UNIT 1. - ELASTICITY Introduction:A rigid body can be defined as one which does not undergo any deformation when external forces act on it. When forces are applied on a rigid body the distance between any two particles of the body will remain unchanged however large the force may be. In actual practice no body is perfectly rigid. For practical purposes solid bodies are taken as rigid bodies. When external force acts on it bring about a change in its Length, volume or shape. Such forces are called deforming force.

What happens to the body when these forces are removed?

The external forces acting on the body compel the molecules to change their position. Due to

these relative molecular displacements, internal forces are developed within the body that

tends to oppose the external deforming force. When the deforming forces are removed, the

internal forces will tend to regain the shape and size of the body.

How does one account for this?

Under the action of the external force, the body changes its form because; the molecules

inside it are displaced from their previous positions. While they are displaced, the molecules

develop a tendency to come back to their original positions, because of intermolecular

binding forces. The aggregate of the restoration tendency exhibited by all the molecules of

the body manifests as a balancing force or restoring force counteracting the external force.

Elasticity is the property by virtue of which material bodies regain their original shape

and size after the deforming forces are removed.

When this external force is removed, the body regains its original shape and size. Such bodies

are called elastic bodies. Example: steel, glass, ivory, quartz…etc are elastic bodies. The

bodies which do not regain their original shape and size are called plastic bodies. No body is

either completely elastic or completely plastic. The property of elasticity is different in

different substances .steel is more elastic than rubber .liquids and gases are highly elastic.

STRESS AND STRAIN

LOAD: It implies the combination of external forces acting on a body and its effect is to change the form or the dimensions of the body.

STRESS: Restoring force per unit area developed inside the body is called stress. Since the magnitude of the restoring force is exactly equal to that of the applied force, stress is given by the ratio of the applied force to the area of its application.

Normal stress: Restoring force per unit area perpendicular to the surface is called normal stress.

1

D D1 C C1

θθF

Tangential stress: Restoring force parallel to the surface per unit area is called tangential stress.

STRAIN: The change in the form of the body i.e., the deformation produced by the external force, accompanies a change in dimensions.The ratio of change in dimensions to the original dimensions is called strain. But the way in which the change in dimensions is produced depends upon the form of the body and the manner in which the force is applied.

Deformation is of three types, resulting in three types of strains, defined as follows:

i) Linear strain or Tensile strain: If the shape of the body could be approximated to the form of a long wire and if a force is applied at one end along its length keeping the other end fixed, the wire undergoes a change in length.

If x is the change in length produced for an original length L then,

Linear strain = =

ii) Volume strain: If a uniform force is applied all over the surface of a body, the body undergoes a change in its volume (however the shape is retained in case of solid bodies). If v is the change in volume to an original volume V of the body then,

Volume strain = =

iii) Shear strain: If a force is applied tangentially to a free portion of the body, another part being fixed, its layers slide one over the other; the body experiences a turning effect and changes its shape. This is called shearing and the angle through which the turning takes place is called shearing angle (θ).

HOOKE’S LAWIt states that “stress is proportional to strain” (provided strain is small), so that in such a case the ratio of stress to strain is a constant, called the modulus of elasticity or coefficient of elasticity.i.e., stress α strain,

Or,

STRESS - STRAIN RELATIONSHIP IN A WIRE

2

A B

When the stress is continually increased in the case of a solid, a point is reached at which the strain increases rapidly. The stress at which the linear relationship between stress and strain ceases to hold good is referred to as the elastic limit of the material. Thus, if the material happens to be in the form of a bar or a wire under stretch, it will recover its original length on the removal of the stress so long as the stress is below the elastic limit, but if this limit is exceeded, it will fail to do so and will acquire what is called as ‘permanent set’. The relationship between stress and strain is studied by plotting a graph for various values of stress and the accompanying strain. This is known as stress-strain diagram.The straight and slopping part OA of the curve clearly shows that the strain produced is directly proportional to the stress applied or the Hook’s law is obeyed perfectly up to A and that, therefore, on the removal of the stress, it will recover its original condition of zero strain, represented by O.As soon as the elastic limit is crossed, the strain increases more rapidly than the stress, and the graph curves along AB, the extension of the wire now being partly elastic and partly plastic. Hence, on being unloaded here, say, at the point B, it does not come back to its original condition along AO, but takes the dotted path BC, so that there remains a residual strain OC in it, which is permanent set acquired by the wire.Beyond the point B, for practically little or no increase in stress(or the load applied) there is a large increase in strain (i.e., in the extension produced) up to D, so that the portion BD of the graph is an irregular wavy line, the stress corresponding to D being less than that corresponding to B. This point B where the large increase in strain commences is called the yield point, the stress corresponding to it being known as the yielding stress.The yielding ceases at D and further extension, it becomes plastic, can only be produced by gradually increasing the load so that the portion DF of the graph is obtained, the cross sectional area is called the ultimate strength or the tensile strength of the wire and is also sometimes termed the ‘breaking stress’. The extension of the wire goes on increasing beyond F without any addition to the load, even if the load is reduced a little, and wire behaves as though it were literally ‘flowing down’. This is because of a faster rate of decrease of its cross-sectional area at some section of its length where local constriction, called a “neck” begins to develop, with the result that, even if the load be not increased, the load per unit area or the stress becomes considerably greater there, bringing about corresponding increase in strain or extension in the wire. The load is there, decreased, i.e., the stress reduced at this stage and the wire finally snaps or breaks at E, which thus represents the breaking point for it.Corresponding to the three types of strain, we have three types of elasticity:

