now maximum intensity 2 2

maxI R A

The condition 0 means that the maximum intensity is formed at op .

This maximum intensity is known as Principal maxium.

Minimum Intensity Positions

Resultant amplitude sin

R A

Intensity I will be minimum when sin 0 .

i.e. when R=0, I will be minimum

now sin 0

, 2 , 3 , 4 ,......, m

But sine

m

sine m -------(9)

Where m=1,2,3,….

Therefore we get the points of minimum intensity on either side of principal maximum.

For m=0, sin 0. This correspondents to Principal Maximum.

1Note: When ‘n’ no. of S.H.M. are acting at a point simultaneously, having equal amplitude ‘a’ and same phase

difference ‘d’, then the resultant amplitude is given by vector addition as

sin2

sin / 2

nda

Rd

Intensity Distribution: The variation of intensity with report to is shown in figure (4).

The diffraction pattern consist of a central principal maximum for 0

There are secondary maxima of decreasing intensity on either sides of it at positions

3 5,

2 2

.

Between secondary maxima there are positions of minima at , 2 , 3 ,......

Figure (4): Intensity DistributionDiffraction Grating

7

2

3

5

2

2

3

2

0

3

2

2

5

2

3

7

2

020201210

011 5

2

5

2

0 0

2

3

2

5

2

| | | | | | | | | | | |

I

Y

X

Diffraction Grating: Diffraction grating is an arrangement which consists of a large number of parallel slits of the

same width. These parallel slits are separated by equal and opaque spacings, known as

diffraction grating.

Fraunhofer used the first grating consisting of large number of parallel wires placed side

by side very closely at regular intervals.

The gratings are designed by ruling equidistant parallel lines on a transparent material

such as Glass with a fine diamond tip.

The ruled lines are opaque to light while the space between any two lines is transparent to

light and act as a slit. This is shown in figure (1).

Usually gratings are designed by taking the cost of an actual grating on a transparent film

like that of cellulose acetate.

Figure (8): Diffraction Grating

Now solution of cellulose acetate is poured on the ruled surface and allowed to dry, for the

formation of a thin film. This thin film is easily detachable from the surface. These impressions

of a grating are preserved by mounting the film between two glass plate thin.

Let e be the width of each line.

Let d be the width of the slit.

Now e d is known as grating element.

If „N‟ is the number of lines per inch on the grating, then

N e d grating elements are there per inch.

i.e. N e d 1" 2.54cms

2.54

e d cmN

(e+d) Transparent (slit)

Ruled lines

Opaque d

e

Transparent

material glass

(e+d) = grating

element

Usually there will be 15,000 lines per inch (or) 30,000 lines per inch on the grating. Due to the

narrow width of the slit, it is comparable to wavelength of light.

When light falls on the grating, the light is diffracted through each slit.

As a result, both diffraction and interference of diffracted light gets enhanced and forms a

diffraction pattern. This pattern is known as Diffraction pattern.

LASERS

Laser an acronym for light amplification by stimulated emission of radiation.

In 1958 Schalow and Townes put forward the idea of constructing a laser using the

process of stimulated emission.

In 1960 Maiman of Hughes Research Laboratory obtained pulsed laser action at 6943 Å

in the Red region of the spectrum using a ruby crystal as the active medium.

Characteristics of laser

1. Directionality: The laser beam is highly directional. For example a laser beam ray 10

cm in diameter when beamed at the moon surface, which is 3,84,000 km away is not

more than 5 km wide. A conventional light source emits light in all directions due to

spontaneous emission. Due to stimulated emission of radiation the laser light is highly

directional. The directionality is measured in angular divergence . The leave light of

wavelength emerges through a laser source aperture diameter d, then it propagates as a

parallel been up to 2d

and gets diverged.

Figure (1) Divergence of laser beam

2 1

2 1

d d

S S

Where 2d and 1d are the diameter of the laser beam spots at distances of 2s and 1s

respectively from the laser source.

For a laser beam 310 radians.

The spread in laser beam is less than 0.01 mm for a distance of 1 m.

2. Manochromaticity

The laser light is highly mono chromatic i.e. the output light is having only one single

color or single wavelength.

d d1 d2

S1

s2

The spread in spectral width is very narrow. In a laser, all the photons emitted between

discrete energy levels and hence they have same wavelength. Let the spread in frequency

be .

The spread in frequency is related to its wavelength spread y as

2

C

.For a laser, 0.001nm . Hence a laser light is highly mono chromatic.

Also far a stable laser 50Hz and 145 10 Hz

For any laser

or

is very small.

The degree of non-monochromaticity

13

14

5010

5 10

The laser is highly monochromatic.

For a conventional sodium monochromatic source of light, the degree of non-

chromaticity is about 10-13

.

3. Coherence

Laser light is highly coherent i.e. the light waves coming from the laser source will

be in phase or will have a constant phase difference over a period of time and space.

Coherence is the prediction of amplitude and phase at any point on the wave

knowing the amplitude and phase at any other point on the same wave.

If laser light is to be coherent, it should be temporally coherent and spatially

coherent.

Temporal coherence

Temporal coherence is the ability to predict amplitude and phase over a period of time

t between initial and final observations.

In this time interval. The wave train must maintain a constant phase difference.

Longer this time, greater is the coherence.

Here amplitude and phase can be predicted at a point on the wave with respect to

another point on the same wave over a period of time t .

For a laser radiation, all the emitted photons are in phase, the result and radiation

will have temporal coherence.

Spatial coherence

The relative phrases between two points in space, on the wave front must remain

constant over some long interval of time.

The spatial coherence refers to the correlation of phrase between two light fields at

two different points in space will maintain a constant phrase difference over a period of

time t , then they are said to be spatially coherent.

For higher coherence v

v

must be small.

4. Intensity

In a laser beam more light energy is concentrated in a small region.

The concentration of energy exists both spatially and spectrally. Therefore high intensity

of laser beam. Now let there be „n‟ number of coherent photons of amplitude „a‟ in the

emitted laser radiation. These photons reinforce together and the amplitude of the

resulting wave becomes na.

Since the intensity is proportional to 2 2n a , the laser light will have high intensity.

Also the number of photos delivered from a laser per second per unit area is given by

22 34

210 10l

PN

h r photons 2 1m s

Here h= 1910 Joule, Power p 3 910 10 w

Radius r= 0.5 x 10-3

m

According to Planck‟s theory of Black body radiation, the number of photons emitted per

second per unit area by a body with a temperature T is given by

16

0 4

1

2 110

T

h

K

CN d

e

Photons 2 1m s

. Here T =1000T, 6000 Å

This shows that laser is highly intense.

5. Brightness

Laser light will have higher brightness.

This is due to the fact that laser light is highly intense, temporally coherent and spatially

coherent.

Spontaneous and stimulated emission of radiation

When the incident Radiation (Photons) interacts with the atoms in the energy levels then

three district processes can take place.

Before Emission After Emission

Figure (2) spontaneous emission

Consider a two level energy system. The energies of the levels are E1 and E2. Here E2>E1.

The population of the energy levels E1 and E2 are N1 and N2. This is shown in figure (2).

Photon emitted

N2

N1

E2

EE

EE

EE

EE

2

E1

E2

E1 N1

N2

The excited atom s in the higher energy level con stay up to 10-8

seconds. This is called

life time.

The life time of an atom is the average time it exists in an excited state before it

makes spontaneous transition to a lower energy state.

Immediately, after the life time of the excited atoms it makes a transition to the

lower energy level E1 by emitting a photon. Energy is the emitted photon.

2 1E E h ,

2 1E E

h

The process of emission of radiation by the transition of an excited atom to the lower

energy level on its own is known spontaneous emission. The no. of spontaneous emission

2N

21 2A N

Where A21 is a constant of proportionality known as Einstein‟s A coefficient of

spontaneous emission.

Stimulated absorption

Let us consider a two level energy system with energies E1 and E2. Here E2> E1.

Let N1 and N2 are the populations of the energy levels E1 and E2. This shown in the

figure (3).

Fig. 3(a) Before absorption b) After absorption

Fig(3) Stimulated Absorption

Stimulated absorption

The incident radiation consists of photons of energy equal to the energy difference

between E1 and E2.

The number of photons per unit volume of incident radiation is known as

Radiation density .

The incident photons interact with the atoms present in the lower energy level E1.

The energy of photons is absorbed by the atoms in E1. After absorbing energy the atoms

make a transition to the upper energy level E2.

This process of exciting the atoms to higher energy level by the absorption of

stimulating incident photons energy is known as stimulated absorption of radiation

N2

N1

E2

E1

E2

E1 N1

N2

Incident

Radiation

The number of stimulated absorption depend upon the number of atoms per unit volume

N1 in E1 and the incident radiation density

Number of stimulated absorptions 1N

1

12 1

N

B N

Where B12 is a constant of proportionality known as Einstein‟s B coefficient for

stimulated absorption of radiation.

If the atoms are excited from E1 to E2, makes a transition to lower energy level E1, then

radiation is emitted.

The emission of radiation takes place in two forms one spontaneous emission and

Second stimulated emission.

Stimulated emission

When a photon having energy equal to the energy difference between the two

energy levels interacts with the atom in the upper state and causes it to change to the

lower state with the creation of a second photon. This process is converse of absorption.

This is known as stimulated emission of Radiation.

This is shown in figure (4).

E2

Figure 4(a) Before emission 4 (b) After emission

Figure (4) stimulated emission

During the transition a photon is emitted out in addition to the incident photon.

The frequency of emitted photons will have 2 1 ,E E

h

2 1E E h

The number of stimulated emissions depends on the number of atoms in the

energy level 2E i.e. 2W and the radiation density of incident photons P( )

Number of stimulated emission 2N

N2

N1

E2

E1

E2

E1 N1

N2 E

h

,

E

h

E

h

Number of stimulated emission

Number of stimulated emissions 2N

Number of stimulated emissions 21 2B N

21B is a constant of proportionality.

21B is known as Einstein‟s B coefficient for stimulated emitted of radiation.

The following are the points.

1) The photon produced by stimulated emission is of almost equal energy to that

which caused stimulated emission.

Here the light waves associated with them must be of nearly the same

frequency.

2) The light waves associated with the two photons are in phase, they are said to be

coherent.

Difference between spontaneous emission and stimulated emission

Spontaneous emission Stimulated emission

1. This was proposed by Neil‟s Bohr.

2. Incoherent radiation.

3. Less intensity .

4. Polychromatic radiation.

5. Emission of light photon takes place

immediately (10-8

sec) without any

inducement during the transition of

atoms from higher energy level to

lower energy level.

6. Less directionality

7. More angular spread during

propagation Ex. Light from a

sodium or mercury vapor lamp.

1. This was proposed by Einstein.

2. Coherent radiation

3. High intensity

4. Highly monochromatic radiation.

5. Emission light photon takes place by

inducement. A photon having energy

equal to the energy difference between

two energy levels interacts with the

atom in the upper level and censes it to

make a transition to the lower energy

level.

6. High directionality.

7. Less angular spread during propagation.

Ex. Light from a laser source.

Population inversion

Consider a two level energy system. Also consider that there are N atoms per unit

volume exist in a given energy state.

This N is known as population and is given by

Boltzmann‟s equation N2-------E2

/

0

E KTN N e --- (1) N1-------E1

A two level system.

Where N0= Population in the ground state

K= Boltz mann‟s constant

T= Absolute temperature

And E= Energy of the level with population N.

From the above it is clear that population is the maximum in the ground state.

Population decreases exponentially as we go to higher energy states.

This experimental decrease is shown in figure (5).

i.e. At the ground level the population is

high and at the higher level population is

low.

Let N1 = population in the energy

state E1.

N2= Population in the energy state E2.

Note that E2>E1.

From Bottzmann‟s law, we have

2 /

2 0

E KTN N e

----(2)

1 /

1 0

E KTN N e ----(3)

Now

2

1

/

2

/

1

T

T

E K

E K

N e

N e

Fig. (5) Exponential decrease of

population

2 1 /2

1

TE E KNe

N

2 1 /

2 1TE E K

N N e

/

2 1TE K

N N e

-----(4)

Where 2 1E E E

From the Boltzmann‟s low / TE K

oN N and equation (4) it is clear that 2 1 1 2N N N N

Since 1 2N N , when ever and electromagnetic radiation incidents on the system, there is a

net absorption.

For laser action to take place, it is important that stimulated emission predominate over

spontaneous emission.

i.e. The system will act like an absorptive system rather than an emissive system.

For predominance of stimulated emission over spontaneous emission, we should have the

condition N2>N1.

That is the upper level should be more populated then the lower levels.

This stimulation where N2>N1 is called

population inversion.

This concept can be best illustrated by

considering a three level energy system.

Consider a system with three energy levels E1,

E2, E3…. When the system is in equilibrium the

uppermost state E3 is populated least and the

lower state E1 is populated most as shown in the

figure (6).

Fig. (6) Exponential decrease of

population

E3 N3

N2

N1 E1

O

E2

Energy

This is a Boltzmann distribution curve. Since the population in the various states is such

that N3<N2<N1 the system is absorptive rather than emissive.

But an excitation by outside energy, it is possible that N2 exceeds N1.

This is possible if E2 happens to be a metastable state. I.e. An energy state with a large

time and the transition probability between levels 3 and 2 is very high.

The population inversion is achieved and is shown in figure (7).

N2>N1

Population

Usually E3 is very close to E2. E2 and E1 are wall separated.

The life times are shown in the diagram.

Conditions for population inversion

The important conditions for population inversion are

1) There must be at least a pair of energy levels in the system.

2) The energy must be supplied continuously to the system.

Usually population inversion is achieved by a process called pumping

Ruby laser

In the year 1960 Maiman constructed a laser using a Ruby crystal. Ruby is a synthetic

material.

Fig. (7) Population inversion in a three

level system

E3

N3

N2

X

E1

O

E2 Energy

Y

Life time 10-8 sec

Life time 10-3 sec

Ruby is a synthetic Aluminum oxide (Al2O3) with 0.05% weight of chromium oxide

Cr2O3 added to it. The chromium ions (Cr+3

) are the active medium, the aluminum and

oxygen atoms are interest.

Construction

Ruby consists of a matrix of Aluminum oxide in which some of aluminum ions are

replaced by chromium ions.

Between the energy levels of chromium ions only losing action takes place.

The ruby crystal cut into a cylindrical rod. The length of ruby rod is around

2-20 cm and diameter around 0.1-2 cm.

The ruby crystal is Al2O3 which is doped with 0.05% weight of chromium oxide

(Cr2O3).

The ends of the Ruby rod are made flat and parallel. On end of the ruby rod is

fully silvered, the other end is made partially reflecting and partially transmitting.

i.e. one end will at like totally reflecting surface and the other end is 90% reflecting and

10% transmitting in order to obtain same output from the device.

The Ruby rod is enclosed in an envelope. The entire system is surrounded by a helical

Xenon flash lamp. The helical Xenon flash lamp is supplied with a high voltage DC

source.

The DC high voltage source is connected to a resistance R and a capacitor „C‟ as shown

in the figure (8).

Due to C-R element in the power circuit, a pulsating voltage will be supplied to the flash

lamp. The two ends of the Ruby rod will act as an optical resonator.

R

Figure (8) The Ruby laser

Working :

When the power supply is switched on, due to C-R element, a pulsating voltage is applied

to the xenon flash lamp.

Due to flashing a xenon flash lamp, an intense white light is produced.

The intense white light falls on the Ruby rod. The ruby rod absorbs light falling on it.

The chromium ions (Cr+3

) in the ground state absorbs radiation in the wave length

regions of 4000Å and 6600Å.

Cooling liquid

inlet

Fully silvered

+ -

Partially

silvered

Cooling liquid

outlet

Laser beam

Xenon flash

lamp

Glass tube

C

Chromium ions are excited to the higher energy levels E2 and E3 as shown in the figure

(9).

The energy levels E2 and E3 are containing bonds of energy levels.

Energy levels E2 and E3 accommodate all the chromium ions pumped from the ground

level.

The chromium ions excited to the energy levels E2 and E3 decays rapidly through non

radiative transition to a metal stable state in a time of 10-8

sec.

The meta stable state M accumulated with chromium ions, since the life time is

around 10-3

sec.

If energy is supplied continuously to the system, a stage is reached where the population

inversion takes place between E1 (ground state) and the metastable state M.

The stimulated emission of radiation dominates over spontaneous emission due to

NE1<NM or NM >NE1. This results in the emission of laser radiation of wavelength

6943Å.

This output of laser is in the red region of the electromagnetic spectrum. Due to rapid

non-radiative transmission from E2, E3 to M, heat will be liberated.

This liberated heat will be absorbed by the surrounding Ruby lattice.

To avoid heating of the Ruby rod, the device is cooled in liquid nitrogen.

310 sect

M

Figure (9) The energy level diagram of ruby laser with chromium ions.

E3 Rapid decay

X

E1

E2 E

T=10-8 sec

Laser beam

6943 Å

M=Metastable state

6600 Å 4000 Å

Ground level

Due to metastable characteristic of level M. population in M will be building up and

inversion is achieved.

The output of the laser is pulsating since charging and discharging of capacitor takes

place through the resistor.

Helium – Neon Gas laser

Helium – Neon Gas laser is a mixed gas laser. The first continuously operating

laser was constructed in 1960 by Javan, Bennet and Herriot and the Bell telephone

laboratories.

In this laser the actuallaser action takes place between excited levels of Neon.

Helium Gas is present to excite the Neon Atoms to a higher level.

Construction of Helium – Neon Gas laser

The Helium – neon gas laser consists of a quartz discharge tube of 100 cm length.

The internal diameter of the discharge tube is around 2-8 mm.

The discharge tube is filled with a mixture of Helium at 1 torr pressure and Neon at 0.1

torr pressure. Helium and Neon gases are mixed in the ratio 10:1. The length of the

discharge in the tube is nearly about 80 cm.

The important components of the He-Ne gas laser are shown in the figure (10).

One end of the tube is arranged with 100% reflecting concave mirror and the other end is

arranged with a partially reflecting and partially transmitting concave mirror.

From the second end, we get laser output.

The end windows are maintained at the Brewster angle and have they are known as

Brewster windows.

The discharge tube is having two electrodes. The electrodes are connected with a high

voltage source of 1kv – 2 kv, through a resistor.

1 torr = 1 mm of mercury

1 torr = 133.32 pascal

Figure : (10) Helium – Neon Gas laser

Working of the laser

When a high voltage dc sensor is switched on, an electrical discharge is passed through

the gas.

During this discharge, electrons are accelerated down the discharge tube.

The electrons collides with Helium and neon atoms. Helium and Neon atoms are in the

ratio of 10:1.

Helium atoms are excited to higher energy levels. The energy level diagram of the laser is

shown in figure (11).

This diagram shows the energy levels of Helium and Neon Atoms.

+ - R

100%

Reflecting

mirror

Partially

Reflecting and

transmitting

mirror

He+ Ne

Quartz discharge tube

End windows maintained

at Brewster Angle

The Helium atoms tend to accumulate at energy levels F2 and F3 due to their long life

times.

(10-4

and 5x156 secs).

Helium atoms collide with electrons and are excited to higher energy levels F2 and F3.

Through atom – atom inelastic collisions

Hence energy is transferred between helium and Neon atoms. Therefore neon atoms are

excited to higher energy levels.

The levels of Neon E4 and E6 have almost same energy as that of F2 and F3.

Hence the excited Helium atoms colliding with neon atoms in the ground state excite

neon atoms to E4 and E6.

Since the pressure of Helium is ten times that of neon, the levels of E4 and E6 are

selectively populated as compared to other levels of Neon. The collision reaction is

shown below.

*

1 2

* *

He e He e

He Ne He Ne

In the above equation e1 and e2 are electrons

*He excited Helium Atoms

*Ne excited Neon Atoms

Transitions between E6 and E3 produces the 6328Å line of the He-Ne laser in the Red

region.

Neon atoms deexcite through spontaneous emission from E3 to E2.

The level E2 is metastable and thus collect atoms. The atoms from this E2 level fall

back to ground level through collision with the walls of the tube. The other two important

wavelengths from the He-Ne laser are

i) 1.15 from which corresponds to E4 E3 transition.

ii) 3.39 from which corresponds to E6 E5 transition.

Here a perfect population inversion is achieved between the energy levels

E6 and E3.

Neon energy levels

Figure (11) Energy level diagram of Helium – Neon laser

The emitted laser wave consists of two components called perpendicular polarized wave

and parallel polarized wave.

To avoid the perpendicular polarized component, the end windows are maintained at

Brewster Angle.

The Brewster angle B is given by

1 2

1

B

nTan

n

10-8 sec

19 –

--

17 --

--

15--

--

13--

--

11--

Helium Energy

levels

F3

F2

Excitation by

collision with

electrons

E6

E4

10-7sec

1.15µm

3.39 µm

6328Å

E5

E3 10-8s

Spontaneous Emission (~6000Å)

Laser

Through atomic

collisions

E6

E4

E2

E1

Deexcitation

by collision

Helium ground

level

Neon ground level

Where 1n Refractive index of the gas mixture.

2n Refractive index of glass

The perpendicular polarized wave is completely attenuated by the windows plate.

The parallel polarized wave is transmitted by the window is same direction.

The parallel polarized wave is repeatedly reflected by the resonator mirrors situated

behind the Brewster windows. Here correspondingly the light passes repeatedly through

the active medium.

Advantages

1. The laser light emitted by the Gas lasers is highly monochromatic and directional

when compared to solid state lasers.

2. He-Ne laser emits continuous wave of laser light.

3. Due to the presence of Brewster windows at the ends, the output laser light is linearly

polarized.

4. In put power is 5-10 watts.

5. Output power is 1-50 mw.

Semi conductor PN junction laser

GaAs and GaAsP lasers were the first PN junction semi conductor lasers built in 1962.

When a PN junction is forward biased at emits coherent radiation.

Principle

When a PN junction is formed between P and N materials of a semiconductor, depletion

layer is formed across the junction.

When the junction is forward biased, the width of depletion layer decreases. Due to this

electrons will flow from will flow from N side the P side of the junction, Here electron –

hole recombination takes place.

Due to this recombination of electrons with holes, light is emitted out from the junction.

The Pn junction which is forward biased and the energy bond diagram showing in the

figure (1)a and (1) b.

Depletion layer

+ -

Forward Bias

P n

Radiation

Valence Band

Conduction band

Emitted light Eg

Fig. 1)a Forward biased Pn junction

Fig. 1)b Energy band diagram

The energy band diagram, showing the movement of carriers, is shown in the figure (2)

Conduction band

N region

Valence band

Figure (2) Energy level diagram of Pn junction laser device.

Electron

hole

Laser

P region

Conduction band

Laser

Valence band

Electrons

Holes

The valence bond in P-region has holes and the conduction bond in N-Region

has free electrons -. When the junction is forward biased, current flows. The electrons

from the conduction bond of N-region make a transition to the valence Band of P-Region.

During this transition, electrons recombine with holes, emitting radiation corresponding

to the energy gap.

This process is called Radiative combination. During this process radiation is emitted out.

When current is increased beyond threshold current, stimulated emission occurs. This

ensures a laser light beam.

The energy of the emitted radiation is given by

E h Eg

The frequency of the emitted laser light is given by

Eg

h

We know that c

C

C Eg

h

The wavelength of emitted laser light is given by

hc

Eg

Where h= planck‟s constant

C= Velocity of light

Eg= Energy band gap of the semi conductor.

From Equation (1), it is clear that, the wave length of the emitted laser light

depends on the energy gap of the semi conductor.

Usually, GaAs semi conductors is used as a direct Band gap semi conductors.

Construction

The Basic structure of a pn-junction semiconductors laser is shown in figure (3). A

GaAs semi conductors is taken and is doped with impurities such that a p and n regions

are formed in the GaAs. Semi conductor.

A pair of parallel planes is cleaved or polished perpendicular to the plane of the

junction. The two remaining sides of the diode (front and rear face) are roughened to

eliminate lasing. The lasing action takes place in one direction only i.e. perpendicular to

the plane of polished surface.

Fig(3): Basic structure of PN junction semiconductor Laser

Front roughened

surface (Rear also)

Active region

Metal contact

I Terminal

I Terminal Metal contact

P-type

Optically flat and

parallel faces

N-type

This structure is called a Fabry- perot cavity. The others two sides are used for Metal

contacts. One metal contact serves the purpose of heat sink. Here the junction is formed

between P and n materials in the same host lattice. In the semi conductors laser doping

concentration levels are high.

Two flat polished parallel planes will serve the purpose of optical resonator.

Working

When P-type is connected to the positive terminal of a Battery and N-type is connected to

the negative terminal then the pn junction will be is forward bias condition.

Due to forward Bias, a current flows in the diode.

Initially at low current there is spontaneous emission in all directions.

When the forward bias increases, eventually a threshold current is reached at which the

stimulated emission occurs. A highly name chromatic radiation is emitted from the

junction. Here electron – hole recombination takes place across the junction.

The source of excitation is in the Battery (Forward Bias). The actual pumping process is

direct conversion.

The output of the semi conductor laser is in the infra red region wavelength range of

9000Å.

Advantages

1. The efficiency of the laser is high.

2. Laser output can be modulated by modulating the junction current.

3. The lasers output is tunable to a continuous wave or pulsed wave.

Applications of Lasers

Industry

1. Two dissimilar metals can be weld using a laser.

2. Laser used to cut glass and quality.

3. Lasers are used to drill holes in Quartz and ceramics.

4. Lasers are used for heat treatment in the tooling and automotive industry.

Medicine

1. To attach a detached retina, it is used in ophthalmology.

2. Lasers are used in correcting short sight.

3. Used for cataract removed.

4. Lasers are used in bloodless surgery.

5. Lasers are used in cosmetic surgery called mammoplasty.

6. Lasers are used in Angioplasty for the Removal of artery Block.

7. Used in the diagnosis of cancer therapy.

8. For removing stones in Kidneys and Gall Bladder.

Science

1. Lasers are used in Isotope separation.

2. Recording and Reconstruction of Holograms.

3. Used to create plasma.

4. Used to produce chemical reactions.

5. To study internal structure of micro organisms and cells.

6. To study the structure of molecules.

FIBRE OPTICS

Introduction

Fibre is a material that can be drawn into a number of threads.

The thin like fibre are bundled and used as carriers of light energy.

Optical fibre is a thin transparent medium which carries information in the form of

light.

The propagation of light through the optical fibre will be in the form of multiple

total internal reflections.

The fibre basically consists of two regions namely core and cladding.

The core region of the fibre having higher refractive index carries most of the

light. The core is surrounded by a cladding of lower refractive index.

These fibers improved the efficiency of transmission, reduced cross talk between

fibers.

The optical signals will have frequency of light; therefore fibres can be used as

carriers of information.

Advantages of optical fibres in communication

1. Fibres are having higher information carrying capacity i.e. band width is high.

This means that a greater volume of information or messages can be carried over

in a fibre optic system.

This is because the rate at which information can be transmitted is directly related

to signal frequency. Light has a frequency in the range of 1014

-1015

Hz, compared to radio

frequency of 106Hz and microwave frequencies 10

8-10

10Hz.

Therefore a transmission system that operates at the frequency of light can

theoretically transmit information at higher rate than systems that operate at radio

frequencies or micro wave frequencies.

2. They are small in size and are very light in weight.

3. No possibility of internal noise and cross talk generation along with immunity to

ambient electrical noise or electromagnetic induction.

4. No short circuit hazards as in the case of material wires.

5. In explosive environments, it can be used safely.

6. Immunity to adverse moisture and temperature conditions.

7. The cost of fibre optic cable is low when compared to copper / G.I. cables.

8. No need of additional equipment to protect against grounding and voltage problems.

9. The installation cost is nominal.

10. Fewer problems in space applications such as space radiation shielding and line to

line data isolations.

Principle of optical fibre – total internal reflection

When ever a ray of light travelling from a medium of high refractive index to a

medium of low refractive index, the light ray bends away from the normal.

When a ray of light is travelling from a denser medium to rarer medium, making

an angle of incidence i, it will be refracted into the air medium ,with angle of refraction

r. this is shown in the figure (1) a. If the angle of incidence further increases, the angle of

refraction also increases. This is shown in figure (1) b. At the interface, when the ray of

light incidents at an angle called critical angle, the ray will not be reflected, but it will

graze the interface. This is shown in fig (1)C.

When i>c , the ray will be totally reflected back internally into the same medium.

This is shown in figure (1) d.

Figure (1) Light ray suffering total internal reflection.

Applying Snell‟s law for the the ray of light suffering total internal reflection,

21 2

1

sin sin sin sinn

n i n r i rn

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

In the case of a fibre, the ray of light travelling from a denser medium to rarer medium,

will be totally internally reflected into the same medium (i.e. into core).

Now consider the incident ray for which r=900

(i.ei=c) then 02

1

sin sin90c

n

n

21 2

1

sin c

nn n

n ---------------(2)

therefore for any ray of light whose angle of incidence is greater than this critical angle,

total internal reflection takes place.

Fibre construction

An optical fibre consists of a thin central thread of transparent plastic or glass,

which is surrounded by a second dielectric. The thin thread of central cylindrical material

is called the core. The core is surrounded by another material called cladding. The

refractive index of the core is slightly greater than that of cladding material such that the

guidance of the light is only through the fibre of the core material. The refractive index of

the core and cladding materials decides the properties of communication fibres. The size

of the core and cladding also determines the characteristics of a fibre to some extent. The

buffer Jacket (protective jacket) over the optical fibre is made of plastic and protects the

fibre from moisture and abrasion. In between the buffer jacket and optial fibre, there is

silicon coating. Due to this further isolation is achieved. Surrounding the buffer jacket

there is a layer of strength member (Kevlar) which provides toughness and tensile

strength. Here the fibre optic cable withstands without any brittleness during hard pulling,

bending, stretching or

rolling, through the fibre is

made from brittle glass.

Finally the cable is

covered by black

polyurethane outer

Jacket.

Figure (2) Fibre construction.

The fibre structure is shown in figure (2) for a typical fibre. Usually fibres are made with

either plastic or glass. Thus there are two types of fibres. 1. Glass fibre 2. Plastic fibre

Glass fibre

Glass fibres are made by fusing mixtures of metal oxides and silica glass.

The most common material used in glass fibre is silica (oxide glasses). It has a refractive

index of 1.458 at 850 nm. For producing two same materials having slightly different

refractive indices for the core and cladding, either fluorine or various oxides such as

B2O3, Ge2O2 or P2O5 are added to silica.

Examples of Glass fibre compositions

1. 2 2 2;GeO SiO Core SiO Cladding

2. 2 5 2 2;P O SiO Core SiO Cladding

3. 2 2 5 2;SiO Core P O SiO Cladding

Another type of silica glasses are made with low melting silicates. Such optical fibres are

made of soda-silicates, germane silicates and borosilicate.

Plastic fibre : The plastic fibres are typically made of plastics, are cheap and can be

handled without special care due to their toughness and durability.

Examples of plastic fibres.

1. A Polystyrene core (n1=1.60) and methyl methacrylate cladding (n2=1.49).

2. A Polymethylmethacrylate core (n1=1.49) and a cladding made of its co-polymer

(n2=1.40).

Propagation of light in fibres

Consider the light

propagating in an optical fibre.

Let us consider a ray of light

which is incident on the

entrance aperture of the fibre

making an angle of incidence i

with the axis, as shown in

figure (3). PQ is the incident

ray, making angles of

incidence i. QR is the refracted ray.

The refracted ray makes an angle with the normal (axis of the fibre). TU is the ray that

emerges from the fibre. The refractive index of the core is n1 and that of the surrounding

medium is n2 (n1 > n2).The surrounding medium is air and its refractive index is

0 1 0n n n

For all practical purposes refractive index of air is taken as unity.

Applying Snell‟s law for the ray of light going from air to core,

0 1 sinn Sini n

1

0

sin

sin

ni

n

11 0

0

sin sin ;n

i n nn

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

If the ray QR has to suffer total internal reflection at the core-cladding interface (at the

point R).

2

1

sinn

n ------------------------- (2)

Now QRM is a right angle Triangle. 90

2

1

sin sin 90n

n

2

1

sin cos (3)n

n

Fig.(3): Propagation of light in optical fibre

But, 1/2

2 2 2sin 1 cos sin 1 cos

1/22

2

2

1

sin 1n

n

--------------------- (4)

From equations (1) and (4), we get

2

1/22

1

2

0 1

sin 1nn

in n

1/22 2

1 1 2

2

0 1

sinn n n

in n

1sin

ni

2 2 1/ 2

1 2

0 1

( )n n

n n

1/22 2

1 2

2

0

sinn n

in

If 2 2 2

1 2 0n n n , then for all values of i, total internal reflection will occur.

We know that for air, n0=1.

The maximum value of i for which the ray of light guided through the fibre is given by

acceptance angle A.

1/ 2

2 2

1 2sin A n n

Acceptance angle 1/2

1 2 2

1 2sinA n n

Acceptance cone

A cone obtained by rotating a ray of light at the end face of optical fibre, around

the fibre axis with acceptance angle is known as accepnace cone.

Acceptable angle

It is the maximum angle with which a ray of light can enter one end of the fibre

and guided through the fibre with total internal reflection. The acceptance angle is

denoted by A.

1/2

1 2 2

1 2sinA n n

It is a measure of light gathering power of the fibre.

Numerical aperture (NA)

Numerical aperture is the light gathering power of an optical fibre.

Numerical aperture is defined as the sine of acceptance angle.

Numerical Aperture = sinA

1/2

2 2

1 2. .N A n n

Types of optical fibres

Depending on the variation of refractive index of core of an optial fibre, the fibres are

classified into two types.

1. Step index fibre

2. Graded index fibre

Again basing on the number of modes (paths) available for the light rays

propagating inside the core, the fibres are classified into.

i) Single mode step index fibre.

ii) Multimode step index fibre.

Differences between single mode step index fibre and multimode step index fibre

Single mode step index fibre Multimode step index fibre

1. The refractive index of the core is

uniform throughout. In such a fibre

the refractive index profile abruptly

changes or step changes at the

cladding boundary.

2. The diameter of the core is 8-12 µm

and that of cladding is 125µm.

3. In a single mode fibre only one

mode or path can propagate through

the fibre.

1. The refractive index of the core is

uniform throughout. The refractive

index profile abruptly changes or step

changes at the cladding boundary.

2. The diameter of the core is 50-200 µm

and that of cladding is 125-400µm.

3. Multimode fibre allows a large

number of paths or modes for the light

rays travelling through it.

4. Index profile diagram for the single

mode step index fibre is shown

below.

1& 2 are cladding regions.

Fig. (1) Index profile diagram for step

index single mode fibre.

5. The light rays are propagating in the

fibre as shown below.

Fig. 2. Propagation of light in a

single mode step index fibre (or)

Fig.2. prorogation of light in a

4. Index profile diagram for the single

mode step index fibre is shown below.

1& 2 are cladding regions.

Fig. (1) Index profile diagram for step

index single mode fibre.

5. The light rays are propagating in the

fibre as shown below.

Fig. 2. Propagation of light in a

multi mode step index fibre (or)

Since the core is wider, greater

number of light rays enters into the

fibrefrom input signal and takes

multiple paths, as shown in fig(2).The

light ray (1) which makes greater

angle with the axis of the fibre,

suffers more number of reflections

through the fibre.It takes more time to

single mode step index fibre

Since it has got are mode of

propagation, only one ray of light

enters into the fibre and traverses a

single path or the light ray

transverses along the axis of the

fibre.

Here the light ray takes only one

path, hence there is no signed

distortion at the output end.

6. Signal distortion is shown below

(No. distortion)

Distortion the single mode step

index fibre

7. The difference between the refract-

tive indices of the core and cladding

is very small.

8. Value is small.

9. These fibres are more suitable for

communication. This is because of

less distortion.

10. Projection of light in to single mode

fibres and joining of two fibres are

very difficult.

11. Fabrication is very difficult and so

the fibre is costly.

12. It is a reflective type of fibre.

13. The light rays travel in the form of

meridional rays.

reach the exit end.Here light travels

more distance in the fibre.The ray (2)

makes smaller angle with the axis of

the fibre, it suffers less number of

reflections in a short time.Light ray(2)

traverses a short distance through the

fibre. Ray(2) reaches the exit end

quickly. Due to a path difference

between these two rays, they

superimpose at the output end. Hence

signals are overlapped.

6.Signal distortion is shown below

(Having distortion)

Distortion the single mode step index

fibre

7. The difference between the refractive

indices of the core and cladding is very

large..

8. Value is large.

9. These fibres are less suitable for

communication. This is because of large

distortion.

10. Projection of light into multi mode

fibres and joining of two multimode

fibres are very easy.

11. Fabrication is less difficult and so the

fibre is not expansive.

12. It is a reflective type of fibre.

13. The light rays travel in the form of

meridional rays.

Differences between step index fibre and graded index fibre

Step index fibre Graded index fibre

1. The refractive index of the core is

uniform throughout. In this fibre the

refractive index profile abruptly

changes or step changes at the core

cladding boundary.

2. Here we have two types of fibres,

namely single mode and multimode

step index fibres.

3. In the case of single mode step index

fibre, the core diameter is about 10µm

and that of cladding is 125µm. In the

case of multimode step index fibre,

the core diameter is 50-200µm and

that of cladding is 125-400µm.

4. The index profile diagram is shown

below in fig. (1).

Figure (1) step index fibre. Index

profile diagram.

5. It is a reflective type of fibre.

6. The propagation of light rays are in

the form of MERIDIONAL rays or

zig-zag rays. These rays croses the

axis of the fibre a number of times.

7. The Meriodional rays are shown

below.

1. The refractive index of the core is not

uniform. In a graded index fibre the

refractive index varies continuously

across the core. It is maximum at the

centre of the core and decreases radially

towards the outer edge. i.e. the refractive

index of the core changes in a parabolic

manner.

2. Here we have only one type of fibre,

namely multimode fibres.

3. The diameter of the core is about 50 µm

and that of cladding is about 125 µm.

4. The index profile diagram is shown

below in fig. (1).

Figure (1) Graded index fibre. Index

profile diagram.

5. It is a refractive type of fibre.

6. The propagation of light rays are in the

form of SKEW rays or helical rays.

These rays will never cross the axis of

the fibre.

7. The skew rays are shown below.

Fig (2) step index fibre meridional

rays.

In the case of step index fibre, the

light rays propagate through the fibre

by way of total internal reflection.

8. In a step index fibre the light rays are

propagated as shown in figure (3).

Fig.3 propagation of light in a step

index fibre

The two rays will not reach the output

and simultaneously there is

intermodal distortion. The output and

input signals are shown in the figure

below.

Fig (2) Graded index fibre skew rays.

In a graded index fibre, the refractive

index of the core decreases from the

fibre axis to the cladding interface in a

parabolic manner. When a light ray

enters into the core and moves towards

the cladding interface, it encounters a

more and more rarer medium due to

decrease of refractive index.

As a result, the light ray bends away

from the normal and finally bends

towards the axis of the fibre. Now it

moves towards the core-cladding

interface at the bottom.

Again the light ray bends in the upward

direction. Thus due to continuous

refraction on the light ray takes

sinusoidal (or) helical path.

8. In a graded index the light rays are

propagated as shown in figure (3)

Fig.3 propagation of light in a graded

index fibre

When the two light rays (1) and (2) enter

into the fibre, by making different angles

with the axis of the fibre, their velocities

will change continuously and reach the

output end simultaneously at the same

time i.e. the light rays will come to focus

at the same point.

Here there is no intermodal dispersion.

The output and the input signals are

shown in the figure below.

Fig.( 4) Input and output signals in a

step index fibre

9. Distortion is more due to intermodal

dispersion.

10. Here there is no self refocusing effect.

11. Numerical aperture is more in a

multimode step index fibre.

Fig. (4)Input and output signals in a

step index fibre

9. Distortion is less.

10. Self refocusing effect takes place due to

continuous refraction. Hence the helical

path for the light rays.

11. Numerical aperture is less in a graded

index fibre.

Fibre optic communication system

The block diagram of a fibre optic communication system is shown in figure (1).

Figure (1) block diagram optical fibre communication system.

The fibre optic communication system consists of the following.

i) Transmitter

ii) Repeaters (or) fibre repeaters and

iii) Reciever iv) couplers and connectors v) fibre cable.

I) The transmitter : or an optical transmitter consists of an Encoder, a source of light

and modulator. The input signal in the form of speech or song is fed to an encoder. The

encoder converts the analog signal into a digital signal.

The digital signal is given to the source of light. The source of light can be a light

emitting diode (LED) or a pn junction laser diode.

The optical carrier signal is now finally fed to modulator. The modulator modulates the

signal depending on the requirement.

The type of modulation may be amplitude or frequency or phase modulation.

The optical signal finally coupled to the optical fibre with the help of couplers.

The couplers launche the optical signal in the fibre without any distortion.

The fibre is connected to the repeater with the help of connector.

ii) The repeater: It consists of an amplifier and a regenerator.

During the transmission of the signal, along the optical fibre, there will be loss in the

signal due to dispersion in the fibre. As a result we get a weak signal at the output end of

the fibre. To minimize the losses, repeaters are employed at regular intervals along the

fibre. Now in the repeater the amplifier amplifies the signal and is reconstructed through

the fibre.

Finally the optical signal is fed to the receiver.

iii) The receiver consists of a photo detector. The photo detector consists of a PIN diode

or avalanche photo diode.

From the fibre the optical signal is fed to the photo detector. The photo detector detects

the optical signal and converts it into an electrical signal.

The electrical is the amplified by the amplifier the amplified signal is fed to the

Demodulator. The demodulator demodulates the signal to get a digital signal. This digital

signal is decoded by a Decoder.

The output of the decoder is a pure form of the original signal. This is taken as final

output.

Applications of optical fibres

1. Optical fibres are used in fibre optic communication systems.

2. Optical fibres are used in exchange of information between different terminals in a

network of computers.