3

D

O

B E

Elastic Behavior

Elastic Limit

C

A

STRESS

S T R A I NPermanent set

Yield point

Breaking point

FPlastic Range

Q

S

B B1Q1

P1

θR

a) Linear Elasticity or Elasticity of length called Young’s modulus, corresponding to linear or tensile strain:

When the deforming force is applied to the body only along a particular direction, the change per unit length in that direction is called longitudinal, linear or elongation strain, and the force applied per unit area of cross section is called longitudinal or linear stress. The ratio of longitudinal stress to linear strain within elastic limit is called the coefficient of direct elasticity or Young’s modulus and is denoted by Y or E. If F is the force applied normally, to a cross-sectional area a, then the stress is F/a. If L is original length and x is change in length due to the applied force, the strain is given by x/L,so that,

Y = = N/m2

b) Elasticity of volume or Bulk modulus: When the deforming force is applied normally and uniformly to the entire surface of a body, it produces a volume strain (without changing its shape in case of solid bodies). The applied force per unit area gives the normal stress or pressure. The ratio of normal stress or pressure to the volume strain without change in shape of the body within the elastic limits is called Bulk modulus. If F is the force applied uniformly and normally on a surface area (a) the stress or pressure is F/a or P and if v is the change in volume produced in an original volume V, the strain is given by v/V and therefore

K = N/m2

Bulk modulus is referred to as incompressibility and hence its reciprocal is called compressibility (strain per unit stress).

c) Modulus of Rigidity (corresponding to shear strain) : In this case, while there is a change in the shape of the body, there is no change in its volume. It takes place by the movement of contiguous layers of the body, one over the other. There is a change in the inclinations of the co-ordinates axes of the system or the body.Consider a rectangular solid cube whose lower face DCRS is fixed, and to whose upper face a tangential force F is applied in the direction as shown. Under the action of this force, the layers of the cube which are parallel to the applied force slide one over the other such that point A shifts to A1, B to B1, P to P1 and Q to Q1 that is the planes of the two faces ABCD and PQRS can be said to have turned through an angle . This angle is called the angle of shear or shearing strain. Tangential stress is equal to the force F divided by area a of the face APQB.

Hence tangential stress = θ = PP1 / PS =

The rigidity modulus is defined as the ratio of the tangential stress to the shearing strain.

4

D

A1

PA

θL

F

C

A A1

A A1 B B1

L

Rigidity modulus η = = N/m2

Poisson’s Ratio (): In case of any deformation taking place along the length of a body like a wire, due to a deforming force, there is always some change in the thickness of the body. This change which occurs in a direction perpendicular to the direction along which the deforming force is acting is called lateral change.

Within elastic limits of a body, the ratio of lateral strain to the longitudinal strain is a constant and is called Poisson’s ratio.

If a deforming force acting on a wire of length L produces a change in length x accompanied

by a change in diameter of d in it which has a original diameter of D, then lateral strain =

and Longitudinal strain α =,

Poisson’s ratio, σ = =

There are no units for Poisson’s ratio. It is a dimensionless quantity.

RELATION BETWEEN THE THREE MODULI OF ELASTICITY:

When body undergoes an elastic deformation, it is studied under any of the three elastic modulii depending upon the type of deformation. However these modulii are related to each other. Now, their relation can be understood by knowing how one type of deformation could be equated to combination of other types of deformation.

EQUIVALENCE OF SHEAR TO COMPRESSION AND EXTENSION:

Consider a cube whose lower surface CD is fixed to a rigid support. Let ABCD be its front face. If a tangential force F is applied at the upper surface along AB, its cause’s relative displacements at different parts of the cube, so that A moves to A1 and B moves to B1 through a small angle. In this way the diagonal AC will be shortened to A1C and diagonal DB will be increased in length to DB1.let θ be the angle of shear which is very small in magnitude.

Let length of each side of the cube = L.AA1=BB1= l.As θ is small, θ = tanθ = l / L.As diagonal DB increases to DB1and diagonal AC is compressed to AC1, We have Extension strain along DB = NB1/DB

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L θ

ll l

θθM N

D C

A BF

Compression strain along AC=MA1/CAIf L is the length of each side of the cube, then, AC = DB= L (By Pythagoras theorem).

Now, NB1=BB1cos45 = and AM=AA1 cos45 =

Elongation strain along DB = NB1/DB=

And compression strain along AC = MA1/CA=

From the above two relation it is clear that a simple shear θ is equivalent to an extension strain and compression strain at right angles to each other and each of value θ/2.The converse is also true i.e., simultaneous equal compression and extension at right angles to each other are equivalent to shear.

Elongation strain + compression strain = , the shearing strain.

EQUIVALENCE OF SHEARING STRESS TO A COMPRESSIVE STRESS AND A TENSILE STRESS:

Compressive stress (or compression) is the stress state caused by an applied load that acts to reduce the length of the material in the axis of the applied load.

Tensile stress is the stress state caused by an applied load that tends to elongate the material in the axis of the applied load, in other words the stress caused by pulling the material.

Consider a section of cube ABCD having o as centre and whose lower face CD is clamped and let a tangential force F being applied on the upper face AB in the direction shown by arrow. If L is the length of each side of the cube, then tangential force per unit area will be F/l2 = T. The face CD experiences equal and opposite forces and these two forces acting along the face AB and CD form a couple which has a tendency to rotate the cube in the clockwise direction. As the cube is unable to rotate, a couple is called in to action which counter balances the applied couple and this couple is called restoring couple.

The restoring couple consists of forces acting tangentially on the faces CB and AD and each of magnitude F. under the action of these four forces (along AB, CD, BC and AD) each of F the cube is deformed as these four forces are in equilibrium the cube does not move. The force acting along AB can be resolved along AO and OB i.e., into two equal components of magnitude f .similarly the remaining three forces

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CD

F

F

F

can be resolved into three pairs of components each of f. Thus we have four forces of value F each acting tangentially on the faces of the cube which are resolved in to eight forces each of value f acting along the diagonals as shown in the figure. The force f acting along AC tends to compress the diagonal and those acting along DB tend to produce extension along that diagonal.

Therefore the area of the triangular face ABC cut parallel to AC and perpendicular to the plane of the paper. = DB x AO = L x L= L2 .

As DB = = = L .

Since the force acting on either side of the section normal to it is 2f. Therefore

compressive stress along AC = =

But f = F cos45 =

Compressive stress along AC = = = T = Tangential stress---------------- (1)

Similarly tensile stress along BD = = = T = Tangential stress---------------- (2)

From (1) and (2) it is clear that a shearing stress is equivalent to an equal linear tensile stress and an equal linear compressive stress at right angles to each other.

RELATION BETWEEN Y AND α

Consider a cube of unit side subjected to unit tension along one side. Let α be the elongation per unit length per unit tension along the direction of the force. Therefore,

Stress = = 1

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L

2f

B D

A

Similarly, linear strain = = = α

Y = =

RELATION BETWEEN K, α AND βConsider a cube ABCDEFGH of unit volume, as shown in the diagram. Let Tx be the stress acting on the faces ABEF and CDGH. Let Ty be the stress which acts perpendicular to the faces ABCD and EFGH. Similarly, Tz is acting perpendicularly between faces ACEG and BDFH. Ty Tz

Tx Tx

Tz Ty

Each stress produces an extension in its own direction and a lateral contraction in the other two perpendicular directions. Let α be the elongation per unit length per unit stress along the direction of the forces and β be the contraction per unit length per unit stress in a direction perpendicular to the respective forces. Then stress like Tx produces an increase in length of α Tx in X-direction: but since other two stresses Ty and Tz are perpendicular to X-direction they produce a contraction of β Ty and β Tz respectively in the cube along X-direction .Hence, a length which was unity along X-direction becomes , 1+ α Tx - β Ty - β Tz.Similarly along Y and Z directions the respective length become, 1+ α Ty - β Tz - β Tx. 1+ α Tz - β Tx - β Ty.

Hence the new volume of the cube is = (1+ α Tx - β Ty - β Tz) (1+ α Ty - β Tz - β Tx) (1+ α Tz - β Tx - β Ty)

Since α and β are very small, the terms which contain either powers of α and β, or their products can be neglected.