3. They are used to carry information and exchange information in cable television

networks. Space vehicles and submarines etc.

4. Optical fibres are used in industry in security alarm systems, process control and

industrial automation.

5. Optical fibres are used in optical fibre gyroscopes and are used in automotive

navigation systems.

6. They are used in pressure sensors in biomedical applications.

7. Used in pressure sensors in Engine control applications.

8. Optical fibres are used in medicine in the fabrication of fiberscope in endoscopy.

The endoscopy is used to visualise internal parts of the body.

9. They are used in fuel tanks to sense the liquid levels as a liquid level sensor.

10. They will be used as chemical sensors.

Unit – II

CRYSTAL STRUCTURES

AND X-RAY DIFFRACTION

AND

ULTRASONICS

Crystal Structures and X-Ray Diffraction

Matter exists in three different states, viz. solid state, liquid state and gaseous state.

In gaseous and liquid states the atoms or molecules will be moving from one place to other place.

The positions of the atoms or molecules are not fixed in them.

i.e. In liquid and gaseous states there is no proper orientation of atoms or molecules.

In solids the positions of the atoms or molecules are fixed, but they may have or may not be

having regular periodic arrangement.

If the atoms or molecules in a solid are periodically arranged at regular intervals in three

dimensional space then the solid is known as crystalline solid.Ex.Iron, NaCl.

If the atoms or molecules in a solid do not have periodical arrangement then the solid is known

as amorphous solid.Ex.Plastic, polymers

When the periodicity of atoms or molecules is extended throughout the solid then the solid

is known as single crystalline solid.

If the periodicity of atoms or molecules is extended up to small regions called grains, such a

solid is called polycrystalline solid.

Space lattice

A space lattice is defined as an infinite three dimensional arrangement of points (i.e.

atoms or molecules or ions) in which every point has surroundings or environment identical to

that of every other point in the array.

The crystal structure can be studied in terms of a space lattice.

Consider the two dimensional array of points as shown in figure (1).

Fig (1) A two dimensional array of lattice points.

2

2

B 1

1

1

1 A

X

Y

O

b

Fig (2) A two dimensional array of points.

Consider the point „O‟ as the origin. Join this origin to a successive lattice points along x and y

directions. Let the position vectors of these points be a and b .

When a is repeated regularly in the x-direction, then we get the lattice points along the x-

direction i.e. 2a , 3a , 4a ,……

Similarly when b is repeated regularly in the y-direction, then we get lattice points along y-

direction i.e. 2 b , 3b ,4b ,…….

Here a and b are used and repeated regularly to get the lattice points in space lattice, they are

known as fundamental translation vectors or primitive vectors.

This shown in figure (1) and (2).

Now the position of the point P can be given by a position vectorT .

i.e. T a b

Similarly the position vector of the lattice point Q is given by

2T a b

For a two dimensional space lattice, the position vector of any lattice point is given by

T na mb

Where n, m are integers.

For a three dimensional lattice, the position vectors of any lattice point is represented by

T na mb pc

T Q

a

O

P

Where n, m and p are the integers.

a ,b and c are the translational vectors along x, y and z directions.

From figure (1), consider a lattice point A. the point

A has got two nearest points at a distance of 1 unit in the x,y

directions. Diagonally it is having nearest point at a distance

of 2 units.

Let us consider another point B in the lattice array. B

is also having two nearest points at a distance of 1unit in x

and y directions. Diagonally it is having a nearest point at a

distance of 2 units.

i.e. Here Both A and B are having the same

environment. Hence figure (1) represents a two dimensional

space lattice.

In this same way we can represent a three dimensional lattice.

In three dimensions a space lattice can be represented by the

figure (3).

Unit Cell

The unit cell is the smallest block or smallest geometric figure, from which the crystal is

formed when repeated regularly in three dimensions.

It is also defined as the fundamental elementary pattern, which when repeated again and again in

three dimensions forms a lattice structure of crystal.

Let us consider a two dimensional crystal lattice with periodic arrangement of atoms, as shown

in figure (4). Here ABCD represent the smallest geometrical figure. This smallest geometrical

figure is a square. When this is repeated again and again regularly in XY space, we get a square.

In three dimensions, a unit cell is shown in figure (5). ABCDEFGH is the cubical unit cell. When

this is repeated in three dimensions regularly, we get a three dimensional crystal structure.

A B

C D

X

Y

Atoms

A

D

E

G H

B

F

C

Atoms

Fig (3) three dimensional lattice

Fig. (4) Two dimensional crystal

structure – unit cell

Fig. (5) Unit cell – three dimensional

crystal structure

Parameters of the unit cell.

Crystallographic Axes.

The lines drawn parallel to the lines of

intersection of any three faces of the unit cell

which do not lie in the same plane are called

crystallographic axes.

From the figure (6) OX, OY and OZ are the

three crystallographic axes.

Interfacial angles

The angles between the three crystallographic

axes are known as interfacial angles.

In the figure (6), , and are the

interfacial angles.

Primitives

The three sides a, b and c of a unit cell are known as primitives.

From figure (6), a, b and c are the three primitives.

Primitive cell: The unit cell formed by the primitives a, b and c and having only one lattice

point is called primitive cell.

Example: A unit cell containing one lattice point.

Coordination Number (N)

The coordination number is defined as the number of equidistant nearest neighbors that

an atom has in the given structure.

Greater is the coordination Number, the more closely packed up will be the structure.

Nearest neighbors distance (2 r)

The distance between the centers of two nearest neighboring atoms is called nearest

neighbor distance. If r is the radius of the Atom, nearest neighbor distance will be 2 r.

Atomic Radius (r)

Atomic radius is defined as half of the distance between nearest neighbors in a crystal of

pure element.

Atomic radius2

2

rr .

Atomic packing factor (APF)

The fraction of space occupied by atoms in a unit cell is known as atomic packing factor

or packing factor.

It is the ratio of volume of the atoms occupying the unit cell to the volume of the unit cell

relating to that crystal structure.

Volume of the Atoms present in the unit cell

APF=Volume of the unit cell

Y

B

o

ᵞ c

a A

X

Z

C

b

Fig (6): Lattice parameters

Density

Density is defined as Ratio between Mass and Volume.

3

A

nMDensity =

a N

Where n = No. of atoms per unit cell.

M = Molecule Weight of the unit cell.

a = Lattice constant

NA = Avogadro‟s number.

Void (or) interstitial space Figure (7) Void

The empty space between the Atoms in a unit cell is known as void or interstitial space.

Void is shown in figure(7) above.

Basis

A Basis is an assembly of atoms with identical composition, arrangement and orientation.

A crystal structure is formed by associating every lattice point with a unit assembly known as

basis. For a lattice to represent a crystal structure every lattice point must be linked with one or

more atoms called the Basis or pattern.

When the Basis is repeated with exact periodicity in all directions, it gives the actual

crystal structure.

Usually Lattice + Basis Crystal structure.

The basis is real and the crystal structure is real. The lattice is imaginary.

The following diagram illustrates basis representing each lattice point.

From the figure(8), it is clear that the basis consists of two different atoms.

Basis

Lattice point

Fig.( 8) Basis

Void

In crystalline solids like copper and sodium, the basis is a single atom.

In NaCl and CsCl, the basis is diatomic.

In crystals like2CaF , the basis is triatomic.

Bravais Space Lattices

The atoms can be grouped together to form different crystalline structures.

Here the unit cells containing atoms or molecules are repeated in a space lattice, we get

crystal structures. The scheme of repetitions of atoms or molecules in a space lattice is limited in

number.

There are 14 ways of arranging points in three dimensional space. These 14 space lattices

are known as Bravais space lattices or Bravais lattices.

Note: The space lattices formed by unit cells are marked by the following symbols.

Primitive Lattice - P or S (simple Lattice)

Body centered lattice - I

Face centered lattice - F

Base centered lattice - C

Simple Crystal Systems

All the crystals based on the geometrical shapes of their unit cells classified into seven

crystal systems.

They are

1. Cubic crystal system.

2. Tetragonal Crystal system

3. Orthorhombic crystal system

4. Rhombohedral crystal system (Trigonal)

5. Monoclinic Crystal System

6. Triclinic Crystal system Figure(9)a

7. Hexagonal crystal system

1. Cubic Crystal system

In this system, the sides of the cube are equal, b

I.e. a=b=c

The crystallographic axes are perpendicularly to one another.

The interfacial angles are equal to090 .

i.e. 090 c

Cubic lattices may be simple, body- centered or face centered.

The diagrams are shown in the figures.

α

a

Simple cubic (p), Ex: Polonium

In the case of simple cubic lattice, the atoms are present at the corners of the unit cell.

Simple cubic lattice is shown in figure(9)a.

Figure(9)b Body centered Cube-I Figure(9)c Face centered cube F

Example: Ba, Fe, Na, Cu20, W Example. Al, Ag, Au, Pb, Cu

In the case of Body centered lattice, there will be a body centered atom and eight corners atoms

as shown in the figure(9)b.

In the case face centered lattice, atoms are present at all corners and at all face centers. This is

shown in figure(9)c.

2. Tetragonal Crystal System

In this case, two sides are equal. i.e. a=b and ,a c b c

All the interfacial angles are equal to 900. i.e. 090

Here the crystal axes are perpendicular to each other. Tetragonal lattices may be simple

or body-centered. They are shown in figures(10)a and (10)b.

Figure (10)a Simple (or) primitive lattice Figure (10)b Body centered lattice (I)

Ex: TiO2, SnO2, Indium etc. Ex: KH2P04, NiSO4

c

900

900

900

b

a

c

b

a

3. Orthorhombic Crystal system

In this system the three sides of the unit cell are unqual.i.e. a b c . The crystal axes are

perpendicular to each other. All the interfacial angles are equal to 900i.e. 090 .

Orthorhombic lattices may be simple, base centered, body centered or face centered. They are

shown in figures(11)a,(11)b,(11)c and (11)d.

4. Rhombohedral crystal system

This is also known as Trigonal system. In this crystal system,

the three sides of the unit cell are equal .i.e. a=b=c.

The interfacial angles are not equal to 900.

i.e. 0 090 120 . 0 0 090 , 90 and 90

The trigonal lattice is only a primitive cell,shown in fig(12).

900

900

900 a

b

c

Figure11(a)Simple lattice (p)

Ex: hydro carbons of High

molecular weight

Figure(11)b.Base centered (c)

Ex: PbCo3,BaSo4

Figure(11)c body centered (I)

Ex: - Sulphur

Figure (11)d Face centered (F)

Ex: KN03, K2So4

b

a

a

α

c

a

a

α

c

Example: 4 , , , ,CaSo As Sb Bi Calcite.

5. Monoclinic Crystal system In this crystal system, the sides of the unit cell are unequal. Here the two interfacial angles are

equal to 900. The third interfacial angle is not equal to 90

0. 090 , 090 and a b c

Monoclinic crystal lattices may be primitive or base centered. They are shown in figures (13)a,

and (13)b.

c

6. Triclinic Crystal System

In this crystal system, the three sides of the unit cell are not equal.

i.e. a b c and .All the three angles are different.

All the interfacial angles are not equal to 900

. i.e. 090

Example: 4 2.5CuSo H O 2 2 7, K Cr O

This crystal lattice is only a primitive lattice,shown in figure(14).

c

7. Hexagonal Crystal system

In this crystal system two sides the unit cell are equal.

a b , a c and b c . However the two interfacial angles

are equal to 900.

Third angle is equal to 1200. i.e. 090 ,

0120 . This is shown figure(15).

This crystal lattice is a primitive lattice only.

Example: Quartz, Zinc, Magnesium and SiO2.

This is only a primitive Lattice. b

900

090

900 a

b

b

a

c

Figure(13)a Simple (or) primitive

lattice

Ex: Na2So4, Gypsum

Figure(13)b Base centered

Ex: K2MgSo4, 6H2O

c

a 900

1200

900

1200

900 b

c

b

a a

Fig (14) Simple lattice

β α

b

a

The 14 Bravais Lattices are now summarized as follows.

S.No Name of the crystal

system

No. of bravais lattices Types of bravais

lattices

1. Cubic 3 P,I,F

2. Tetragonal 2 P,I

3. Orthorhombic 4 P,I,F,C

4. Rhombohedral or Trigonal 1 P

5. Monoclinic 2 P,C

6. Triclinic 1 P

7. Hexagonal 1 P

Simple Cubic Crystal Structures (SCC)

The simplest structure to describe is the simple cubic crystal structure. Figure (1) shows the unit

cell of simple cubic structure.

Usually in a simple cubic lattice, there is only one lattice point in the unit cell. At all the corners

there will be corner Atoms. If we take an Atom at one corner as a reference atom, it is

surrounded by six equidistant nearest neighbours.

Coordination Number of SCC, N=6.A simple cube has eight corner atoms.

Each Corner atom is shared by eight surrounding unit cells.

Share of an Atom to each corner of unit cell = 1

8 of an Atom.

i.e. Each corner of the unit cell = 1

8 of an Atom. No. of corners of the unit cell = 8.

Total Number of Atoms per unit cell of SCC = No. of corners of the unit cell Share of each

corner.

Total Number of Atoms per unit cell of SCC. 1

88

n 1n

Effective number of lattice points in simple cubic unit cell is one.

Fig (1) Unit cell of simple cubic

crystal structure.

1

8

of an

Fig (2) Top view – touching of the corner

atoms along the edges

r

r

a

Thus a simple Cubic unit cell is a primitive cell. The top view of the SCC structure is shown in

fig (2). Here the corners atoms touch each other along the edges.

From the figure (2) nearest neighbour distance = 2r.

But 2r a

Lattice constant a = 2r

Atomic Radius2

ar .

Atomic Packing Factor:

Let r be radius of each atom. Volume of each Atom 34

3r Number of atoms per unit cell of

SCC, n=1.

Volume of all the Atoms present in the unit cell of SCC = 341

3r --------------(1)

Side of the unit cell of SCC = a.

Volume of the unit cell = a3 -------------------(2)

Now, APF Volume of all the Atoms present in the unit cell

=Volume of the unit cell

From (1) and(2), we get

33

3 3

41

43APF=a 3 2 2

ra a

ra

4

33 a

3a

80.52

62

i.e. 52% of the Volume of the unit cells is occupied the Atoms in the unit cell.

Also void space or Interstitial space = 48%.

Density = 3

A

M

a N (Number of Atoms,n=1)

Example : Polonium.

Body centered cubic structure (BCC)

The unit cell of BCC is shown in figure (1). In this unit cell a=b=c, 0= = =90 .

In this structure of the unit cell, there are eight corners. At all the corners, we have corner atoms.

In addition to the corner atoms, there will be a body centered Atom at the body centre of the unit

cell.

Now if we take an Atom as a reference Atom, it is surrounded by 8 unit cells.

Share of a corner Atom to each unit cell 1

8 of corner Atom.

i.e. Each corner of the unit cell 1

8 of an Atom.

No. of corners of the unit cell = 8.

Total No. of Atoms contributed by 8 corners.

= No. of corners Share of each corner

1

88

=1

Now Total Number of Atoms per unit cell of BCC =

One body centered Atom + contribution made by the corner Atoms.

11 8 1 1 2

8

Coordinate Number (N)

Any corner reference Atom is surrounded by 8 number of unit cells. For any corner Atom the

nearest neighbouring Atom is the Body centered Atom. Each unit cell is having one body

centered Atom. Hence the coordination number N= 8.

Lattice Constant (a) and Atomic Radius (r)

The corner Atoms will not touch each other. The corner atoms touch the Body centered

atom along the body diagonal.

This is shown in the figure (2).

aR

Fig (2) Touching of the corner atoms with the

body centered atom along the body diagonal

Fig (1) Unit cell of BCC structure

1

8

of an atom

Body centered

atom

Now from the figure (2), Triangle ACD is a Right angle Triangle.

2 2 2AD AC CD 2 2 2AD AC a ----------- (1)

Also from the Right angled triangle ABC,

2 2 2AC AB BC

2 2 2AC a a ----------- (2)

From (1) and (2), 2 2 2 2AD a a a

2 23AD a

But AD = 4r.

2 24 3r a

4 3r a

Atomic Radius 3

4r a .

Also lattice constant 4

3

ra

Also nearest neighbour distance = 2r. i.e. 3

22

r a

Atomic packing factor (APF)

Let r be the radius of each Atom. Volume of each Atom 34

3r

Number of Atoms per unit cell of BCC = 2.

Volume of the two Atoms present in the unit cell

342

3r -----------------(3)

Let a be the side of the unit cell of BCC.

Volume of the unit cell of BCC = 3a ----------------- (4)

Now Atomic Packing Factor (APF).

Volume of all the atoms present in the unit cell

=volume of the unit cell

APF

3

3

42

3=

r

a

3

3

8 3=

3 4a

a

8

=3 3a

3 3

64

3

8

a

3

= 0.688

68% of the volume of the unit cell is occupied by the atoms present in the unit cell.

Now void space (or) interstitial space = 32%.

Density : Density 3

A

nM

a N and for BCC structure, n=2,

3

2

A

M

a N

Examples: Barium ,Iron and Sodium etc.

Face Centered Cubic Crystal Structure (FCC)

The face centered cubic crystal structure is shown in the figure (1)

In this FCC structure, the unit cell is a cube i.e. a=b=c=a and 090 At all the eight corners, we will have corner

atoms and in addition to them at each and every face centre,

there is a face centered atom.

The unit cell structure of FCC crystal with their atoms at the

corners and at the face centre are shown in figure (2).

Total Number of Atoms per Unit Cell (or) Effective Lattice Points

If we take any corner atom as a reference atom, it is surrounded by

eight number of unit cells. Therefore a corner Atom is shared by

eight number of surrounding unit cells.

Also every face will have a face centered Atom.

Share contributed by an Atom to each corner 1

8 of an Atom.

Total Number of corners = 8.

Contribution made by all the eight corners.

= Total corners share of each corner.

= 1

88

=1

Share of a face centered Atom to each face

1

2 Of an Atom.

Total number of faces = 6

Contribution made by all the six faces = 1

6 3Atoms3

Effective Number of atoms per unit cell of FCC 1 1

6 82 8

= 3+1

= 4 Atoms.

Fig (1) : FCC Crystal structure

– Unit cell

a

1

8

of an

Face centered atom

(½ of an atom )

Fig (2) Unit cell structure

of FCC crystal

Atomic Radius (r)

In a FCC cell there are eight corners and eight corner atoms. There will be one atom at

the centre of each face of the unit cell. The corner atoms touch the face centered atoms along the

face diagonal AC as shown in figure (3).

The atoms A and N are the nearest neighbouring Atoms.

By definition 2

ANr

OR AN = 2r and AC = 4r

From the figure triangle ABC is a right angled triangle.

2 2 2

2 2 2

2 22 , but AC=4r

AC AB BC

AC a a

AC a

2 24 2

4 2

2

4

2 2

r a

r a

r a

ar

Lattice constant (a)

Now lattice constant a = 2 2 r i.e. 4

2

ra

Coordination Number (N)

In the FCC unit cell. The nearest neighbours of any corner atom are face centered atoms.

Consider the atom at the face centre as origin; it can be observed that this face is common to two

unit cells.

If we take one corner atom as a reference atom, it is surrounded by 8 unit cells.

Here there are twelve atoms surrounding corner atom situated at a distance equal to half

the face diagonal of the unit cell.

Thus the coordination number of FCC lattice is 12

i.e. Any corner atom has Four face centered atoms in its own plane, four in a plane above it and

four in a plane below it.

Atomic Packing Factor (APF)

Let r be the radius of each Atom.

Volume of one atom 34

3r

Number of atoms per unit cell of FCC = 4

Fig (3) Touching of the corner

atoms with the face centered

atom along the face diagonal.

Volume of all the atoms per unit cell of FCC = 344

3r ---------- (1)

FCC = 344

3r ------------ (1)

Let a = side of the unit cell of FCC

Volume of the unit cell of FCC 3a ------------ (2)

Atomic packing factor is given by

Volume of all the atoms occupying the unit cellAPF =

Volume of the unit cell

3

3

44

3APF =

r

a

But 4r

a=2

APF 3

3

16

43

2

r

r

3

3

16

643

2 2

r

r

16

3r 2 2

3 644

3

2

r

2

6

=0.74

74% volume of the unit cells is occupied by the Atoms present in the unit cell.

Void space : The free space is 0.26. 26% Volume of the unit cell is free space.

Density : Density 3

A

nM

a N . For FCC structure n=4

3

4

A

M

a N

Examples : Copper, Aluminum, Lead, Gold, Silver etc.

X-RAY DIFFRACTION

X-Rays are electromagnetic waves like ordinary light. Similar to light they exhibit

interference and diffraction.

The wave length of X-rays is of the order of 0.1 nm, so that ordinary devices such as

ruled diffraction gratings don‟t produce observable effects with X-Rays.

Laue suggested that a crystal with a three dimensional array of regularly spaced atoms

could serve the purpose of a grating.

The crystal differs from the plane grating in the sense that the diffracting centers in the

crystal are lying in different planes. Hence the crystal acts as a space grating rather than a plane

grating.

Friedrich and knipping succeeded in diffracting X-rays by passing them through a thin

crystal of Zinc Blend (ZnS).

The diffraction pattern obtained consists of a central spot and a series of spots arranged in

a definite pattern around the central spot.

This symmetrical pattern of spots is known as Laue pattern and it proves that X-Rays are

electromagnetic in nature.

A simple interpretation of the diffraction pattern was given by W.L. Bragg.

According to Bragg, the spots are produced due to the reflection of some of the incident

X-Rays from the various sets of parallel crystal planes.

These sets of parallel planes contain a large number of atoms. These planes are called

Bragg’s Planes.

Bragg’s Law

When a beam of monochromatic X-Rays falls on a crystal, the X-Rays are scattered by

individual atoms present in the set of parallel planes. When path difference between two

reflected rays equal to integer multiple of incident wavelength of X-rays, constructive

interference occurs. This is called Braggs law.

Bragg assumed that the combined scattering of X-Rays from the planes can be looked

upon as a reflection from the planes.

Hence Bragg‟s scattering is known as Bragg‟s reflections and the different planes are

known as Bragg‟s planes.

Let us consider a crystal containing a set of parallel planes with interplanar spacing d.

When X-Rays of wavelength fall on these crystal planes, the atoms in the planes

diffract the X-Rays in all directions.

Let the X-Ray PQ incident at a glancing angle with the plane XY. This PQ X-Ray is

reflected by the atom at Q in the direction QR. This is shown in figure (1).

Another X-Ray ' 'P Q is reflected by the atom at 'Q in the Direction ' 'Q R .

i.e. PQR and ' ' 'P Q R are the two parallel X-Rays incident and diffracted from the atoms Q and 'Q .

X Y

d

Figure (1): Braggs Law of X-Ray Diffraction

These diffracted X-Rays will interfere constructively or destructively depending on the path

difference between the X-Rays PQR and ' ' 'P Q R .

Now for finding the path difference, normal QS and QT are drawn from Q on to ' 'P Q

and ' 'Q R .

Now the path difference = ' 'SQ QT n ------------ (1)

Here constructive interference takes place. Where n = 1, 2, 3…

If the two reflected waves are out of phase, then they produce zero intensity spots.

Now the triangle 'QTQ and 'QSQ are two similar right angled triangles.

Hence ' 'SQ QT

Also from the figure, triangle 'QTQ and 'QSQ are right angled triangles.

'' '

'sin sin

SQSQ QQ

QQ --------------- (2)

Also from the right angled triangle 'QTQ , '

'sin

QT

QQ

' ' sinQT QQ --------------------- (3)

Q

Q1

T S

P1 P R R1

Normal

Now for maximum intensity, the two reflected rays must be in phase.

Also the path difference should be equal to integer multiple of for constructive

interference.

From equations (1), (2) and (3), we get

' 'sin sinQQ QQ n

'2 sinQQ n

But 'QQ d

2 sind n --------------- (4)

This is known as Bragg‟s Law.

Here d = Interplanar spacing

= Angle of Diffraction

n = order of Diffraction

= Wavelength of incident X-Rays.

Corresponding to n=1, 2, 3… we get 1st order, 2

nd order and third order diffraction sports.

Now maximum value of sin is 1, from equation (4), we have

2

2 ,d

n dn

For any order, the wavelength shouldn‟t exceed two times the interplanar spacing.

Significance of Bragg’s Law

1. From Bragg‟s Law 2d sin n , we get 2sin

nd

Knowing the wavelength of X-Rays, angle of diffraction and order of diffraction n,

d can be calculated.

2. Knowing the value of d, the lattice constant „a‟ can be calculated for a cubic crystal, using

2 2 2

ad

h k l

3. For 1st order maximum, n=1, 1sin

2d

For 2nd

order maximum, n=2, 2

2sin

2d

For 3rd

order maximum, n=3, 3

3sin

2d

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

Here the intensity goes on decreases as the order of spectrum increases.

3. If we know lattice constant a, density and molecular weight M of the crystal. Then

number of atoms (or) molecules in the unit cell can be calculated.

4. For a cubic crystal, 2 2 2

ad

h k l

If we know the values of d and a, the value of 2 2 2h k l can be calculated.

5. Depending on the value of2 2 2h k l , we can classify the crystals as SCC, BCC, and

FCC etc.

Laue Method of X-Ray diffraction

The Laue method is one of the X-Ray diffraction techniques used for the study of crystal

structures.

The white X-Rays beam incidents on a stationary single crystal as shown in the figure (2).

Crystal is held stationary with the help of a stand. The white X-Rays are allowed to fall on

the crystal. Fine beam of X-Rays can be obtained by passing the X-Rays through pin holes of

lead diaphragms. These X-Rays of fine beam now allowed to fall on the crystal.

The crystal planes in the crystal diffract the X-Rays satisfying Bragg‟s Law. The diffracted

X-Rays are allowed to fall on the photographic plate.

The diffraction pattern consists of a series of bright spots corresponding to interference

maximum for a set of crystal planes satisfying the Bragg‟s law 2 d sin = n for a particular

wavelength of incident X-Rays.

A set of planes diffracts a particular wavelength which satisfies the Bragg‟s law for particular

dhkl values.Here we get a number of diffracted beams, each corresponding to a set of planes

for a particular wavelength.

Laue pattern can be obtained either by transmission method or back reflection method. Both the

methods are shown in figure (2) and figure (3).

Figure (2) Laue Transmission method of X-Ray Diffraction

Crystal

Lead of diaphragms

Pin hole

White X-Ray Beam

Photographic plate

Laue spot

Figure (3): Laue Back reflection method of X-Ray diffraction

Here the wavelength of X-Rays = 0.20

2 A . The dimensions of the crystal are usually less

than 1mm. The distribution of spots on the photographic plate depends on the symmetry of

the crystal and its orientation with respect to X-Ray beam.

In the case of transmission method, the spots lie on Ellipses. The Bragg angle for any

transmission Laue spot is given by

12r

TanD

----- (1)

Where 1r = Distance between the centre of the film and the diffraction spot.

D= Distance between the specimen and the film or photographic plate.

D is usually 5 cm.

In the case of back reflection method, the spots lie on Hyperbolas. The Bragg angle for any

spot on a back reflection pattern is given by 2(180 2 )r

Tand

---- (2).

Where 2r = Distance between the centre of the photographic plate and the diffraction spot.

D= Distance between the specimen and photographic plate.

D is usually 3 cm.

Back reflections give weak spots. Here the photographic plate requires longer exposure.

The Laue spots near the centre of the photographic plate correspond to the 1st order reflections

from the planes inclined at very small angles to the incident beam. In this case only lower

wavelengths can satisfy the Bragg condition.

White X-Ray Beam

Photographic plate

Lead diaphragms Laue spot

Crystal

The Laue photograph for a simple cubic crystal is shown in figure (4).

Figure (4) Laue photograph for a simple cubic crystal

Merits

1. The Laue method is suitable for the rapid determination of Crystal Structures.

2. This method is also useful for studying the crystal defects under mechanical and thermal

treatment.

Demerits

1. The Interpretation of Laue photograph requires the concept of reciprocal lattice, which is

cumbersome.

2. This method is not convenient for the study of actual crystal structure ,because X-Rays

are diffracted in different order from the same plane and they superimpose as a single

Laue spot.

POWDER METHOD OF X-RAY DIFFRACTION

Many materials that are used in industry are polycrystalline materials. They are used in

powder form. The powder patterns are simple and are used for the characterization of crystals.

In the Laue method, we get spots. The powder photograph of X-Ray diffraction consists

of arcs.

The powder method is an X-Ray diffraction technique used to study the structure of

crystals in the form of powder. This method gives information regarding the size and orientations

of the crystallites in the powder.

This method is also known as Debye and Scherrer method. In this method a cylindrical

camera will be employed. The cross sectional view of the camera is shown in figure (1).

X-Xrays

Filter

Figure. (1) Debye- Sherrer cylindrical camera

The inner curvature of the camera is mounted with a photographic film. The width of the film is

1 inch.

The polycrystalline material (or) the given crystal which is made into fine powder and is stacked

into a capillary tube.

The material of the capillary tube should not diffract X-Rays.

The diameter of the capillary tube is around 0.3 to 0.5 mm. The powder will have a

particle size equal to 1-3 m .

The sample of powder can also be arranged on a thin wire and bound with a small

quantity of binding material.

The binding material and thin wire should not diffract X-Ray beam.

Now the sample is placed exactly at the centre of the cylindrical camera.

The camera contains a filter and diaphragms containing pin holes.

The filter allows only one wave length of X-Rays. The other wavelength will be cut off.

P

Photographic film

Cones of diffracted

X-Rays

Luminescent screen

P= Powder sample Lead diaphragms

Monochromatic

X-Rays

The X-Rays are allowed to pass through the filter such that monochromatic X-Rays are

allowed to pass through the pinholes present in lead diaphragms. Now the emergent beam is a

fine narrow and sharp X-Ray beam.

Usually the powder is prepared by crushing polycrystalline material. Now the powder

consists of crystallites.

These crystallites are randomly oriented so that they make all possible angles with the

incident X-Rays.

Thus all orders of reflections from all possible atomic planes are recorded at the same

time.

The fine beam after passing through the entry hole falls on the capillary tube P containing

the powdered crystals.

The powder consists of randomly oriented particles, all possible and d values are

available for diffraction of incident X-Rays.

The diffraction takes place for those values of and d, satisfying the Bragg‟s law of

diffraction 2 sind n . For a particular value of glancing angle , different orientation of a

particular set of planes is possible.

The diffracted X-Rays corresponding to certain values of and d lie on the surface of a cone

with its apex at the sample p. The semi vertical angle of the cone is 2 .

Different cones are observed for different sets of and d for a particular order n.

Different cones of X-Ray diffraction are also formed for different combinations of and n for a

particular value of d.

The powder method is shown in figure (2).

Figure (2): Powder photograph Method

The transmitted X-Rays come out through the exit hole.

The diffracted cones make impression on the film in the form of arcs on either side of the exit

and entry holes. Here the centers of the arc coincide with the holes.

S P

Mono chromatic X-

Ray Beam Pin hole

X-Rays

Lead diaphragms Entry hole Exit hole

X-Rays

2 2

Filter

Film

The diffraction pattern is shown in the figure (3).

Figure (3): Diffraction pattern

The angle corresponding to a particular pair of arcs in relation to the distance„s‟ between the

pair of arcs is given by

S Arc

4θ Radians = Angle = R Radius

Here R = Radius of the cylindrical camera.

S = Distance between a pair of Arcs.

Now 4 θ (Degrees) = S 180

R

4 θ (Degrees) = 57.3 S

R

Where is the angle of diffraction.

From the above expression can be calculated. Now the interplanar spacing for first order

diffraction is given by

2sind

By knowing all the parameters, the crystal structure can be studied.

Entry hole Exit hole

Arcs on the film

ULTRASONICS

Introduction to Ultrasonics

(1) The word ultrasonic combines the Latin roots ultra, meaning ‘beyond’ and

sonic, or sound.

(2) The sound waves having frequencies above the audible range i.e. above

20000Hz are called ultrasonic waves.

(3) Generally these waves are known as high frequency waves.

(4) The field of ultrasonics has applications for imaging, detection and

navigation.

(5) The broad sectors of society that regularly apply ultrasonic technology are

the medical community, industry, and the military and private citizens.

Properties ultrasonic waves

(1) They have high energy content. Since E= hυ and frequency is very high.

(2) Just like ordinary sound waves, ultrasonic waves get reflected, refracted and

absorbed.

(3) They can be transmitted over large distances with no appreciable loss of

energy.

(4) If an arrangement is made to form stationary waves of ultrasonic‟s in a liquid,

it serves as a diffraction grating. It is called an acoustic grating.

(5) They produce intense heating effect when passed through a substance.

(6) When they travel in a medium, the particles of the medium will have two

modes of vibrations.

They are longitudinal mode of vibration and transverse mode of vibration.

(7) Velocity of Ultrasonic waves is constant in a homogeneous medium.

Production Of ultrasonics

Ultrasonic waves are produced by the following methods.

(1) Magneto striction generator or oscillator

(2) Piezo electric generator or oscillator

Piezo electric method

Principle: Inverse piezo electric effect

Piezo electric effect:

If mechanical pressure is applied to one pair of opposite faces of certain crystals

like quartz, equal and opposite electrical charges appear across its other faces. This

is called piezo electric effect. The converse of piezo electric effect is also true.

Inverse Piezo electric effect:

If an electric field is applied to one pair of parallel faces, the corresponding

changes in the dimensions of the other pair of parallel faces of the crystal are

produced. This is known as inverse piezo electric effect or electrostriction.

The circuit diagram is shown in Figure (1)

Fig(1)Cicuit diagram

Piezo electric oscillator

The quartz crystal is placed between two metal plates A and B.The plates are

connected to the primary (L3) of a transformer which is inductively coupled to the

electronics oscillator.

The electronic oscillator circuit is a base tuned oscillator circuit. The coils L1 and

L2 of oscillator circuit are taken from the secondary of a transformer T.

The collector coil L2 is inductively coupled to base coil L1. The coil L1 and

variable capacitor C1 form the tank circuit of the oscillator.

Working

When H.T. battery is switched on, the oscillator produces high frequency

alternating voltages with a frequency

Due to the transformer action, an oscillatory e.m.f. is induced in the coil L3. This

high frequency alternating voltages are fed on the plates A and B.

Inverse piezo electric effect takes place and the crystal contracts and expands

alternatively. The crystal is set into mechanical vibrations.

The frequency of the vibration is given by

n = 2

P Y

l

Where P = 1,2,3,4 … etc. For fundamental, first over tone, second over tone

etc.

Y = Young‟s modulus of the crystal and

ρ = density of the crystal.

The variable condenser C1 is adjusted such that the frequency of the applied AC

voltage is equal to the natural frequency of the quartz crystal, and thus resonance

takes place.

The vibrating crystal produces longitudinal ultrasonic waves of large amplitude.

Advantages

(1) Ultrasonic frequencies as high as 5 x108 or 500 MHz can be obtained with this

arrangement.

(2) The output of this oscillator is very high.

(3) It is not affected by temperature and humidity.

Disadvantages

(1) The cost of piezo electric quartz is very high

(2) The cutting and shaping of quartz crystal are very complex.

Note: (1) Very large electric fields are needed to produce very small strains.

(2)An electric field of 104 Vm-1 produces a strain of about 1 in 108 only.

1 1

1

2f

LC

Applications of Ultrasonic Waves in Engineering

1) Detection of flaws in metals (Non Destructive Testing –NDT)

Ultrasonic waves are used to detect the presence of flaws or defects in the form

of air cracks, cracks, blowholes porosity etc., in the internal structure of a

material

By sending out ultrasonic beam and by measuring the time interval of the

reflected beam, flaws in the metal block can be determined.

(2) Ultrasonic Drilling

Ultrasonics are used for making holes in very hard materials like glass, diamond

etc.

(3) Ultrasonic welding

• The properties of some metals change on heating and therefore, such metals

cannot be welded by electric or gas welding.

• In such cases, the metallic sheets are welded together at room temperature by

using ultrasonic waves. This is called cold welding.

• (4) Ultrasonic soldering

• Metals like aluminum cannot be directly soldered. However it is possible to

solder such metals by ultrasonic waves.

• (5) Ultrasonic cutting and machining

• Ultrasonic waves are used for cutting

• (6) Ultrasonic cleaning

• It is the most cheap technique employed for cleaning various parts of the

machine, electronic assembles, armatures, watches etc., which cannot be

easily cleaned by other methods.

• (7) SONAR

• SONAR is a technique which stands for Sound Navigation and Ranging.

• It uses ultrasonics for the detection and identification of underwater objects.

• The method consists of sending a powerful beam of ultrasonics in the

suspected direction in water.

• By noting the time interval between the emission and receipt of beam after

reflection, the distance of the object can be easily calculated.

• The change in frequency of the echo signal due to the Doppler Effect helps to

determine the velocity of the body and its direction.

Measuring the time interval (t) between the transmitted pulses and the

received pulse, distance between the transmitter and the remote object is

determined using the formula,

2

v td

Where v is the velocity of sound in sea water.

The same principle is used to find the depth of the sea.

Applications of SONAR

• Sonar is used in the location of shipwrecks and submarines on the bottom of

the sea.

• It is used for fish-finding application.

• It is used for seismic survey. Measuring Earths properties by using magnetic,

electric, gravitational, thermal and elastic theories.

• Obstetric ultrasound is primarily used to:

• Date the pregnancy

• Check the location of the placenta

• Check for the number of fetuses

• Check for physical abnormities

• Check the sex of the baby

• Check for fetal movement, breathing, and heartbeat.

Ultrasonics in Research

• Scientists often use in research, for instant to break up high molecular weight

polymers, thus creating new plastic materials.

• Ultrasound is used to determine the molecular weight of liquid polymers, and

to conduct other forms of investigation on the physical properties of materials

such as density and viscosity.

• Ultrasonics can also speed up certain chemical reactions. Hence it has gained

application in agriculture, that seeds subjected to ultrasound may germinate

more rapidly and produce higher yields.

Piezo electric Effect:

If mechanical pressure is applied to one pair of opposite faces of certain crystals

like quartz, equal and opposite electrical charges appear across its other faces. This

is called as piezo electric effect. The converse of piezo electric effect is also true

Explanation: As shown in the figure a mechanical pressure is applied on pair of

Faces. Mechanical pressure is defined as force per unit area. Now it said to be stressed.

Now a change in dimensions of the crystal takes place. Due to this electronic

structure of the crystal changes. As a result of this opposite charges will be

induced on pair of other faces. This phenomenon is called Piezoelectric effect.

Inverse Piezo electric effect:

If an electric field is applied to one pair of faces, the corresponding changes in

the dimensions of the other pair of faces of the crystal are produced. This is

known as inverse piezo electric effect or electrostriction.

Explanation: As shown in the figure an electric field is applied on a pair of

opposite faces of the Quartz crystal. Due to this electronic structure of the

crystal changes.

Here the crystal is strained and there is a stress in the crystal. Due to this

ultrasonic waves are generated.

UNIT-III

QUANTUM MECHANICS

And

SEMICONDUCTORS

QUANTUM MECHANICS

The physical concept of a particle is characterized by mass and velocity.

Experiments dealing with particles, usually interpreted in terms of mass and

Velocity by using Newton‟s Laws of Motion. This classical approach is not

sufficient to describe some experiments. The results of some of the experiments are

contrary to Newton‟s Laws

The experiments like Photo Electric Effect, Black Body Radiation and

Compton Effect confirmed the particle nature of Photons.

Interference, diffraction and Polarization of light confirmed that light is

having wave nature.

Louis De Broglie extended the idea of dual nature of radiation to matter.

According to De Broglie matter possesses wave as well as particle characteristics.

The concept of dual nature of radiation can be understood by knowing relationship

between particle as well as the wave and their characteristics.

Waves and Particles:

PARTICLES WAVES

1. A particle occupies space

2. A particle will have a definite mass

3. The particle will have position

4. Due to change in position of the

particle, it will have velocity.

5. Due to Mass and Velocity , the

particle posses momentum

Momentum P= m X v

6. A Particle will have Energy

1. The transmission of disturbance

from one point to other point in a

material medium is known as

Wave.

2. A Wave will have amplitude

3. It will have time period

4. It will have frequency

5. It will have wave length

6. It will have phase

7. It will have intensity 2AI

The particle and wave nature can be explained by using Planck‟s Quantum Theory.

According to this theory emission of radiation is in the form of photons. A photon

will have velocity of light and mass which is in motion. i.e., it will have both

momentum and energy. Thus a photon behaves like a particle. The energy of a

photon is given by

nhE Where n=1, 2, 3… i.e., the energy of the photon is quantized.

h= Planck‟s constant, Frequency of radiation

Therefore in addition to frequency, the other parameters attribute wave nature to a

photon. i.e., a photon will have dual nature.

The De Broglie Hypothesis:

The dual nature of light possessing both wave and particle properties was

explained by combining plank‟s expression for the energy and Einstein‟s Mass –

Energy relation.

The Energy of a photon according to Plank is given by

E = h ----------------- (1)

Einstein‟s Mass – Energy relation is given by

2mcE ------------------ (2)

Here h= Plank‟s constant

= Frequency of Radiation

m = Mass of Photon

c = Velocity of light

From equations (1) & (2)

2h mc ------------------ (3)

Also the velocity of light is given by

c = λ

=

c ------------------------ (4)

From (3) & (4) we have

2mchc

mc

h

p

h ------------------------------- (5)

Where = wave length of the Photon

p = Momentum of Photon

Using this concept, De Broglie proposed the concept of matter waves.

According to this the material particle of mass „m‟ moving with a velocity

„v‟ should have an associated wave length .

This wave length is called the De Broglie wave length.

Now = momentum

h

= mv

h =

p

h ------------------ (6)

Where h = Plank‟s constant, p=momentum

Equation (6) is known as De Broglie wave equation and is called De Broglie

wave length.