New volume of the cube = 1 + α(Tx+ Ty+ Tz)- 2β(Tx+ Ty+ Tz), = 1+ (α-2β)(Tx+ Ty+ Tz)

If Tx= Ty= Tz= T Then the new volume =1+ (α-2β) 3TSince the cube under consideration is of unit volume, increase in volume = [1+3T (α-2β)]-1 = 3T (α-2β)

If instead of outward stress T, a pressure P is applied, the decrease in volume = 3P (α-2β).

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C

F

EG

H

A A1 B B1

L

Volume strain = =

K =

RELATION BETWEEN Y, K, AND α

K = ( and Y =

RELATION BETWEEN AND

Consider a cube with each of its sides of length L under the action of tangential stress T. let tangential force F be applied to its upper face. It causes the plane of the faces perpendicular to the applied force F turn through an angle θ. as a result diagonal AC undergoes contraction and diagonal DB1undergoes elongation of equal amount. Now, shearing strain occurring along AB can be treated as equivalent to a longitudinal strain, along AB1 and an equal lateral strain along the diagonal A1C i.e., perpendicular to DB. Let α and β be the longitudinal and lateral strains per unit length per unit stress which is applied along AB , since T is the applied stress, extension produced for the length DB due to tensile stress = T. DB.α and extension produced for the length DB due to compression stress = T.DB.β.

Total extension along DB = DB.T.(α+β), It is clear that the total extension in DB is approximately equal to B1N when BN is perpendicular to DB1.

B1N= DB.T(α+β),B1N=( ).T(α+β), ( DB cos45 =L)

Now, =45 and since θ is always very small,ABD ,

B1N = BB1cos45 = ( )

9

l l

θθM N

D C

Or,

Or,

RELATION BETWEEN Y, K and

We know that and K

Rearranging which we get,

and =1-2σ

Adding the above we get, ,

Or

Relation between K , and σWe have the relations Y = 2 (1 + σ), and Y = 3K (1-2σ)Equating the above equations we get,2η + 2ησ = 3K – 6Kσ,

2ησ + 6Kσ = 3K - 2η,Or , σ(2η + 6K) = 3K - 2η,

Or, σ =

Torsion of a cylinder A long body which is twisted around its length as an axis is said to be under torsion. The twisting is brought into effect by fixing one end of the body to a rigid support and applying a suitable couple at the other end. The elasticity of a solid, long uniform cylindrical body under torsion can be studied, by imagining it to be consisting of concentric layers of the material of which it is made up of. The applied twisting couple is calculated in terms of the rigidity modulus of the body.

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Expression for the Torsion of a cylindrical rod: Consider a long cylindrical rod of length ‘L’ and radius ‘R’ rigidly fixed at its upper end. Let OO' be its axis. Imagine the cylindrical rod is made up of thin concentric hollow cylindrical layers each of thickness ‘dr’.

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The rod twisted at its lower end, and then the concentric layers slide one over the other. This movement will be zero at the fixed end and gradually increased along the downward direction. Let us consider one concentric circular layer of radius ‘r’ and thickness ‘dr’ . Any point ‘X’ on its uppermost part would remain fixed and a point like ‘B’ at its bottom moves to ‘B'’. Now the gives the angle of shear. Since is also small, the movement length . Also, if , the length B B' = rθ. Now, the cross sectional area of the layer under consideration is 2r dr . If ‘F’ is the shearing force, then the shearing stress T is given by

Shearing force F = T(2rdr) Rigidity modulus n = Shearing stress/shearing strain.

The moment of the force about

This is only for the one layer of the cylinder.

Therefore, twisting couple acting on the entire cylinder

Couple per unit twist is given by C=Total twisting couple / angle of twist.

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O

O1

dr

φ

θ

X

Torsional PendulumTorsional pendulum consists of a heavy metal disc is suspended by means of a wire. When the disc is rotated in a horizontal plane so as to twist the wire, the various elements of the wire undergo shearing strain. The restoring couple of the wire tries to bring the wire back to the original position. Therefore disc executes torsional oscillations about the mean position.Let θ be the angle of twist made by the wire and C be the couple per unit twist.Then the restoring couple per unit twist = Cθ.