If the particle is moving with a velocity comparable to the velocity of light

then the mass of the particle is always changes. The mass „m‟according to theory

of relativity is not an invariable entity as in Newtonian Physics.The relativistic

mass „m „ is given by Here 0m rest mass of the electron

2

2

0

1c

v

mm

c = Velocity of Light

From equation (6) it is found that if the particles are accelerated to various

velocities, we can produce waves of various wave lengths.

Higher the electron velocity, smaller the De Broglie wave length and vice versa.

Ralation between DeBroglie Wavelength and KiniticEnergy of the particle

Let us assume m is the mass of the particle. Now the particle is moving with

velocity „v‟

Kinetic Energy of the particle 2

2

1mvE

m

pE

2

2

mEp 22

mEp 2 ---------------------------- (1)

But according to De Broglie hypothesis p

h ----------------------- (2)

From (1) & (2) mE

h

2 -----------------------------(3)

Where h = Plank‟s constant

m = Mass of the particle and E = Kinetic energy of the particle

Relation between de Broglie wave length and the applied potential difference:

(De Broglie wave length of electrons)

Let m be the mass of the electron. This electron is applied with a potential

difference of V volt.

Here the work done (energy) on the electron is given by eV.

Here e = charge on the electron

V= applied potential difference in volts.

The work done is converted into Kinetic Energy of the electron.

i.e. eVmv 2

2

1

Here v = velocity acquired by the electron now

eVm

p

2

2

meVp 22

meVp 2 ---------------------------- (1)

Now the De Broglie wave length associated with the electron is given by

p

h ---------------------------- (2)

From (1) & (2) meV

h

2 ----------------------------(3)

Ignoring the relativistic considerations, m = rest mass of the electron

eVm

h

02

Here m=mass of an electron

v=linear Velocity of Electron

r=Radius of the orbit

form (1)&(2) ,now we have

L = mvr = 2

nh………(3)

mv

nhr 2

p

nhr 2

…………..( 4)

mvp , momentum of electron.

34 10

31 19

6.625 10 12.26 10

2 9.1 10 1.602 10

X X

VX X X X V

meter

026.12

AV

V = Applied Voltage in Volts

Matter Waves:

According to Debroglie concept that a moving particle is associated with

wave nature. This can be explained by Bohr‟s atomic model.

The angular momentum (L) of a moving electron in an Atomic orbit of radius „r‟ is

quantized.

i.e. L=2

nh ………………………………(1)

n=1, 2, 3, 4……………

Now Angular momentum L=m v r ………... (2)

In the equation (4), r2 is the circumference

length of the orbit in which the electron is revolving.

Figure (1) Bohr’s orbit and deBroglie

Waves of an electron in the orbit

This circumference is equal to the „n‟ times the wave length of the associated wave

of a moving electron in the orbit.

i.e nr 2 …………………………. (5)

This is shown diagrammatically for n=4 and n=6 in figure (1).

According to the deBroglie, a moving particle will have both particle and wave

nature. The waves associated with a moving material particle are called matter

waves or deBroglie waves. The deBroglie waves are associated with materialistic

particles such as electrons, protons, neutrons etc.

Properties of Matter Waves:

1. DeBroglie waves are not electromagnetic waves.

They are called pilot waves.

The waves that guide the particles are called matter waves or pilot waves.

2. Matter waves consist of a group of waves or a wave packet associated with a

particle. The group has the velocity of particle.

3. Each wave of the group travel with a velocity known as phase velocity given

by Vph =K

where = Angular frequency, K = Wave vector or wave

Number.

4. These waves cannot be observed.

5. The wave length of matter waves is given by

p

h h = Planck‟s constant, p = momentum of the particle

m

h

6. Lighter the particle, greater will be the wave length associated with it.

7. Smaller the velocity of the particle, longer will be the wave length.

8. When ,0V . Also if 0, V

9. Matter waves can be produced whenever the particles in motion are charged

or uncharged.

10. Matter waves travel faster than velocity of light.

11. The wave nature of the matter introduces uncertainty in the location of the

position of the particle.

HeisenBerg’s Uncertainty Principle:

Usually the moving particle must be regarded as a deBroglie wave group rather

than a localized particle.

This suggests that there is a fundamental limit to the accuracy with which we can

measure its particle properties.

According to classical Mechanics, a moving particle at any instant has a fixed

position in space and a definite momentum which can be determined

simultaneously with accuracy.

But we know that a moving particle is similar to a wave, we cannot determine the

position and momentum simultaneously, accurately.

The measurement of position and moment of a moving particle is impossible

Let x denotes the error in the measurements of the position of the particle along

x-axis and p represents the error in the measurement of momentum, then

2

))((h

px Here h=Planks Constant.

If we locate the particle exactly 0x only at the expense of imparting to it

an infinite momentum p

The uncertainty principle can also be written as 2

))((h

tE

Applications:

1. It explains the absence of electrons in the nucleus.

2. It gives proof for the existence of protons and neutrons inside the nucleus

3. Explains uncertainty in the frequency of highest emitted radiation by an

Atom

4. It explains Energy of an electron in an Atom

Differences between Matter waves and Electromagnetic Waves:

Matter Waves Electromagnetic waves

1. These waves are associated with the

moving particles.

2.wavelentgth depends upon mass of the

particle mv

h

P

h

3. Can travel with a velocity greater

than velocity of light.

4. These waves are not electromagnetic

waves.

1.Oscillating charged particles gives

electromagnetic

radiation(electromagnetic waves)

2. Wavelength depends upon the energy

of the photon.

hE ,

cc

hcE ,,

E

hc

3. These waves travel with with a

velocity of light.

C= sm /103 8

4.In this wave electric and magnetic

fields oscillate perpendicular to each

other.

Note on Simple Harmonic Motion :

If a particle executing simple Harmonic motion, then its motion is periodic,

acceleration is directed towards an equilibrium point and acceleration is

proportional to displacement. ( a -x)

The general equation of motion for SHM is given by

y = A Sin (t - ) ……………………………( 1 )

Here y = displacement of the particle executing simple harmonic motion

A = Amplitude of the particle executing simple harmonic motion

ω = Angular frequency

Φ = Phase difference

Now Phase difference =

2 X Path difference

x

2

……………………………………… (2)

From equations (1) & (2) we get

)2

( xtASiny

)2

2( xtASiny

)(2

x

tASiny

Since we have to solve problems by Schrödinger‟s time independent Wave

Equation, we choose wave equation involving no time.

2 ( )x

y ASin

)(2

x

ASiny

)(2

x

ASiny .

Schrodinger’s Wave equation

Let us consider a particle of mass m, moving with a velocity along the positive

X-direction.

The wave function for a particle moving freely in the positive x-direction

has the same form as the wave equation for simple harmonic motion and simple

harmonic waves in the positive x-direction.

)(2

x

ASin …………………….(1)

Here is a function of x only.

Differentiating equation (1) with respect to x once and two times, we get

xA

dx

d 2cos

2

Again differentiating, we get

xA

dx

d 2sin

42

2

2

2

.

xA

dx

d 2sin

42

2

2

2

But

xA

2sin

2

2

2

2 4

dx

d ……………………………..(2)

DeBroglie wavelength associated with the particle is

mv

h

h

m

1

2

2

2

22 )2/1(21

2 h

mvm

h

vm

………………….(3)

Let E be the total energy of the particle and V be the potential energy of the

particle and T be the kinetic energy.

Then total energy ,E=T+V

T, KE = 2

2

1mv = E – V ……………………………….(4)

Substituting the above value of K.E. in Equation (3), we get

2

)(212 h

VEm

……………………………(5)

From equations (5) and (2), we get

)(8 2

22

2

VEh

m

dx

d

0)(8

2

2

2

2

VEh

m

dx

d

0)(

4

2

2

22

2

VE

h

m

dx

d

0)(

)2

(

2

22

2

VE

h

m

dx

d

0)(2

22

2

VEm

dx

d

( )

2(

h ) ………………….(6)

This is the Schrödinger‟s time independent one dimensional wave equation.

Wave number:

In Spectroscopy In wave mechanics

wave Number of an Electromagnetic

wave is given by

1K , But

p

h

h

P

P

hK

)(

1

12 mh

mEK

For the special case of an

electromagnetic wave,

2K .

hC

hE

E

hC

hC

EK

2

C

EK

Wave velocity:

Wave velocity is defined as the velocity with which a particular

crest or trough or a particular phase of a wave advances in a medium.

The wave Velocity u of the matter waves can be obtained from the energy of

photon.

Now hE ……………………………….(1)

Or frequency, h

Xm

p

h

mv

h

E 1

2

)( 222

1

Multiplying and dividing the numerator and the denominator by h, we get

Frequency, 22

2 1

22 X

m

h

h

pX

m

h ……………..(2)

There fore, the wave velocity,

u = frequency X wave length

u

XXm

h2

1

2

The wave velocity of the electron

m

hu

2 ……………………………(3)

Physical Significance of Wave Function

The wave function is a Complex function. This does not have a direct

physical meaning. The square of its absolute magnitude 2

can be taken as

definite meaning by considering the case of an electromagnetic wave.

The intensity of a light wave is proportional to the square of the amplitude.

(2AI )

2 Is the probability density of the particle associated with the deBroglie wave

described by the wave function .

That is the probability of finding a particle is proportional to2

at the point x,

and at any instant of time t.

The wave function is given by

( )x a ib , * is its complex conjugate,

* = iba

Now * = ( iba ) ( iba ).

* = 22 ba ,

* is denoted by P

2 is called the probability density.

The probability of finding a particle is real.

The probability of a particle being present in a volume dx dy dz is 2

dx dy dz.

The total probability of finding the particle somewhere is unity.

Since the probability of finding a particle somewhere in the space is certain.

12

dxdydz

Or 1*dxdydz

The triple integral extends overall possible values of x, y and z.

A wave function satisfying the above relation is known as normalized wave

function.

Particle in a One-Dimensional Potential Box:

(OR Electron in a Potential Well): Consider an electron of mass „m‟ this is bound to move in a one dimensional

crystal of length L.

The electron is prevented from leaving the crystal by the presence of a large

potential energy barrier at its surface.

Though the barriers extend over a few atomic layers near the surface, these are

taken infinitely large for the sake of simplicity. The problem is similar to that of

an electron moving in a one-dimensional potential Box.

This is represented by a line and is bounded by infinite potential energy as shown

in figure (2).

Fig (2) Electron in a one dimensional

Potential well.

The potential energy within the crystal or box is assumed to be zero

Thus we have

V(x) = 0 for 0<x<L ………………… (1)

V(x) = for Landxx 0

The wave function n of the electron occupying the nth

state is given by

0)(2

22

2

nnn VE

m

dx

d

…………… (2)

Here also En= Total energy of the electron in the nth state.

V= Potential energy.

Inside the box, v=o

02

22

2

nnn E

m

dx

d

-------------------------- (3)

02

2

2

nn k

dx

d

------------------------------- (4)

Where 2

2

2n

mk E

nEmk

2 --------------(5)

Equation (4) is a differential equation. The general solution of the equation (4) is

given by

BCoskxASinkxxn )( ……………………………….. (6)

In equation (6), A and B are arbitrary constants,

These constants are to be determined from the boundary conditions.

Since the electron is constrained by infinitely high potential barriers at x=0 and

x=L, v . We assume that (0) 0 ( ) 0n nand L

The product V(x) n (x) in equation (2) also approaches infinity.

Thus in order that the wave function n (x) may be continuous, the kinetic energy

must also become infinite which is not feasible.

Hence n (x) must vanish for x=0 and x=L.

For x=0 equation (6) gives B=0

0=A SinK (0) +B CosK (0)

0=A (0) +B (1)

B=0

Now equation (6) becomes

n (x)=A SinK(x) -------------------------------- (8)

Also since n (L) =0, equation (8) becomes

A SinKL=0

0A But SinKL=0 nKL

Or K=L

n …………………………………….(9)

Where n=1, 2, 3…

Thus the expression for the allowed wave function becomes.

n (x)=A Sin (L

n ) x ……………………. (10)

Eigen Energy Values:

The allowed energy values can be obtained from equations (5) & (9) as

nEmk

2

L

nEmk

n

2

2

2

2

L

nEm n

22

2

L

n

mEn

2

222

2 L

n

mEn

2

22

2

2

24 L

n

m

hEn

2

22

8mL

hnEn …………………… (11)

i.e En n2

Some Features:

1. The lowest energy of the particle is given putting n=1

2

2

18mL

hE

1

2EnEn

This known as zero point energy.

2. For n=1, 2, 3…We get discrete energy values of the particle in the one

dimensional box.

2

2

18mL

hE

12

22

2 48

2 EmL

hE

12

22

3 98

3 EmL

hE

…………………….

3. It is apparent from equations (10) and (11) that the allowed wave functions

n (x) and the allowed energy values nE exist only for integral values of n.

The number n is called quantum number.

4. The spacing between the nth energy level and next highest energy level

[n+1] th level is given by

1

2

2

22

1 )1(8

)1(En

mL

hnEn

1

2

2

22

8En

mL

hnEn

1

2

1

2

1 )1( EnEnEE nn

1

2

1

2

1 )21( EnEnnEE nn

11 )12( EnEE nn

5. The energy spectrum consists of discrete energy levels. The spacing between

the levels is determined by the values of n and L.

The Spacing decreases with increase in L.

If L is of the order of a few centimeters, the energy lever from almost a continuum.

But if L has atomic dimensions, the spacing between the levels becomes

appreciable.

The energy levels corresponding to n=1, 2, 3 and 4 are shown in fig (3).

E4 n=4

E3 n=3

E2 n=2

E1 n=1

E0 n=0

Fig (3) First four energy levels of an electron in a one dimensional Box.

Determination of constant A in n (x)=A Sin (L

n )x (Normalization of the

wave function):

The constant A in n (x) =A Sin (L

n )x is determined by using the condition that

the probability of finding an electron some where on the line is unity

1)()(..0

* dxxxei n

L

n

1|)(|0

2 dxx

L

n

10

22

dxx

L

nSinA

L

10

22

dxx

L

nSinA

L

12

21

0

2

dx

xL

nCos

A

L

12

220 0

22

dxx

L

nCos

Adx

AL L

Now 02

0

dxx

L

nCos

L

12

0

2

dxA

L

12

][2

2

0

2

LA

xA L

L

A22

LA

2

Now from the equation xL

nSinAxn

)(

xL

nSin

Lxn

2)( …............ (1)

This is the normalized wave function. The first four wave functions and the wave functions of the

electron in a one dimensional Box are shown in the figure (4)

Fig (4) First three wave functions Fig.(5) the probability density of

of an electron in a one an electron in a one dimensional

dimensional box

The probability density of the particle in the one dimensional Box.

The probability of finding the particle in a small length dx along x is given

bydxAdxxP nn

2||)(

dxxPn )( dxxL

nSin

L

22

Also Probability Density xL

nSin

LxPn

22)(

This is maximum when ,2

5,

2

3,

2

x

L

n

,2

5,

2

3,

2 n

L

n

L

n

Lxor

For n=1, the most probable positions of the particle is at x=2

L

For n=2, the most probable positions are at x=4

L and

4

3L

The probability density of the particle in the one dimensional Box is shown for

various values of „n‟ in the figure (5).

Fermi-Dirac Distribution:

According to free electron theory, Electrons in a solid move in all possible

directions like gas molecules in a container. These free electrons contribute for

electrical conduction.

The free electron model of a metal has survived to the actual situation in

metals, particularly the monovalent atoms such as Alkali metals.

Quantum mechanics requires that all valence or free electrons should be specified

by the three quantum numbers nx , ny , nz together with the spin.

The spin can have either 2

1 or

2

1

The Pauli Exclusion Principle does not permit more than one electron to have same

four quantum numbers.

Many of the occupied states in a metal containing 1023

free electrons must be

described with fairly large quantum numbers.

Now it is most convenient to discuss the metallic state with statistical mechanics.

The probability that a particular quantum state having an energy E is occupied is

given by Fermi-Dirac function.

TK

EfEEf

B

exp1

1)(

Here f (E) is called probability of occupying a state

EF is the energy of the Fermi level.

E is the energy of the state in which the electron is occupied at T K0 .

BK is the Boltzmann constant.

Properties:

1. The Fermi-Dirac function also valid for semiconductors. In a

semiconductor, the probability of occupancy of states by electrons is given

by the F-D distribution function.

TK

EfEEfEP

B

e

exp1

1)()(

2. The distribution function is valid only in equilibrium.

3. The Fermi level is absolutely valid in equilibrium only.

4. Fermi-Dirac distribution function is valid for all the particles obeying

Pauli‟s exclusion principle. This is equally applicable regardless of the type

of the solid, doping of the semiconductor, etc.

Any particle obeying F-D distribution function is called Fermions.

The Fermi-Dirac distribution function considers statistically the entire collection of

fermions in the volume.

Thus it considers all electrons in the semi conducting solid and not merely

electrons in a Band.

5. An empty electron state is called a HOLE. The Fermi-Dirac distribution

function for holes in the solid would correspond to the statistical distribution

of vacant sites.

The hole distribution function is denoted as

Ph (E) =1-PFD (E)

TK

EEEP

B

f

h

exp1

1)(

6. AT E = Ef,

2

1

11

1)()(

EPEP eh

i.e. the probability of occupancy of the electron or hole is 2

1

This also gives a definition for the Fermi level.

7. Fermi level is the energy level where the probability of occupation is 2

1

8. At 00 K, Pe (E) =1 for E E f

And Pe (E) = 0 for E > E f

This implies that at 00 K all states up to the Fermi level are completely occupied by

the electrons. All the states above the Fermi level are empty.

9. The distribution function is a strong function of temperature only at energies

close to Ef.

Plots of Pe (E) and E at different temperatures are shown in figure (6).

Y T

T2>T1>T in 0 K

T1

P(E)

T2

1/2

X

O E Ef

Fig (6). Pe (E) versus E for various T values. At all temperatures, the curves

passes through the point 1

,2

fE

.

Note on Fermi-Dirac Distribution function: The Fermi-Dirac distribution for

Electrons is given by

TK

EEEfEP

B

F

e

exp1

1)()(

Now at the Absolute zero (T = 00K ), there are two situations

(i) For E < EF ,

TK

EEEP

B

F

e

exp1

1)(

)0(1

1)(

BK

Xe

e

EP

eEPe

1

1)(

e

EPe 11

1)( But 0

1

e

Pe (E) = 1, for E < EF at T = 00

(ii) For E > EF

TK

EEEP

B

F

e

exp1

1)(

)(exp1

1)(

oK

XEP

B

e

eEPe

1

1)(

1

1)(EPe

1)(EPe

0)( EPe

This means that no electrons have energy greater than EF at 00 K.

i.e the Fermi energy EF is the maximum energy that a free electron in the metal can

have at absolute zero.

SEMICONDUCTORS

Semiconductors and classified basing on their conductivities and resistivity‟s.

Electrical resistivity of semi conductors lies in between those of conductors and resistors.

In semiconductors, there are two types of carriers namely semi conductors and holes.

Hence semiconductors are bipolar.

The current in semiconductors is due to two types of carriers namely electros and holes.

Pure semi conductors are known as intrinsic semiconductors.

Example : Silicon and Germanium.

The electrical conductivity can be enhanced by a process called doping. i.e. the number of

carriers can be increased by a process called doping. Doping is the process of adding an

impurity to a pure semi conductor. By adding a suitable impurity to an intrinsic semi conductors,

it will become an extrinsic semi conductor. The transportation of charge carriers (movement)

takes place due to drift and diffusion.

The extrinsic semi conductors are widely used in solid state electronic devices and semi

conductor electronic devices.

To study electronic devices, it is important to study the fundamental electronic transportation

properties in semi conductors.

Intrinsic semi conductors

Usually pure semi conductors are known as intrinsic semi conductors. Examples are

Silicon (Si) and Germanium (Ge) Silicon (Si) and Germanium belongs to IV group of periodic

table.

Atomic Number of Silicon is 14.

Electronic configuration 1s2 2s

2 2p

6 3s

2 3p

2

Atomic number of Germanium is 32.

Atomic number of Germanium is 32.

Electronic configuration is

1s2 2s

2 2p

6 3s

2 3p

2 3p

6 3d

10 4s

2 4p

2

In Silicon and Germanium, there are four valence electrons. Bonding in these semiconductors is

covalent bonding.

Each silicon Atom forms four covalent bonds with the surrounding from neighboring Silicon

atoms in the silicon Semiconductor crystal.

Here no electrons are available freely for conduction and the semi conductor acts like an

insulator.

The conduction process can be understood with the help of energy bond diagram.

In the energy bond diagram, we have conduction bond and valence bond.

The conduction bond and valence bond are separated by a forbidden energy gap Eg., known as

energy and gap. The bond representation and the energy band structure is shown in figure (1) at

O0K.

At O0K, all valence electrons are tightly bound to their atoms and are taking part in covalent

bond formation.

For Silicon Eg = 1.12 ev.

Germanium Eg = 0.69 ev.

In the figure (1) a Ec= Energy level corresponding to Bottom of the conduction bond.

Ev = Energy of the energy level corresponding to the top of the

Valence bond.

Ef = Fermi energy level.

At O0K, the valence bond is completely filled and the conduction bond is empty.

At O0K, the semi conductors behave like insulators.

Above O0K (i.e. At Room temperature), the valence electrons acquire sufficient amount of

thermal energy. Due to this they break the covalent bonds and make themselves available as free

electrons. Against to creation are free electron, a vacancy is created in its initial position in the

crystal structure. This vacancy is known as a hole.

The hole is a virtual positive charge, having the magnitude of charge of the electron.

The free electrons after acquiring sufficient thermal energy, and closes the energy gap.

These electrons will enter into the conduction band from valence bond and occupy

energy levels in the conduction bond.

The electrons leaving the valence bond create holes in its original place.

Now the valence bond will have holes and the conduction bond contains electrons.

The crystal structure and energy bond structure above O0K is shown in figure (2).

Si

Si

Si

Si

Si

Si

Si

Si

Empty

EC

EF

EV

Egg

Conduction bond

Valence bond

(filled)

Fig. (1)a: Intrinsic silicon at O0k –

Two dimensional Representation.

Fig (1)b: Energy band structure of Intrinsic

Semiconductor Silicon at O0k.

In an intrinsic semi conductor,

Number of holes = No. of electrons = ni.

n=p=ni ;

n = Number of electrons per unit volume (or) electron concentration (or) electron density.

p = Number of holes per unit volume (or) hole concentration (or) hole density.

n; = Intrinsic concentration.

Now np = ni2

Intrinsic carrier concentration

Above O0K, in an intrinsic semi conductor, each broken bond leads to generation of two carriers.

They are electron and hole.

At any temperature T, the number of electrons generated will be therefore equal to the number of

generated Holes.

Let n = Number of electrons per unit volume or electron concentration in the

conduction bond.

P = Number of holes per unit volume or Hole concentration in the valence bond.

For an intrinsic semiconductors;

n = p = ni ; --- (1)

Where ni = intrinsic carrier concentration.

Now the electron concentration in the conduction band is given by

/

/

c F B

F C B

E E K T

c

E E K T

c

n N e

n N e

--- (2)

The Hole concentration in the valence bond is given by

/

/

F V B

V F B

E E K T

v

E E K T

v

P N e

n N e

--- (3)

Here ,c VN N are known as pseudo constants, depends on temperature.

Free electron

Broken bond

Vacant site (hole)

Si

Si

Si

Si

Si

Si

Si

Si

Si

Figure (2) a: Two dimensional crystal structure of

intrinsic semi conductor silicon above O0K

EC

EF

EV

Electron

Hole

Electron

Conduction bond

Eg

Figure (2) b: Intrinsic Silicon – Energy band

structure above O0K.

BK = Boltzmann constant

T = Temperature in 0K of the intrinsic semiconductor.

Now 2

in np

/ /2 .F C B V F BE E K T E E K T

c Vni N e N e

2

F C V F

B

E E E E

K T

cni N e

2

2

2

F C

B

F C

B

g

B

E E

K T

c V

E E

K T

c V

E

K T

c V

ni N N e

ni N N e

ni N N e

Where c vE E Eg

1/ 2

2c v

B

Egni N N e

K T

---- 4

From equation (4), It is clear that

i) Intrinsic carrier concentration is independent of fermilevel.

ii) Intrinsic carrier concentration in is a function of temperature T.

iii) Intrinsic carrier concentration in is a function of Energy gap Eg.

Fermi level expression

The Fermi level is the top most occupied energy level. The Fermi level indicates the probability

of occupation of energy levels of the electrons in conduction and valence bands.

In intrinsic semiconductors, electron and hole concentrations are equal.

i.e. it indicates that the probability of occupation of energy levels in conduction bond and

valence bond are equal.

Usually in an intrinsic semiconductors, the Fermi level lies in the middle of the energy gap Eg.

For an intrinsic semiconductors, n=p.

Now

/

/

c f B

c f B

E E K T

c

E E K T

c

n N e

n N e

---(1)

/f v BE E K T

vp N e

/V f BE E K T

vp N e

--- (2)

Equations (1) and (2) represent electron and hole concentrations for intrinsic semiconductors.

Since n=p. / /f c B v f BE E K T E E K T

c VN e N e

f c

V BT

v fC

BT

e E E

N K

E ENe

K

.

f c v fV

C BT BT

e E E E ENe

N K K

f c v fV

C BT

e E E E EN

N K

2 /f c v BTE E E KV

C

Ne

N

--- (3)

Taking Naparian Logarithm an both sides.

2 /log C BTe EF E EV KV

e e

C

NLog

N

2log

CVe

C B

EF E EVN

N K T

2 log

2 log

log2 2

VF C V B

C

VF C V B

C

C V VBF

C

NE E E K T

N

NE E E K T

N

E E NK TE

N

For an intrinsic semiconductor * *me mn

Hence V eN N

log 12 2

2

C V BF

c VF

E E K TE

E EE

Therefore Fermi level lies exactly midway between

conduction bond and valence bond.

Expression for intrinsic conductivity

Let us consider intrinsic semiconductors. This is

applied with a potential difference of V volts. Electric field

Fig. 3 Conduction in an intrinsic semiconductor

Electron flow

Hole flow

+ -

v

Due to the applied voltage an electric field E

will be established as shown in the figure.

Now the charge carriers drift as indicated in figure (3). This constitutes an electric current I.

The drift velocity acquired by the charge carriers is given by.

dV E ---- (1)

Where = Mobility of charge carriers.

Also the current density due to drift of electrons is given by

n dJ nev ---- (2)

Where n= electron concentration

E = charge on the electron.

dv = drift velocity of the electrons.

Also /n nJ ne E ---- (3)

Where n = Mobility of electrons.

Current density Current I

J= =Area ^

Also the holes will drift in a direction opposite by electrons, the hole current density is given by

pJ e Pp E ----- (4)

Where p = Hole concentration.

e = charge on the hole.

p = Mobility of holes.

Now the total current density is given by

n p

n

n

J J J

J ne E Pe pE

J n P p eE

----- (5)

But according the classical theory, ohms law is given by

J E ---- (6)

Where = Electrical conductivity

From equations (5) and (6), we have

n

n

E n P p eE

n P p e

But according to law of mass Action, for an intrinsic semiconductors in p n

n ini n p e

i nn e n p ---- (7)

Where in = Intrinsic concentration.

But 1/ 2 / 2 BEg K T

i C Vn N N e ---- (8)

Substituting (8) in (7), we get

1/ 2 / 2 BEg K T

C V n pN N e e

Electrical conductivity for intrinsic semi conductors is given by / 2 BEg K TAe ---- (9)

Where 1/ 2

C V n pA N N e

A = a constant

Determination of Energy Gap (Eg) for intrinsic semiconductors

The energy gap between the conduction Band and the valence bond is represented as band gap

Eg. For intrinsic semi conductors, the energy gap is given by

/ 2 BEg K TAe ---- (1)

Where A = a constant

Eg= Energy band Gap.

KB= Boltzmann constant.

T = Absolute scale of temperature.

Let P = Electrical Resistivity.

1

P

/ 2

1BEg K T

PAe

/ 21BEg K TP e

A

/ 2 BEg K TP Be ---- (2)

Where1

BA

, a new constant.

Taking Neparian logarithm an both sides,

/ 2log log BEg K T

e ep Be

/ 2log log log BEg K T

e e ep B

ln ln2 B

Egp B

K T

ln ln2 B

Egp B

K T --- (3)

Where m = shape of the straight line

From figure (1), 2 B

Eg ym

K x

2 B

yEg K

x

---- (4)

If a graph is plotted between 1

T an X-axis and ln

an y-axis, a straight line graph is obtained. The

straight line graph is shown in figure (4).

Extrinsic semi conductors

O X

Y

∆y

∆x

ln

1/T

Fig 4: Plot of 1/T and ln

Extrinsic semi conductors are an impure semiconductors. With the addition of impurities, a pure

semi conductors becomes an extrinsic semiconductors, a pure semi conductors becomes an

extrinsic semi conductors.

An extrinsic semi conductor shows good conducting properties due the presence of impurities.

Depending on the type of impurity present in the intrinsic semi conductors, extrinsic semi

conductors are classified into two types.

1) N – type extrinsic semi conductors. 2) P - type extrinsic semi conductors.

N-Type semi conductors

For silicon if a small amount of pentavalent impurity such as phosphorous, arsenic or antimony

or Bismuth is added, we get N-type semiconductors.

Four valence electrons of phosphorous form covalent bonds with the adjacent four silicon atoms.

The fifth electron is left free. It cannot form bond with any other electron in the lattice structure.

This is shown in figure (1).

At O0k, this fifth electron is bond to phosphorous with 0.045 ev.

The corresponding energy Band diagram and lattice structure are shown in figure (1) at O0k. At

O0k, the valence Bond and the conduction band are separated by an Energy Gap Eg.

The donor energy level Ed lies below the bottom of conduction band. This donor energy level

contains phosphorous atoms. Which denotes electrons at T>O0k The donor energy level in

shown in the figure (1) b. Above O0k, when temperature is increased. The 5

th bond electron

becomes a free electron. This free electron enters into the conduction band. Due to this the Donor

Atoms will get ionized, by denoting an electron to the conduction band. When temperature is

further increased, the covalent bonds will break down. Here electron hole pairs will be generated.

Electrons will move from valence bond to conduction bond, leaving holes in the valence bond.

At higher temperatures, the energy band diagram of N-type silicon is shown in figure (2)

The Fermi level varies as shown in fig (2) b at 3000k.

Figure (5)a: N-type silicon at O0k

Now the concentration of electrons increases in the conduction bond when compared to holes.

Hence the electrons become the majority charge carriers and holes the minority charge carriers.

The variation of Fermi level is also shown in figure 2( b).

P-type semi conductors

For silicon if a small amount of trivalent impurity such as indium, Gallium, Thallium or

Aluminum or Boron is added, we get an Fi- Type semi conductors.

Three valence electrons of Boron form covalent bonds with the adjacent three silicon Atoms.

There is not fourth electron to form a covalent bond with the neighboring silicon atom. This is

like a missing bond. This is represented as a missing electron or vacant site.

5th

electron

Si

Si

Si

Si

Si

Si

Si

P

Si

P=Phosphorous

Conduction bond

Valence bond

Ec

Ev

vc

Ed Donor levels

Eg

Ef

Figure (1) b: Energy band diagram of N-type

semiconductor at O0k

Figure (2)b: Energy band diagram of N-type silicon at

T=3000K and above

Figure (2)a: Energy band diagram of N-type semiconductor

at T>O0k

+ve

Donor

ions

Conduction Band

Ec

Valence bond

Ed

Ei

Ev

Ef

+ve

Donor

ions

Conduction Band

Ec

Valence bond

Ed

Ei

Ev

Ef

This is shown in figure (1)

Fig. (1)a: P-type silicon at O0K fig.(1)b: Energy band structure of P-type

silicon at 00k

This missing electron is called Hole. The energy B and structure of P-type semi conductors is

shown in figure (1) b.

At O0k, the conduction Band is empty and the valence B and contains electrons.

The acceptor energy level EA is just above the top of the valence Band.

Acceptor energy level EA contains the acceptor atoms.

Here cE = Bottom of the conduction band.

VE = Top of the valence band.

iE = Intrinsic energy level.

The Energy band structure of P-type silicon is shown in figure (2) a above O0k.

When the temperature is above O0k, the covalent bonds with the silicon are broken down.

Si

Si

Si

Si

Si

Si

Si

B

Si

Vacant site

Vacant site

(hole)

Hole

Conduction bond

Valence bond

EC

EA

EV

Acceptor level

Ei

Acceptor level

contains acceptor

atoms

Ef

Eg

Fig.(2)a: Energy band structure of P-type silicon at O0k

Here same electrons are released and the acceptor atoms accept three electrons and there by they

become negatively charged ions. There are called negative acceptor ions. Here the Fermi energy

level lies just above the top of the valence bond and below the acceptor level.

The energy band diagram of P-type semi conductors is shown in the figure (2) b. at T = 3000k.At

3000k, the bands in silicon with further breakdown and the electrons will move from valence

band to conduction band. Therefore electrons are available in the conduction band. At 3000k the

Fermi level varies as shown in the figure.

Fig.(2)b: Energy band diagram of P-type silicon at 3000k

Law of Mass Action

The electron concentration in intrinsic semi conductors is given by

/c fE E

c Bn N e K T

/f cE E

c Bn N e K T

---- (1)

Similarly in an intrinsic semiconductors, the hole concentration is given by

/f cE E

v BP N e K T

/v fE E

v BP N e K T

---- (2)

Where cN and vN are pseudo constants.

Conduction band

Ec

EA

Ev

Valence band

Ei

EF

Holes

Eg

EF= Fermi energy level

Conduction band

Ec

EA

Ev

Valence band

Ei

EF

Holes

Eg

EF= Fermi energy level

BK is the Boltzmann constant.

T is temperature in 0K.

EF is the energy of Fermi level.

EC is the bottom of the conduction band.

Ev is the top of the valence band.

In an intrinsic semiconductors n=p=ni

/ /.f c B v f BE E K T E E K T

c vnp N e N e

/ /2

/2

/2

.

Where

f c B v f B

f c B

B

E E K T E E K T

c v

E E K T

c V

Eg K T

c V c v g

np ni N e N e

np ni N N e

np ni N N e E E E

1/ 2 / 2 BEg K T

c vni N N e --- (3)

The above relation shows that for any arbitrary value of Eg the product of n and p is a constant.

This is known as Law of Mass Action

For an extrinsic semiconductors, the electrons and hole concentrations are given by expressions

similar to Equations (1) and (2)

For an N-type semiconductor

/f c BE E K T

n cN N e

----- (4)

/v f BE E K T

n vP N e

----- (5)

Where nn Electron concentration.

nP Hole concentration.

Now / /

.f c B v f BE E K T E E K T

n n C V vN P N N e N e

/f c BE E K T

n n C Vn P N N e

/g BE K T

n n C Vn P N N e

------ (6)

Where c v gE E E

2

n nn P ni ---- (7)

The above expression (7) is known as Law of Mass action for N-type semi conductors.

For P-type semi conductors, the law of mass action is given by

2

p pp n ni ----- (8)

Equations (7) and (8) imply that the product of majority and minority carrier concentrations in

extrinsic semi conductors at a given temperature is equal to the square of Intrinsic carrier

concentration at that temperature.

The law of mass action is very important in conjunction with charge neutrality condition.. This

enables us to calculate minority carrier concentration. This law states that the addition of

impurities to intrinsic semi conductors increases the concentration of one type of carrier, which

consequently becomes majority carrier and simultaneously decreases the concentration of the

other carriers, which is known as the minority carrier.

The minority carriers decrease in number below the intrinsic value.

This is because there is an increase of majority charge carriers Recombination rate.

According to the law of Mass action, the product of majority and minority carriers remains

constant in an extrinsic semi conductors and it is independent of the amount of donor and

acceptor impurity concentrations. When the doping concentration levels are high, the minority

carrier concentration will be law and the majority carrier concentration will be high when the

doping concentration levels are low, the majority carrier concentration is low and the minority

carrier concentration is high.

Hall Effect

Some times it is necessary to determine whether a material is n-type or p-type. Measured

conductivity of a specimen will not give this information since it cannot distinguish between

positive hole and electron conduction.

The Hall Effect can be utilized to distinguish between the two types of carriers, and it is also

useful in the determination of density of charge carriers.

Hall Effect definition

“If a piece conductor (metal or semiconductor) carrying current is subjected to a transverse

magnetic field, an electric field is generated inside the specimen in a direction normal to both the

current and the magnetic field”

This phenomenon is known as Hall Effect. The generated voltage is known as Hall

Effect. The corresponding electric field is known as Hall Effect field.

Let us consider a sample having thickness t and with b. the sample is a rectangle sample,

as shown in the figure.

Figure (1) Hall effect

Assuming that the material is an n-type semiconductor, the current flow consists of

almost due to electrons, moving from right to left.

This corresponds to the direction of conventional current from left to right as shown in

figure (1).

I t

y

z

F

I or Ix

Electrons

experience a

force F in the

down ward

direction due to

B

X

B

Face (1)

Face (2)

b

Fig (2) Motion of electrons in an n-type

semiconductors

eEH

Be

Current I is in the positive X-direction and the magnetic field B is applied in the positive Z

direction. According to Flemings, left hand Rule, The electrons experience a force, called

Lorentz force. This Lorentz force acts in the negative Y-direction.

Now Lorentz force vLF B E

sinLF Bev

Where v = velocity of electrons.

Since the velocity of electrons and B are perpendicular downwards in the negative Y-direction

and the positive charges drift upwards in the positive Y-direction. As a consequence, the lower

surface collects negative charge and upper surface becomes positively charged. Due to this an

electric field called Hall electric field will be established between upper and lower surface of the

specimen.

This hall electric field EH establishes a potential called the Hall Voltage VH.

The hall field EH exerts an upward force FH on the electrons as shown in figure (2).

H HF eE ---- (2)

But total force on the electrons, is given by

0HBev eE ---- (3)

The above equation is called Lorentz equation. Under equilibrium conditions.

HE Bv ---- (4)

Now the current density in the X-direction is given by

xJ nev ----- (5)

Now (5) nJv

ne ---- (6)

Here n = electron density (electron concentration)

E = charge on the electrons.

Now from (4) and (6),

xH

BJE

ne ------ (7)

Now the Hall coefficient RH can be described as follows.

For a given semiconductor electron concentration n is constant and charge on the electron e is

constant.

H xE BJ

H H xE R BJ ----- (8)

Where HR is a constant of proportionality.

1

HRne

----- (9)

Here ,HE B and xJ are measurable. Hence hall coefficient HR and carrier density „n‟ can be

found.

Determination of Hall coefficient

Let t be the thickness of the rectangles slab.

b be the width of the sample.

Now the Relation between HE and HV is given by

H HV E t ----- (10)

Also (8) H H xE R BJ ----- (8)

Now from (8) and (10), we get

H H xV R BJ t ----- (9)

But xJ = current density

xx

IJ

A , x

x

IJ

bt (Since A = Area of cross section

A = bt)

Equation (9) becomes

xH H

IV R B

b t t

xH H

I BV R

b

HH

x

V bR

I B ----- (10)

HV , b , xI and B all are measurable and substituting them in equation (10), we can obtain the

value of Hall coefficient HR .

Note that the polarity of HV will be opposite for n and p type semiconductors.

Carrier concentration and mobility

Hall coefficient 1

HRne

1

HRne

(Magnitude)

Electron concentration 1

H

nR e

can be determined.

Now electrical conductivity ne .

Where mobility

Mobilityne

HR

For a P-type material Hall coefficient is positive.

1HR

pe

Where p= hole concentration.

Application of the Hall Effect

1. Useful in determining whether the given semiconductor is n-type or p-type.

2. Hall Effect can be used to find the carrier concentration and mobility of carriers.

3. Hall Effect is used to measure the magnetic field.

4. Hall Effect semi conducting devices are used as sensors to sense the magnetic fields.

5. The Hall Effect is used in magnetically activated electronic switches. They are used as

non contacting key boards and panel switches.

PN Junction

When a P type material is suitably joined with an n type material, a Pn junction is formed.

When an intrinsic semi conductors is simultaneously doped with P-type and n-type

impurities, a Pn junction is formed.

The Pn junction may be formed by crystal growth or alloying or diffusion method.

The plane dividing the two zones is called Pn junction.

The Pn junction is shown in figure (1)

+ ve ionized Donors

- ve ionized Acceptor

+ Hole

- Electron

Fig (1) b: Diffraction of electrons and holes

Fig (1) a: a Pn junction

VB Holes

Electrons

Co

nce

ntr

atio

n

Electrons

1024

1020

1016

X

Y

X

Fig (1) c: Space charge region for an alloy or abrupt

junction

Space charge region

X Fig (1) d: Electric field due to space charge region Electric

field E

X Fig (1) e: Barrier potential (or) contact

potential Voltage V

x2

v1

VB x1

v2

Space

charge

density ---

+

P

n

E

+

+

+

+

+

-

-

+ + -- --

+ + -- - -

-- --

In the p side „+‟ represents holes. In the n side „-‟ represents electrons.

In the n-side there is a high concentration of electrons.

In the P- region there is a high concentration of holes.

Therefore, at the junction there is a tendency for the electrons to diffuse from n-region to p-

region and holes from p-region to n-region. This process is called diffusion. When the free

electrons move across the junction from n-side to p-side. The demotions become positively

charged. Hence a not positive charge is built on the n-side of the junction.

The free electrons that cross the junction uncover the negative acceptor ions by combining with

the holes.

Therefore a not negative charge is established on the p-side of the junction.

This not negative charge n the p-side prevents further diffusion of electrons from n-side to p-

side.

Similarly the net positive charge on the n side prevents further diffusion of holes from p side to n

side.

Due to this a barrier is set up near the junction.

This barrier prevents further movement of charge carriers i.e. electrons and holes. This barrier is

called potential barrier.

It should be noted that outside this barrier an each side of the junction. The material is still

neutral.