Therefore the angular acceleration produced by the restoring couple in the wire. a =

Let I be the moment of inertia of the wire about the axis. Therefore, we have, I = -Cθ

The above relation shows that the angular acceleration is proportional to angular displacement and is always directed towards the mean position. The negative sign indicates that the couple tends to decrease the twist on the wire. Therefore, the motion of the disc is always simple harmonic motion (SHM). Therefore, the time period of oscillator is given by relation,

Uses of torsion pendulum:(i) For determination of moment of inertia of an irregular body:

For determining the moment of inertia of an irregular body the torsion pendulum isFound to be very useful. First, the time period of the pendulum is determinedWhen it is empty and then the time period of the pendulum is determined after fixing a regular at the free end and after this the time period is determined by replacingThe regular body by the irregular body whose moment of inertia is to be determined. If I1 and I2 are the moments of inertia of the pendulum, regular body and irregular body respectively and,T1 and T2 are the time periods in the three cases respectively,then

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T=2π . … . . . . . . (1)

T1=2π . . . . . . . . . .(2)

T2=2π . . . . .. . . .. . (3)

From relations (2) and (1), we have

T12-T2=

And from (2) and (3), we have

T22- T2=

Bending of beams:

A homogenous body of uniform cross section whose length is large compared to its other dimensions is called a beam.

Neutral surface and neutral axis: Consider a uniform beam MN whose one end id fixed at M. The beam can be thought of as made up of a number of parallel layers and each layer in turn can be thought of as made up of a number of infinitesimally thin straight parallel longitudinal filaments or fibres arranged one closely next to the other in the plane of the layer. If a cross section of the beam along its length and perpendicular to these layers is taken the filaments of different layers appear like straight lines piled one above the other along the length of the beam. For a given layer, all its constituent filaments are assumed to undergo identical changes when that layer is strained.

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A BC DE F

A1 B1

C1 D1

E1F1

If a load is attached to the free end of the beam, the beam bends. The successive layers along with constituent filaments are strained. A filament like AB of an upper layer will be elongated to A1B1 and the one like EF of a lower layer will be contracted to E1F1. But there will always be a particular layer whose filaments do not change their length as shown for CD. Such a layer is called neutral surface and the line along which a filament of it is situated is called neutral axis. The filaments of a neutral surface could be taken as line along which the surface is intercepted by a cross section of the beam considered in the plane of bending as shown.

Neutral Surface: It is that layer of a uniform beam which does not undergo any change in its dimensions, when the beam is subjected to bending within its elastic limit.

Neutral axis: It is a longitudinal line along which neutral surface is intercepted by any longitudinal plane considered in the plane of bending.

When a uniform beam is bent, all its layers which are above the neutral surface undergo elongation whereas those below the neutral surface are subjected to compression. As a result the forces of reaction are called into play in the body of the beam which develop an inward pull towards the fixed end for all the layers above the neutral surface and an outward push directed away from the fixed end for all layer below the neutral surface. These two groups of forces result in a restoring couple which balances the applied couple acting on the beam. The moment of restoring couple is called restoring moment and the moment of the applied couple is called the bending moment. When the beam is in equilibrium, the bending moment and the restoring moments are equal.

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A1 B1

C1 D1

E1 F1

θ

Neutral axis

Bending moment of a beam: Consider a uniform beam whose one end is fixed at M. If now a load is attached to the beam, the beam bends. The successive layers are now strained. A layer like AB which is above the neutral surface will be elongated to A1B1 and the one like EF below neutral surface will be contracted to E1F1. CD is neutral surface which does not change its length.

The shape of each layers of the beam can be imagined to form part of concentric circles of varying radii. Let R be the radius of the circle to which the neutral surface forms a part.

CD=Rwhere is the common angle subtended by the layers at common center O of the circles. The layer AB has been elongated to A1B1. change in length =A1B1-ABBut AB=CD=RIf the successive layers are separated by a distance r then,A1B1=(R+r)Change in length=(R+r)-R = rBut original length = AB=R

Linear strain =

Youngs Modulus Y= Longitudinal stress/linear strainLongitudinal stress = Yx Linear strain

= Yx

But stress =

Where F is the force acting on the beam and a is the area of the layer AB.

Moment of this force about the neutral axis=F x its distance from neutral axis.

= F x r = Yar2 /R

Moment of force acting on the entire beam = Σ

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P

P1

M NY

=

The moment of inertia of a body about a given axis is given by Σmr2, where Σm is the mass of the body. Similarly Σar2 is called the geometric moment of Inertia Ig.

Ig=Σar2 =Ak2 , where A is the area of cross section of the beam and k is the radius of gyration about the neutral axis.

Moment of force =

Bending moment =

Single Cantilever

If one end of beam is fixed to a rigid support and its other end loaded, then the arrangement is called single cantilever or cantilever. Consider a uniform beam of length L fixed at M. Let a load W act on the beam at N. Consider a point on the free beam at a distance x from the fixed end which will be at a distance (L-x) from N. Let P1 be its position after the beam is bent.

Bending moment = Force x Perpendicular distance. = W(L-x)

But bending moment of a beam is given by

= W(L-x) ----------------(1)

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

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x (L-x)

But if y is the depression of the point P then it can be shown that

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

where R is the radius of circle to which the bent beam becomes a part.