Only inside the barrier, there is positive charge on n side and negative charge on p-side.

This region is called depletion layer. This is so because mobile charge carriers are

depleted in this region.

It is clear that a potential barrier VO or VB is set up.

As a consequence of this an electric field is established across the depletion layer.

The Barrier potential is about 0.3v for Germanium and 0.72V for silicon.

The depletion layer and the Barrier potential are shown in the fig (1)a and Fig (1)e.

The width of the depletion region is less than 1 m (~0.5 m ). Since the depletion region has

immobile ions which are electrically charged it is known as space charge region. The space

charge region is shown in figure (1) c. the established electric field is shown in figure (1) d.

Hence across the junction no current flows and the system is in equilibrium.

To the left of this depletion layer (in the P side), the carrier concentration is P ~ NA.

To the right of the depletion layer (in the n side), the carrier concentration is n~ ND.

Diode Current Equations

The diode current pertaining to VI characteristics is given by

0 1VT

V

I I e

Where I = Diode current

0I = Diode reverse saturation current at room temperature.

V = External voltage applied to the diode.

= A constant

=1 for Germanium

=2 for Silicon

x=0

TV = Volt equivalent temperature or thermal voltage.

BT

K TV

q

BK Boltzmann constant

BK 1.3806

23 110 JK

q = Charge on the electron

q = 91.602 10 coulomb

T = Temperature of the junction in 0 K .

When the diode is reverse biased, the current equation is given by

0 1T

v

VI I e

Drift current:

In a perfect crystal the periodic electric field enables electrons and holes to move

freely as if in vacuum.

When there is no electric field, there is no net current. This is because charge

movement in any direction is balanced by charge movement in the other direction.

In the presence of the field, the carriers experience directed movement. This is

called drift.

Definition Of drift: Forcible movement of Charge carriers under the influence of

an

Electric field is called drift.

With the field carriers drift and this results in results in current flow through

the

semiconductor

The current density is given by

dJ neV --------------- (1)

Here dV drift velocity.

Also d dV E V E ------------- (2)

Where is called the mobility of the carriers.E= Electric field. From equations

(1) and (2),

Now current density dJ nev ----------- (3)

In semi conductors, the current flow is due to electrons and holes.

Electron current density is given by

n nJ drift ne E ------------- (4)

Hole current density is given by

p nJ drift pe E ----------- (6)

The two charge carriers move in the opposite direction.

Now the total drift current density is given by

n pJ drift J drift J drift

n pJ drift ne E pe E

( ) nJ drift E ne pe p -------- (7)

For an intrinsic semiconductors in p n

( ) i nJ drift En e e p ------------- (8)

Equation(8) gives current density equation.

Diffusion current :

Usually directed movement of charge carriers will give rise to electric current.

The movement of charge carriers may be due to either drift or diffusion.

Usually non-uniform concentration of carriers gives rise to diffusion.

Definition: Movement of charge carriers from high concentration region to low

Concentration region in a semiconductor is known as diffusion.

Let us suppose that the concentration of electrons varies with distance x in the

semi conductors. Here the concentration gradient is given by n

x

.

Ficks low states that the rate at which carriers diffuse is proportional to the

density gradient and the movement is in the direction of negative gradient.

Mathematically, the rate of flow of electrons can be written as

n

nf

x

, Here nf = rate of flow of electrons across unit area.

The rate of flow of electrons is given by

n n

nf D

x

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

Here nD = Diffusion coefficient for electrons.

Partial derivatives are used because n is a function of temperature and distance.

This flow of electrons constitutes an electron diffusion current density. Since

conventional current in the rate of negative charge, we have

Rate of flow of electrons across unit areanJ diffusion e

( )n n

nJ diffusion e D

x

n n

nJ diffusion eD

x

----------- (2)

If an excess hole concentration is created in the same region, hole diffusion takes

place in the same direction at a rate per unit area.

The rate of flow of holes per unit area is given by

p p

pf D

x

------------- (3)

This results in a hole diffusion current density

Now rate of flow of holes across unit areapJ diffusion e

(4)p p

pJ diffusion e D

x

Here pD Hole diffusion coefficient for holes.

Einstein Relations or Einstein Equations

At equilibrium with no field, the free electron distribution is uniform and there is

no net current flow. Any tendency to disturb the state of equilibrium which would

lead to diffusion current creates an internal electric field.

This internal electric field creates a drift current balancing the diffusion current

component.

Under equilibrium conditions, we have therefore the drift and diffusion currents.

These currents are due to an excess density of electrons.

Now driftnJ n eE----------------------- (1)

diffn n

nJ eD

x

--------------- (2)

Under equilibrium conditions, ( ) ( )n nJ drift J diff

(3)n n

nn eE eD

x

The force F on excess carriers restoring equilibrium is given by the product of

excess charge and Electric field.

F ne E --------------- (4)

(3)n

n

nD

xnE

----------- (5)

Now from (4) and (5),we get

n

Dn E nF e

E x

n

e Dn nF

x

------------- (6)

This force F depends on the thermal energy of the excess carriers.

By making can analogy between the excess carriers in a semiconductors and gas

molecules in a low pressure gas, the force F corresponds to pressure gradient.

Pressure gradient = B

nK T

x

nB

n

eDn nK T

x x

nB

n

eDK T

n

n BD K Te

Bn n

K TD

e --------------- (7)

Similarly for holesp

Bp

K TD

e ---------- (8)

7

8

n n

p p

D

D

------------ (9)

Equations (7), (8) and (9) are called Einstein‟s Relations.

UNIT-IV

Magnetism

And

Super Conductivity

Super Conductivity Introduction

Usually materials exhibit electrical resistance and resistivity. This is due to scattering of

electrons by the positive ions present in the materials. When the temperature of certain materials

decreased to lower values, the resistance and resistivity decreases. Hence conductivity increases.

This is because scattering of electrons decreases due to lower energy. The phenomenon of

attaining zero resistivity or infinite conductivity at low temperatures is known as super

conductivity. Here the state of the material is known as super conducting state. The temperature

at which the material transforms into super conductor from normal state is known as super

conducting transition temperature (or) critical temperature (or) transition temperature Tc. This

was first observed by Heike Kammerlingh Onnes in 1911 in mercury. When temperature of

mercury decreased by cooling it in liquid Helium, the resistivity completely disappeared at 4.20k.

This is shown in figure (1).

NC=Normal Conductor

SC= Superconductor

Figure (1) variation of resistivity with temperature in mercury.

The super conductor will have zero electrical resistance below a well defined temperature Tc.

The materials exhibiting super conductivity are called super conductors. Same examples of

super conductors and their Tc values are given below.

Element / material Tc in 0k Element / material Tc in

0k

Aluminum 1.19 Lead 7.175

Cadmium 0.52 Zinc 0.9

Gallium 1.09 Zirconium 0.8

Indium 3.4 Niobium 9.3

Tin 3.72 NbN 16

Mercury 4.12 Vanadium 5.03

Temperature TOK

Resistivity

Y

X O

4.2

Tc

NC SC

Cooled to

General properties 1. The super conductivity is a low temperature phenomenon.

2. The transistor from normal state to super conducting state occurs below the critical

temperature.

3. The transition temperature is different for different materials.

4. The current that is set up in a super conductor persists for a long time due to zero

electrical resistivity.

5. Super conductivity is found to occur in metallic elements in which the number of valence

electrons (Z) lies between 2 and 8.

6. Materials having high normal resistivities exhibit super conductivity.

The condition is n> 106 is a good criterion for the existence of super conductivity.

Here n = number of valence electrons per cc.

= Resistivity in esu at 200c.

7. Super conductors when placed in magnetic field, they do not allow magnetic lines of

force to pass through them. Now they behave as diamagnetic i.e. they expel magnetic

field. This property of expulsion of magnetic field is known as Meissner effect.

8. With increase in magnetic field, the material looses its super conductivity. The field at

which the superconductor looses superconductivity and becomes a normal conductor is

called critical magnetic field Hc.

9. Monovalent, Ferro magnetic and Antiferro magnetic materials are not super conductors.

10. The induced current in a super conductor induces a magnetic field in it. If the magnetic

field is equal to the critical magnetic field then it converts into a normal conductor.

Now the current through the super conductors is known as the critical current Ic.

The critical current is given by Ic = 2 cr H

Where r= radius of the super conductor.

11. Super conductors are not good conductors at room temperature.

12. Good conductors like Cu, Ag, Au when cooled to 0.07, 0.35 and 0.05k; they are

exhibiting still resistivities.

13. In the case of Tin, the variation of electrical resistivity with respect to temperature is

shown in figure (2).

Fig (2). Electrical resistivity of Tin as a function of temperature.

X 10-11 -m

20 –

10 –

0

| | | | |

2 4 6 8 10

T

0K

Effect of Magnetic field (critical magnetic field)

When a magnetic field is applied to a super conductor, then at a particular magnetic field value,

the super conductivity disappears. Here the material becomes a Normal conductor.

At a sufficiently higher magnetic field, the super conductivity will be destroyed. At this stage the

material will restore to their normal conducting state.

The amount of magnetic field that is required to destroy super conductivity is called the critical

field. This is denoted by Hc. The functional relationship of Hc with temperature is given by 2

0 1c

C

TH H

T

Where H0= Field required to destroy the super conductivity at 00k.

H0= is a definite value for every material.

Tc= critical temperature

Hc= the maximum critical field strength at T0k.

* At T=Tc, Hc=0

* when T=00k, then Hc=H0

The variation of critical magnetic field Hc as a function of temperature is shown in figure (3).

+

Figure (3). Variation of Hc as a function of temperature T in 0K

H

cc

T

0K

0

H0

Tc

Normal

State

Super

conducting

state

The Meissner effect – flux exclusion :

Meissner and ochsenfeld in 1933 found that if a long super conductor is cooled in a longitudinal

magnetic field, below the critical transition temperature, then the lines of force or lines of

magnetic induction B are pushed out of the body of super conductor at the transition temperature.

Here the applied magnetic field H must be less than the critical magnetic field value Hc. The

magnetic field exclusion is shown in the figure (4).

Figure(4) Meissner effect

This magnetic field expulsion phenomenon by the super conductor at Tc and below Tc is

called the Meissiner effect. Now inside the super conductor, magnetic induction field

strength B is zero. That is the magnetic flux is excluded from the body of the super

conductor.

Also for a super conductor r = 0

Also we know that magnetic induction B is given by

0B H M -------------------- (1)

But B=0

0 0

0

1

H M

H M

M H

M

H

Cooled

B=0

T<TC

H<HC

Super conducting

T > Tc, , H < Hc

But 1M

xH

Where χ= magnetic susceptibility.

For a super conductor magnetic susceptibility is negative. Hence a perfect super

conductor is a perfect diamagnetic material.

Note: Usually a diamagnetic material will have negative magnetization. Hence it will

have a negative magnetic susceptibility.

Penetration Depth

Let us consider a super conductor. Consider that a magnetic field H0 is applied to one of

its faces as shown in the figure (5).

But according to Meissner effect, it will not allow magnetic lines of force to pass through

it. But in practice a small portion of H0 penetration to a small distance into the

superconductor.

Figure 5. Penetration of field through are face of super conductor

The applied field does not suddenly drop to zero at the surface of the super conductor, but decays

exponentially. The penetration of the magnetic field at a distance x from the surface is given by

/

0

xH H e

Where H = field at a distance x from the surface into the material.

0H = Field at the surface.

= Characteristic length known as a penetration depth.

X= distance from the surface into the material.

is the distance for H to fall from 00

HH H to

e

When x , then 0H H e

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

1

0H H e

0HH

e

x

H0

Mag

net

ic f

ield

Super conductor

The penetration depth is the distance into the super conductor at which the magnetic field is

equal to 1

e

of the applied magnetic field 0H . The magnetic field is likely to penetrate through

a super conductor to a depth of 10-100 nm.

Penetration depth is more in thin film type of superconductors. When compared to bulk type

super conductors. The variation of magnetic field H with the distance x is shown in figure (6).

0H

e

Figure (6).Variation of magnetic field H with distance x into the super conductor

The variation of penetration depth with temperature T is given by

0

1/ 24

(2)

1C

T

T

Penetration depth at 0T K

0 Penetration depth at 00T k

cT Critical transition temperature

The penetration depth varies with applied field. According to pippard, there is only a few

percentage change of with H, even up to Hc.

Y

X O

X

H0

H

This is of the form. 2

0 21 0.02c H

C

HT H

H

(Pippard relation)

The variation of penetration depth with temperature T in Tin is shown in figure 7.

Figure (7) variation of penetration depth in Tin. In equation (2)

The percentage depth of same materials at 00K are given below.

Super conductor (in nm)

Mercury 70

Indium 64

Lead 39

Tin 50

Aluminum 50

Types of super conductors

When a super conductor placed in a magnetic field, at critical magnetic field, it transforms from

super conducting state to normal state.

Basing on this transformation, superconductors are classified into two types.

Type – I super conductors

Type – II super conductors

Types – I Super conductors

Let us consider a cylindrical form of super conductor. Now a magnetic field H is applied along

the axis of the super conductor.

But according to Meissner effect. The super conductor does not allow the magnetic lines of

force to pass through it. Here H =- M; shown in figure (7)a i..e inside the super conductor, the

magnetization acts in a direction apposite to magnetic field H.

When the magnetic field H is equal to the critical magnetic field Hc, then the magnetization (-M)

becomes equal to zero.

| | | |

1 2 3 4

TC

5 --

4 --

3 --

2 --

1 --

0 --

0

Pen

etra

tio

n d

epth

X 1

0-5

cm

T 0k

In this state the magnetic lines of force penetrates through the super conductor completely. Here

the super conductor completely transforms into a normal conductor. This is shown in figure (7)b.

Hc=0.1Tesla

Hc

The variation of – M with respect to H is shown in figure (8). This curve is called Magnetization

curve. Type – I Super conductors exhibit complete Meissner effect.

Type – I super conductors are soft super conductors. Here they are completely diamagnetic and

hence the flux is completely excluded. If the superconductor is an ideal one, the magnetization

curve is reversible. For non ideal superconductors the magnetization curves are irreversible.

Below Hc, the critical field, the material is a super conductor. Above the critical magnetic field

the super conductor becomes a normal conductor. At Hc, the super conductivity in type- I super

conductors suddenly falls to zero.

Examples

Lead, Aluminum, zinc, mercury, cadmium, indium, Sn etc.

This was first observed by Silsbee in 1916.

Type – II Super conductors

Let us consider a spherical form of super conductor. Now let a magnetic field is applied

along the diametrical axis of the super conductor. But according to Messier effect, the super

conductor expels the magnetic lines of force and H=-M. i.e. The magnetization inside the

superconductor opposes the passage of lines of force through it. Now the magnetic lines of force

are concentrated at the two curved ends of the spherical super conductor as shown in figure (9)a.

Here the intensity of magnetic field H is predominant at these ends.

For applied magnetic fields below lower critical magnetic field Hc1, the specimen

behaves like a superconductor. Here the magnetic flux is completely excluded in this range of

field. At HC1 the magnetic flux begins to penetrate the specimen and the penetration increases

until HC2 is reached. At HC2 the magnetization vanishes and the specimen becomes a normal

conductor. HC2 is known as upper critical magnetic field.

Y

Super

conducting state Normal

state

H

C

X O

-

M

H

-M

Figure 7 (b) Normal

conductor H>Hc, T>Tc

Figure (8) Magnetization curve for

Te – I super conductor

Figure 7 (a) a super

conductor H < Hc, T<TC

Figure (7) b Type – I Super conductor

Y

Superconducting

state Normal

state

H

C

X O

-

M

H

Between HC1 and HC2, the penetration of the flux is at the curved ends of the spherical

superconductor. At HC2 above HC2 the specimen completely transforms from super conducting

state the normal state.

Between HC1 and HC2 the material is having super conducting as well as Normal conducting

properties i.e. between HC1 and HC2 the specimen is in the mixed state or vortex state.

Above HC2 the specimen becomes a normal conductor. Here – M = 0. This is shown in figure

(9)c.The mixed state is also shown in figure (9)b.

Figure (9) Type II super conductor

-M -M

-M=0

Fig. (9) c. Normal conducting

H>Hc2.

Figure (9)a Superconductor ,

H<Hc1

Fig. (9) b. Superconducting +

Normal conducting (mixed state)

H<HC2 & H>HC1

The magnetization curve for Type – II super conductors is shown in figure (10).

Figure (10) Magnetization curve for type II superconductor

Figure (11) Magnetic phase diagram of type-II superconductors

In type – II super conductors, the transformation process from super conducting state to

normal state is slow.Ttype-II. Super conductors are hard super conductors.

The magnetic phase diagram of type – II superconductors is shown in figure (11).

-M

Y

X H

Super conducting

State

Vortex state

(or) mined

state

Normal

state

HC1 O

HC2=10 tesla

HC2

O X

T0K

HC2

Y

HC1

Type – II super conductivity was first discovered by Schubnikov in 1930.

Examples : Osmium, Zirconium, Thorium, Thallium, Tantalum etc.

Flux quantization

Let us consider a hollow normal conducting ring in a magnetic field H. This ring now

allows magnetic flux through it.

The magnetic flux will be present inside the hollow space, on the ring and outside the

ring, shown in fig. 12a. When the ring is cooled up to Tc and below Tc. (It‟s critical

temperature), it becomes a super conductor. Now it obeys Messier effect.

In the superconducting ring ,persistent currents will be set up.As a result of Meissner

effect H= - M, Magnetic flux will be excluded.

In this case, it is observed that the flux is present outside and inner hollow space of the

ring only. This is shown in figure (12)b.

Fig. (12)a. Hollow ring normal

conducting in a magnetic field H (T>Tc).

Persistent

current super

Magnetic flux

Fig. (12)b. Hollow supercoducting Ring

in a magnetic field H (T<Tc).

When the applied magnetic field is removed, then the persistent current will exist in the

specimen. Due to this same magnetic flux is retained in the inner walls of the super

conductor. This is shown in the figure (12) C.

Figure (12) C. Hollow super and conducting ring when no field (H) (T<TC)

The trapped magnetic flux will have quantized values. This is given by

, 1, 2,3,....2

nhn

e

Where h = Planck‟s constant

e= charge on the electron.

2 3, , ,...

2 2 2

h h h

e e e

The unit of flux is called fluxoid.

1 fluxoid = 2.07 x 10-15

weber.

1 fluxoid =2

h

e

The flux through the ring is the sum of the flux due to the external source ext and flux due to the

super current flowing through that ring sc

i.e. scext

This flux is quantized.

ext is usually not quantized and sc adjusts such that is always quantized.

Super

current

persistent

current

Josephson Effect

Let us consider two super conductors joined suitably with a thin insulating layer, as shown in

figure (13). This constitutes a junction called Josephson junction.

Fig.(13) Josephson effect

SC I - super conductor I, SC II- super conductor II

Usually in a super conductor, we will have cooper pairs. Cooper pairs are nothing but

electrons occurring in Pairs.

Tunneling of electron pairs across an insulating gap between two superconductors was

predicted by Josephson.

This prediction was experimentally verified by Anderson and Rowell, using lead –

insulation lead sand witches. The cooper pair will penetrate or tunnel across the thin

insulator and causes a small amount of current. This is current is known as super current.

This effect is known as Josephson effect.

The Josephson Effect is of two types.

(i) DC Josephson effect and (ii) AC Josephson effect.

DC Josephson Effect

A DC current flows across the junction in the absence of voltage across the junction.

Here the cooper pairs tunnel across the thin layer of insulating medium. Due to this a DC

current flows across the junction.

This effect is known as DC Josephson effect. This is shown in figure (14).

Figure (14) DC Josephson effect

SC – I SC – II

Thin insulating layer

Thin insulating layer

SC – I SC – II

Copper

pairs

Thin insulator

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

SC = Super conductor

Super currents flow through the junction even in the absence of voltage difference.

The current through the Josephson junction is given by

I= Im sin

Where mI maximum current flowing across the junction.

mI Depends on the thickness of the insulating layer.

is the phase difference in the state of waves describing the cooper pairs on both sides

of the insulator.

2 1

AC Josephson effect

When a DC voltage is applied across the junction of two super conductors separated by a

thin insulating layer, then the cooper pairs oscillates through the insulating layer.

Due to this radio frequency oscillations will be set up across the junction.

This effect is known as AC Josephson effect.This is shown in fig(15)

This frequency of the AC signal produced is given by 2ev

f Hzh

Fig 15. AC Josephson effect

A DC voltage of 1 v produces a frequency of 486.6 MHz.

The applied DC voltage introduces an additional phrase of

Et

Where E= Total energy of the system

t = given time in seconds

Here E= (2e) V0

V0= Applied DC voltage

Since a cooper pair contains „2‟ electrons, the factor 2 appears in the above equation.