Comparing equations 2 and 3

Integrating both sides

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

C is constant of integration

But dy/dx is the slope of the tangent drawn to the bent beam at a distance x from the fixed end. When x=0, it refers to the tangent drawn at M, where it is horizontal. Hence (dy/dx)=0 at x=0. Introducing this condition in equation (4 )we get 0=C1

Equation 4 becomes

Integrating both sides we get

-----------------(5)

where C2 is constant of integration, y is the depression produced at known distance from the fixed end. Therefore when x=0, it refers to the depression at M, where there is obviously no depression. Hence y=0 at x=0. Introducing this condition in equation 5 we get

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At the loaded end, y=y0 and x=L

Therefore

Depression produced at loaded end is

Therefore the youngs modulus of the material of the cantilever is

-----------------(6)

Case (a): If the beam is having rectangular cross-section, with breadth b and thickness d then,

Ig = -----------------(7)

Substituting equation (7) in equation (6) we get

Case(b):

If the beam is having a circular cross section of radius r then,

---------------------(8)

Substituting equation (8) in equation (7) we get

Experiment to determine young’s modulus by single cantilever method.

Single cantilever: Single cantilever is a beam fixed horizontally at one end and the free end to be loaded. Experiment: The given material in the form of uniform bar whose one end is fixed to a rigid support is taken and an empty pan is attached to the other end. A pin is firmly placed on the top of the loaded end. A traveling microscope adjusted to the vertical transverse is focused to the tip of the pin and the reading is taken and entered in the

19

tabular column. A suitable weights are attached to the pan in a equal increments and the reading of the traveling microscope are noted for the corresponding positions of the tip of the pin.

Load (g)m1

TM reading (cm) MeanR1

Load(g)m2

TM readings (cm) MeanR2

Depression for the load (m2-m1)R2R1

Load increasing

Load decreasing

LoadIncreasing

Load decreasing

W W + 40

W + 10 W + 50

W + 20 W + 60

W + 30 W + 70

After the required value of the load in the pan is attained, weights are unloaded by the same amount in each step as they were added while loading. The corresponding readings are

entered in the tabular column. The value of is evaluated from the table. The length of

the beam between the point of emergence of the beam from the rigid support and the point where the pin is mounted is measured. The breadth ‘b’ and the thickness ‘d’ of the beam are also measured. The young’s modulus of the material of the beam is calculated by using the

formula N m-2.

b

l W

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UNIT 2. – QUANTUM MECHANICS

Quantum Mechanics

Newton’s laws describe the motion of the particles in classical mechanics and

Maxwell’s equations describe electromagnetic fields in classical electromagnetism. The

classical mechanics correctly explains the motion of celestial bodies like planets,

macroscopic and microscopic terrestrial bodies moving with non-relastivistic speeds.

However classical theory does not hold in the region of atomic dimensions. The classical

theory could not explain the stability of atoms, energy distribution in the black body radiation

spectrum, origin of discrete spectra of atoms etc. It also fails to explain large number of

observed phenomena like photo electric effect, Compton effect, Raman effect etc. The

insufficiency of classical mechanics led to the development of quantum mechanics. Quantum

mechanics is the description of motion and interaction of particles at the small scale atomic

system where the discrete nature of the physical world becomes important. With the

application of quantum mechanics, several problems of atomic physics have been solved.

Wave and particle duality of radiation

To understand the wave and particle duality, it is necessary to know what is a particle

and what is a wave.

A particle is a localized mass and it is specified by its mass, velocity, momentum,

energy etc. Whereas a wave is a spread out mass and is characterized by its wavelength,

frequency, velocity, amplitude, intensity etc. It is hard to think mass being associated with a

wave. Considering the above facts, it appears difficult to accept the conflicting ideas that

radiation has wave particle duality. However this acceptance is essential because the radiation

exhibits phenomena like interference, diffraction, polarization etc, and shows the wave nature

and it also the particle nature in photo electric effect, Compton effect etc.

Radiation, thus, sometimes behave as a wave and at some other time as a particle, this

is the wave particle duality of radiation. Here it should be remembered that radiation cannot

exhibit its wave and particle nature simultaneously.

de- Broglie’s concept of matter waves.

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Louis de-Broglie in 1924 extended the wave particle dualism of radiation to

fundamental entities of physics such as electrons, protons, neutrons, atoms, molecules etc. de-

Broglie put a bold suggestion that like radiation, matter also has dual characteristic, at a time

when there was absolutely no experimental evidence for wave like properties of matter

waves. de-Broglie hypothesis of matter waves is as follows.

In nature energy manifests itself in two forms namely matter and radiation.

Nature loves symmetry

As nature loves symmetry, if radiation act like wave sometime and like particle at

some other times then the material particles ( like electrons, protons etc) should also

act as waves at some other times.

The waves associated with matter in motion are called matter waves or de-broglie

waves or pilot waves

Wave length of matter waves: The concept of matter waves is well understood by

combining Planck’s quantum theory and Einstein’s theory. Consider a photon of energy E,

fre4quancy γ and wavelength λ.

By Planck’s theory

By Planck’s theory

Where p is the momentum of the photon.

Now consider a particle of mass m moving with a velocity and momentum p.

According to the do-Broglie hypothesis matter also has dual nature . Hence the wavelength λ

of

matter waves is given by

. This is the equation for the de-Broglie wave length

de – Broglie wavelength of an electron

Consider an electron of mass ‘m’ accelerated from rest by an electric potential V. The

electrical work done (eV) is equal to the kinetic energy gained by the electron.

22

therefore wavelength of electron wave

But

Note : Instead of a an electron, if a particle of charge ‘q’ is accelerated through a potential

difference V, then

Properties of matter waves

1. Lighter the particle, greater would be the wavelength of matter waves associated with

it.

2. Smaller the velocity of the particle, greater would be the wavelength.

3. For V = 0, λ is infinity ie, the wave becomes indeterminate. This means that matter

waves are associated with moving particles.

4. Matter waves are produced by charged or uncharged particles in motion. Where as

electromagnetic waves are produced only by a moving charged particle. Hence matter

waves are non – electromagnetic waves.

23

5. In an isotropic medium the wavelength of an electromagnetic eave is a constant

whereas the wave length of a matter wave changes with the velocity of the particle.

Hence matter waves are non- electromagnetic waves.

6. Velocity of any wave is greater by νλ . therefore for a matter wave, wave velocity ‘u’

is given

As the velocity of a particle can’t exceed the velocity of light ‘c’, the velocity of matter

waves is greater than the velocity of light.

7. A particle is a localized mass and a wave is a spread out spark. so, the wave nature of

matter introduces a certain un – certainty in the position of the particle.

8. Matter waves are probability waves.

Heisenberg’s uncertainty principle.

Heisenberg’s uncertainty principle is a direct consequence of dual nature of

matter. In classical mechanics, a moving particle at any instant has a fixed position in

space and a definite momentum which can be determined if the initial values are

known (we can know the future if we know the present)

In wave mechanics a moving particle is described in terms of a wave

group or wave packet. According to Max Born’s probability interpretation the

particle may be present anywhere inside the eave packet. When the wave packet is

large, the momentum can be fixed but there is a large infiniteness in its position. On

the other hand if the wave packet is small the position of the particle may be fixed but

the particle will spread rapidly and hence the momentum ( or velocity ) becomes

indeterminate. In this way certainty in momentum involves uncertainty in position

and the certainty in position involves uncertainty in momentum. Hence it is

impossible to know within the wave packet where the particle is and what is its exact

24

momentum. (we cannot know the future because we cannot know the present).

Thus we have Heisenberg’s uncertainty principle.

According to the Heisenberg’s uncertainty principle “ It is impossible to

specify precisely and simultaneously certain pairs of physical quantities like position

and momentum that describe the behaviour of an atomic system”. Qualitatively this

principle states that in any simultaneous measurement the product of the magnitudes

of the uncertainties of the pairs of physical quantities is equal to or greater than h/4π

( or of the order of h)

Considering the pair of physical quantities as position and momentum, we

have

ΔpΔx ≥ h/4π .….1

Where Δp and Δx are the uncertainties in determining the momentum and the position

of the particle. Similarly we have other canonical forms as

ΔEΔt≥ h/4π ……2

ΔJΔθ≥ h/4π ……3

Where ΔE and Δt are uncertainties in determining energy and time while ΔJ and Δθ

are uncertainties in determining angular momentum and angle.

Experimental illustration of uncertainty principle

As an illustration of Heisenberg’s Uncertainty Principle Neil

Bohr proposed a thought experiment to the inevitable error involved to locate the

position of an electron.

Let OP be the axis of an imaginary γ- ray microscope perpendicular to x-

direction. Consider a γ-photon of wavelength λ and momentum h/λ moving along x-

direction strikes a stationary electron at P and undergoes Compton scattering at an

angle α to the x- direction and the electron recoils though an angle φ as shown in the

following figure. Let λ’ be the wavelength of the scattered photon [according to

Compton scattering there will be a change in the wave length of the scattered photon

as it transfers part of its energy to the free electron]

25

Fig. γ- ray microscope

For the angle α < (90-θ ) and also for α > (90+θ ) the electron will not be seen

at all as it cannot enter the microscope. The scattered photon can enter the microscope

anywhere between PA and PB.