Now 02eV

t

Now the tunneling current can be written as

+ -

V0

SC -1 SC – 2

Thin insulator

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

I= Im sin 0

I= 00

2m

eV tI Sin

0mI I Sin t

Where, 02eV the angular frequency.Current voltage characteristics of a Josephson

junction are shown in figure (16).

i) When V0=0 there is a current flow if DC current Ic through the junction. This

current is called superconducting currents. This effect is called DC Josephson effect.

ii) If V0<VC, a constant dc current flows.

iii) If V0>Vc the junction has a finite resistance and the current oscillates with a

frequency 02eVw

This effect is called AC Josephson effect.

Figure (16) V-I characteristics of a Josephson junction

iv) When V0=0 there is a current flow of DC current Ic through the junction.

This current is called superconducting current. This effect is called DC Josephson

effect.

v) If V0<VC, a constant dc current flows.

vi) If V0>Vc the junction has a finite resistance and the current oscillates with a

frequency 02eV

This effect is called AC Josephson effect.

Applications of Josephson’s effect

1. By using Josephson junction, microwaves can be generated having frequency.

02eV

2. A.C. Josephson effect is used to define a standard voltage.

Current

I

Y

IC

VC Voltage V O

X

3.A.C. Josephson effect is used to measure very low temperatures based on the

variation of frequency of the emitted radiation with temperature.

4.A Josephson junction is used for switching of signals from one circuit to

another circuit. The switching time is of the order of 1PS. They are useful in high speed

computers.

The BCs theory (Bardeen, Cooper and shrieffer theory)

Bardeen, Cooper and Schrieffer proposed a theory regarding super conductors, known as

BCs theory. They investigated the electron – phonon interaction in super conductors.

In normal conductors, the electrons are moving at random in all possible directions.

Usually the positive ion cores, the electrons get scattered in all directions. Due to this the

electrons scattering, the materials will posses some electrical resistance, the materials will

possess some electrical resistance. Here the force between two electrons is always a

repulsive force.

When the normal metal transforms to the super conducting state at and below critical

transition temperature, the scattering of electrons decreases. Hence there is a decrease of

scattering energy. In superconducting state the electrons paired. These paired electrons

are called cooper pairs. This is shown in figure (17). The formation of cooper pairs can be

understood from the following.

Consider an electron passes near an ion core; there is a mutual attraction between the

electron and the ion core.

This is due to columbic interaction and as a result the ion core is set into motion.

Let us consider that another electron now passes nearby. The second electron feels the

effect of the motion of the ion core.

Due to this both the electrons are entering into attractive field region.

The ion core motion has provided the mutual interaction between two electrons despite of

their mutual repulsion.

Under very restrictive circumstances, the attraction exceeds the repulsive interaction.

(i) The electrons entering into such an attractive field are having opposite moments

and opposite spins with equal magnitudes.

i.e. K1= -K2 and S1= -S2

(ii) The temperature is low enough such that T0.

Such paired electrons are called cooper pairs. The attractive interaction between two

electrons takes place in the presence of phonon field.

The interaction between the electron and the distorted lattice (settled lattice) occurs,

which in turn lowers the energy of the second electron.

That is the two electrons interact via the phonon field. This is shown in figure (18).

This results in lowering of energy for the electrons. The lowering of energy implies that

the force between the two electrons is attractive.

Salient features of BCs theory

1) The total energy of the BCs state is lower with respect to the normal state. The total

energy of the BCS state comprises of the kinetic energy and the attractive potential

energy. The attractive potential energy acts to reduce total energy of the BCS state. The

normal state comprises of the kinetic energy. As a consequence of this BCS state is more

stable than the Fermi state.

2)The one particle states are occupied with pairs. If a state with a wave vector K and spin

up is occupied then the state with wave vector – K and spin down is also occupied.

Applications of super conductors

Josephson Junction devices

The emf of standard cells using conventional materials, usually drift (changes).

AC Josephson effect can be used to generate voltage standards.

The DC Josephson effect has been used to fabricate sensitive magnetometers with an

accuracy of 10-11 gauss.

Super conductors can be used to generate and detect electromagnetic radiations from

radio to infrared frequencies using AC Josephson effects.

Electric Generators

Super conducting generators are smaller in size, with less weight.

They consume low energy. The low-loss superconducting coil is rotated in a strong

magnetic field. Due to this an electric current is generated.

This is the basis of new generation of energy – saving power systems.

Electric Generators

Super conducting generators are smaller in size, with less weight.

They consume low energy. The low-loss superconducting coil is rotated in a strong

magnetic field. Due to this an electric current is generated.

This is the basis of new generation of energy – saving power systems.

High field magnets

Super conducting materials can be used for producing high magnetic fields with low

power consumption.

Magnetic bearings

The possibility of flux trapping and flux exclusion suggests that a super conducting

cylinder and disc may be used as magnetic bearing.

The bearing is restricted in its lateral movement but can spin on a cushion of flux.

If the material of the moving part of the bearing is chosen so that no flux will penetrate

into it.

The friction in the bearing will come from the viscosity of the surrounding medium.

Energy storage

A super conducting coil can be used to store energy. A trapped magnetic field of

H oersted stores energy of 2

8

H

ergs / cm

3.

Super conductor fuse and breaker

Some insulating materials exhibit super conductivity at lower temperature. Thin films of

such materials can be used instead of fuse because when more current greater than critical

current pass through them, then they change into normal state.

In normal state, they are insulators they would not conduct current, so it will act as

a fuse.

In breaker, a long thin film of superconductor is used. In normal state, thin film

possess high resistance. In this situation lead is used.

Fig. (18) Electron – Electron interaction through phonons

SQUIDS

SQUIDS are fundamentally superconducting Rings that act as storage devices for

magnetic flux. SQUIDS can be used to store and measure magnetic fields.

SQUIDS are known as super conducting Quantum interference devices

These SQUIDS are used to localize epileptic centers deep within the brain.

The short-circuiting action of an epileptic centre produces electrical currents that generate

a distinctive magnetic signature.

Doctors place an array of a dozen SQUID magnetometers around the patients head.

Computer can analyze the data from all the sensors together a three dimensional picture

activity within the brain.

This technique is known as magnetic Encephalography (MEG).

SQUIDS are used for detecting ORE deposits.

SQUIDS are used for detecting earthquakes.

Cooled

1K q

1K 2K

q

Virtual

phonon

2K q

Fig. (17) a scattering of electrons

in

normal conductor

Fig (17)b. Formation of cooper

pairs in a super conductor

e

e

e e

+ve ion cores

Electrons

Cooper pairs

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

Ion core

Fast electrical switching (Cryotron) Applying a magnetic field greater than (Hc),

changes the super conductor to normal state and when the field is removed it regains the

superconducting state. This principle is used in cryotron switch.

Superconductors in computers

A super conducting ring is used in memory cell. When persistent current passes through a

super conducting state, then it is said to be in „1‟ state.

In normal state, current will not pass through it, and it is in „0‟ state.

Thus a super conducting can be used as a memory device.

Magnetic levitation (Meg Lev Vehicles)

Meg lev vehicles mean magnetically levitated vehicles.

The diamagnetic property of a super conductor (Meissonier effect) is the Basis for

magnetic levitation.

This concept is used for high speed transportation.

Fast electrical switching (Cryotron)

Applying a magnetic field greater than (Hc), changes the super conductor to normal state

and when the field is removed it regains the superconducting state. This principle is used

in cryotron switch.

UNIT-V PHYSICS OF

NANOMATERIALS

PHYSICS OF NANOMATERIALS Introduction:-

Nano meaning that 10-9.

A manometer is one thousand millionth of a meter. i.e. a nanometer equal to 10-9

m. In elements

of atoms are very small. The diameter of a single atom can vary from 0.1 to 0.5 nm depending

on the type of the element. For example carbon atom is approximately 0.15nm in diameter.

Also diameter of Red Blood Cell (RBC) is approximately 7000nm.

Diameter of water molecule is around 0.3 nm. Thickness of human hair is around 80,000nm.

Nanomaterials could be defined as those materials which have structured components with size

less than 100nm at least in one dimension.

Nanoscience:-

Nano science can be defined as the study of phenomena and manipulation of materials at

atomic, molecular and macro molecular scales, where properties differ significantly from those at

a larger scale.

Nano technology:-

Nano technology can be defined as the design, characterization, production and application of

structures, devices and systems by controlling shape and size at the nano scale.

Nano materials’:-

If we take any material it will be composed of grains, which in turn comprise of many atoms.

depending on the size, the grains may be visible or invisible to the naked eye.

Conventional materials have grains of size varying from hundred of microns to centimeters.

Nano materials could be defined as those materials which have structured components with size

less than 100nm at least in one dimension.

Examples:-

One dimensional Nanomaterail:-

Materials that are on nanoscale in one dimension are layers such as thin films or surface coatings

.In these materials, the particles are layered on layers or multilayer‟s.

Two dimensional nano materials:-

Materials that are on nano scale in two dimensions are called two dimensional Nanomaterials.

Ex. nanowires and nano tubes.

This consists of ultrafine grains laid over layers.

Three dimensional nano materials:-

Materials that are on nanoscale in three dimensions are particles or grains.

Examples are precipitates, colloids and quantum dots.

Basic principles of nanometerials:

The properties of namo meterials are different from those of bulk materials.

Two important factors that make the nanometerials to differ significantly from other materials

are increased surface area and quantum effects.

1. Increase in surface area to volume ratio:

Nano materials have a relatively larger surface face area when compared to the same volume (or

mass) of the material produced in a larger form

Let us cosider a shpere of radius 'r'

Surface area = 24 r and Volume =

34

3r

Surface area to volume ratio = 2

3

4 3

4

3

surfacearea r

Volume rr

Thus when the radius of the sphere decreases its surface area to volume ratio increases.

Let us consider another example.

For a cube of 1 unit volume shown in figure (1)

the surface area is 6m2

When it is divided into eight pieces its suface area becomes 12m2.

When the same volume is divided into 27 pieces its surface ares becomes 18m2

When a given volume is divided into smaller pieces the surface area increases.

As particle size decreases, a greater proportion of atoms are fand at the surfcae compared to there

inside. for example a particle of size 30nm has 5% of its atoms on its surface at 10nm 20% of its

atoms, and at 3nm 50% of its atoms on its surface.

Diue to this, nano materials are more chemically reactive. Some materials in bulk form are inert;

when they are in nano form they are reactive.

This affects their properties.

Quantum confinement: When Atoms are isolated the energy levels are discrete. when very

large number of atoms are closely packed to form solid the enery levels split and form bands.

Nano materials represent intermediate stage when the material is of nano size and nano scale, the

energies of the electron changes.

The quantum confinement effect is observed when the size of the particle is too small to be

comparable to the wavelength of electrons. Here motion of electrons is restricted in specific

energy levels.

Here band gap also increases.( Eg 1/ )

This effect is called quantum confinement

Due to this the electrical, optical and megnetic properties of the nanomaterials changes.

Fabrication of nanomaterails:-

The nanometerials can be synthesized by two techniques namely top-down and bottom up

techniques

In the bottom-up approach, the nanomaterials are syntherized by assembling the atoms or

moleculers together to form the nanomaterials

In the top-down approach the bulk solids are disassembled (broken down to pieces) into finer

pieces until the particles are in the order of nanoscale. The schematic representation of the

synthesis and fabrication of nanomaterials are shown in figure above.

I n top down method there are different techniques to fabricate nonmaterial‟s

For example Ball milling and sol-gas are the methods used for fabricating nonmaterials in top-

down approach

The chemical vapor disposition method, and plasma arching method are the methods employed

in the fabrication of nanomaterials in bottom up approach

Fabrication of nanomaterials:-

Ball Milling method

Ball milling method is a top down method. It is also known as mechanical crushing. This is a

simple method to synthesize all classes of nano materials. This method is used to produce nano

crystalline or amorphous materials.

The mechanical attrition mechanism or mechanical crushing mechanism is used to obtain

nanocrystalline strctures.The nano materails are prepared from single phase powders or

dissimilar powders or amorphase materials. The ball milling method is shown in figure above.

Depending on the material to be synthesised refractory balls or steel balls or plastic balls are

used.

When the balls are rotating with certain RPM, the enery is transferred to the powders.

This redues the size to the powder particles to the nanoscaled particles. Here the nano particles

are produced due to shear action between the balls and the metal pieces. The energy transferrred

to the powder from the balls depends on the factors such as rotational speed of the balls, size of

the balls number of balls, milling time, the ratio of the ball to the powder mass and milling

atmosphere.

By using a cryogenic liquid the brittleness of the particles can be increased.

Care should be taken to prevent oxidation during the process of milling. For fabricating softer

materials, usually harder balls will be chosen. Usually alumina and zirconium are used widely as

balls for synthesizing the nanomaterial. This is because they have high grinding restistance

values.

In this method scaling can be achieved up to tonnage quantity of materials. in this technique non

metal oxides cannot be fabricated due to contamination of milling media.

Sol- Gel method:The sol-gel method is a wet chemical technique. That is here chemical solution

deposition technique is used for the production of high purity and homogeneous nanomaterials.

Here in this method metal oxide nano particles are fabricated

The starting material from a chemical solution leads to the formation of colloidal suspensions

know as SOL. Then the sol evolves toward the formation of an inorganic network containing a

liquid phase called the Gel. The removal of the liquid phase from the sol yields the gel.

The particle size and shape are controlled by the sol/gel transition

Usually metal alkoxides and metal chlorides are used as starting (precursor) is diluted or

dissolved in water or dilute acid in an alkaline solvent. This process is called hydrolysis..

From the sol containing a liquid phase, water molecules are removed by a process called

dehydration now a gel is formed.

The gel is dried up rapidly by heating it and the Nan particles are formed.

The schematic representation of the synthesis of nano particles using sol-gel method is shown in

the figure below.

Aero gel is a highly performs material like glass and glass ceramics.

Sol-gel derived particles find applications in optics, electronics, energy, space etc.,

Chemical Vapor Deposition Method: Ina chemical vapor deposition (CVD) the atoms or molecules which are in gaseous state are

allowed to react homogenously or heterogeneously.

In a homogenous CVD, the particles or atoms or molecules in the gas phase are diffused towards

the cold surface. This occurs only due to thermophoreic forces.

The diffused particles can be scrapped from the cold surface to get nano powder

Also the differed particles are deposited into a subtract to form a film know as Particulate Film.

In a heterogeneous CVD, a dense film of nano particles is obtained on the surface of substrate

.In CVD method particle size crystal structure and chemical composition can be controlled

The schematic representation of CVD is shown in figure above. The metal organic precursor is

introduced into the hot zone of the reactor with flow controller.

The precursor is vaporized by using inductive heating or resistive heating. An inert gas like

argon or neon is used as carrier gas. The evaporated matter consists of hot atoms which undergo

condensation into small clusters through a homogenous nucleation,.

Other reactants are added to the clusters to control the chemical reactions. The cluster size is

controlled by controlling rate of evaporation rate of condensation and rate of gas flow. The

condensed clusters are allowed to pass through the cold finger.

The nano particles are collected by using a scrapper this is shown in figure above.

The CVD method is used to produce defect free nano particles.

Properties of Nano materials.

Physcial propereties: Usually for nano particles, surface to volume ratio increases due to this there is a variation in

material properties. The following are the physical properties.

1. The inter atomic spacing decreases with the nano scale. This is because of short range

core-core repulsion

2. Melting paint decreases due to decrease in size of the particle

3. Due to increase in surface area surface pressure decreases.

CHEMICAL PROPERTIES:-

1. When particle size is reduced to nano scale the electronic bands in the metals becomes

narrow. This leads to the transformation of the delocalized electronic states into more

localized molecular bands. This results in the increase of ionization potential.

2. The large surface to volume ratio, the variations in geometry and electronic structures

takes place. This will have a strong effect an catalytic properties

3. The chemical potential increases

4. When particles size decreases the hydrogen absorbing capacity increases.

Mechanical properties:-

Most metals are made up of small crystalline grains. If there grains are nanoscale in size the

interface area within the material greatly increases, which enhances its strength.

For example nano crystalline nickel is as strong as hardened steel.

It low temperatures a reduction in grain size lowers the transition temperature in steel from

ductile to brittle the average grain size and yield strength are given by Hall petch relation

0

K

d

= yield strength.

0 =Friction Stress.

K= Constant of Proportionality.

d= Average grain size.

The relation between Hardness and average grain is given by

i

KH H

d

H= Hardness.

iH = Vickers hardness

K= Constant of Proportionality.

d= Average grain size.

At high temperatures, the nonmaterials behave like superplasitc materials. Super plastic

materials will have extensive tensile deformation without fracture. In nonmaterial‟s the

occurrence of super plastic temperature decreases due to the decrease in grain size. MAGNETIC PROPERTIES:-

In nano magnetic particles the magnetic properties are different from the bulk material.

When the material is at nanoscale, the particle will have only single domain

The coercivityl and saturated value of magnetization values increases with a decrease in

grain size. Nano particles are more magnetic than bulk material. The magnetic moment of

nano cobolt particles of size 2nm is found to have 20% higher value than that of bulk cobalt.

Nano particles of non magnetic solids are found to be magnetic.

At small size the cluster become spontaneously magnetic.

The following table illustrates the behaviour of nano particles.

Metal Bulk Cluster

Na,K Paramagnetic Ferromagnetic

Fe,W,Ni Ferromagnetic Super paramagnetic

Rh Paramagnetic Ferromagnetic

Electrical properties:--

The ionization potential at small sizes is higher than that for the bulk materials.

This is become of quantum confinement effect the electronic bands in metals become narrower

In nano ceranics and magnetic composites the electrical conductivity increases with reduction in

particle size. Bulk silicon is an insulator which becomes a conductor in nano phase.

Usually the electrical conductivity increases with reduction in particle size.

OPTICAL PROPERTIES:-

Nano crystalline systems have novel optical properties. If semi conductor particles are

made small enough quantum effects come into play. This limits the energies of particles at which

electrons and holes can exist.

Golden nano shperes of 100nm size appears orange in color.

Gold nano spheres of 50nm size appear green in color.

Therefore the optical properties can be changes by controlling the particle size.

Nano matericles can be used as large electro chrome devices.

Thermal Properties:

Thermal conductivity increases with decrease in size of particle.

Melting of gold decreases from 1200K to 800K when the size decreases from 300 0A to 200

0

A.

Stable aluminium becomes combustible in nanophase.

Solid gold becomes liquid gold in nano phase.

APPLICATIONS OF NANO MATERIALS:-

Materials technology:-

1. Magnets made of nano crystalline yttrium- samarium-cobalt grains possess unusual

magnetic properties. This is because they are having large interface area. Nono

magnetic crystals will have high coactivity .they are used in motors and analytical

instruments like magnetic resonance imaging (MRI)

2. Nano sized titanium diozide and zincoxide are currently used in sunscreens. They

absorb and reflect ultra violet rays (UV). They are transparent to visible light.

3. Nano engineered membrane could potentially lead to more energy- efficient water

purification processes. They are used in desalination water plants by reverse osmosis.

4. Nano sized iron oxide is used in lipsticks as a pigment

5. Ceramics are hard, brittle and difficult to machine. However with a reduction in a

grain size to the nano scale ductility in ceramics can be increased. Zirconia, normally

a hard, brittle ceramic, and can be rendered super plastic. If can be deformed up to

300%

6. An important use of nano particles and nano tubes in composites carbon fibres and

bundles of multiwall CNTs are used in composites having potential long term

applications.

7. Carbon nano particle act as fillers in a matrix. They are used as a filter to reinforce car

tires.

8. Clay particles based composites containing plastics and nano-sized flakes of clay also

used in the fabrication of car bumpers.

9. Improved control porosity at the nanoscale has applications in textiles. Breathable

water proof and stain resistant fabrics can be fabricated using nano materials.

10. Nano particles are having high surface area. They can be used as catalytically active

agents.

11. Nano magnetic fluids can be prepared by using nano magnetic materials. Smart

magnetic fluids are used as Vacuum seals. Viscous dampers cooling fluids magnetic

separators.

12. Unusual color paints can be prepared by using nano particles; this is because nano

particles exhibit different optical properties.

INFORMATION TECHNOLOGY:-

1. Nano scale fabricated magnetic materials are used in storage of data.

2. Nano crystalline zinc selenide, zinc sulphide, cds and telluride fabricated by

sol-gel technique are the materials for high emitting phosphors and are used in

flat panel displays.

3. Nano particles are used for information storage.

4. Nano dimensional photonic crystals are used in chemical/optical computer.

5. Coatings with thickness controlled at the nano scale are used in optoelectronic

devices.

BIOMEDICALS:-

1. nano crystalline silicon carbide is used for artificial values of heart

because of low weight high strength and inertness

2. Biosensitive nano particles are used for tagging of DNA and DNA

chips.

3. Nano structured ceramics readily interact with bone cells and hence are

used as implant material

4. Controlled during delivery and controlling the decrase are possible with

nano technology.

5. Nano materials are used as agents in cancer therapy

ENERGY STORAGE:-

1. Addition of nano particles (Cerium oxide) to diesel fuel improves fuel

economy by reducing the degradation of fuel consumption over time

2. Nano particles are having high absorbing capacity nano particles of nickel,

platinum are used in hydrogen storage devices

3. Nano particles are used in magnetic refrigeration

4. Metal nano particles are very useful in the fabrication of Ionic batteries.