If the electron is scattered along PA

The x-component of momentum of the scattered photon = ( h/ λ’) cos α

= ( h/ λ’) cos ( 90-θ)

= ( h/ λ’) sin θ

The change in the momentum of the photon = ( h/ λ) - ( h/ λ’) sin θ - - - - - - - - - - (1)

The change in the momentum of the photon is equal to the gain in the momentum of

the electron and it is also the momentum of the electron because its initial momentum

is zero.

The momentum of the electron = ( h/ λ) - ( h/ λ’) sin θ - - - - - - - - - - (2)

If the electron is scattered along PB

The x-component of momentum of the scattered photon = ( h/ λ’)cos ( 90+θ)

= -( h/ λ’) sin θ

The change in the momentum of the photon = ( h/ λ) + ( h/ λ’) sin θ - - - -

(3)

The change in the momentum of the photon is equal to the gain in the

momentum of the electron and it is also the momentum of the electron because its

initial momentum is zero.

As the scattered photon can enter the microscope anywhere between

PA and PB there is an uncertainty in the momentum of the electron is given by

(∆Px )max

Maximum uncertainty in locating the position of the electron

(∆Px )max = [( h/ λ) + ( h/ λ’) sin θ] – [( h/ λ) – ( h/ λ’) sin θ]

(∆Px )max = 2h/ λ’ sin θ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -(4).

6

26

For the photon to be seen by the microscope, there should be a

Compton scattering of the photon and the recoil of the electron. Thus there is a change

in the position of the electron. The smallest distance between any two points that can

be resolved by a microscope is called the limit of resolution and it is equal to λ’/2sin

θ

The minimum uncertainty in the position of the electron = (∆x)min = λ’/2sin θ. - - - -

- (5).

From the equations (4) band (5) we have

∆Px .∆x ≈ λ’/2sin θ. 2h/ λ’ sin θ∆x ≈h

∆Px .∆x ≈h

As h> h/4π, the above equation is in accordance with the Heisenberg’s Uncertainty

principle ∆Px .∆x h/4π.

Schrödinger’s Wave Equation

In1926 Schrödinger starting with de-Broglie equation (λ = h/mv ) developed it into an

important mathematical theory called Wave mechanic which found a remarkable success in

explaining the behaviour of atomic system and their interaction with electromagnetic

radiation and other particles. In water waves, the quantity that varies periodically is the height

of water surface. In sound waves it is pressure. In light waves, electric and magnetic fields

vary. The quantity whose variation gives matter waves is called wave function (ψ). The value

of wave function associated with a moving body at a particular point x in space at a time t is

related to the likelihood of finding the body there at a time. A wave function ψ(x,t) that

describes a particle of entirely un known position moving in a positive x-direction with

precisely known momentum and kinetic energy may assume any one of the following forms:

Sin( wt -kx) , cos( wt -kx), ei(wt – kx), e-i(wt – kx), e-iwt, e-i kx or some linear combination of them.

Schrödinger wave equation is the wave equation of which the wave functions are the

solutions. It cannot be derived from any basic principles. It is an equation arrived at,

consistent with previous theoretical and experimental results and is able to predict new

verifiable results.

Time independent, one dimensional Schrödinger wave equation.

In many situations the potential energy of the particle does not depend on time

explicitly; the force that acts on it, and hence the potential energy vary with the position of

the particle only. The Schrödinger wave equation for such a particle is time independent

27

wave equation. Let ψ(x,t) be the wave function of the matter wave associate with a particle of

mass m moving with a velocity v. The differential equation of the wave motion is as follows.

The solution of the Eq.(1) as a periodic displacement of time t is

ψ(x,t) =ψ0(x) e-iwt …..(2)

Where ψ0(x) is amplitude of the matter wave.

Differentiating Eq.2 partially twice w.r.t. to t, we get

ψ0(x) e-iwt

ψ0(x) e-iwt

ψ0(x) e-iwt

- ψ …….(3)

Substituing Eq.3 in Eq.1 ……………………..(4)

We have

Substituting this in Eq4, we get

Substituting the wavelength of the matter waves λ=h/mv in Eq.6 we get

If E and V are the total and potential energies of the particle respectively then

The kinetic energy of the particle

28

Substituting this in Eq.7, we get

This is the Schrödinger time- independent, one dimensional wave equation.

Physical significance of the wave function

The wave function ψ(x,t) is the solution of Schrödinger wave equation. It gives a quantum-

mechanically complete description of the behaviour of a moving particle. The wave function

ψ cannot be measured directly by any physical experiment. However, for a given ψ,

knowledge of usual dynamic variables, such as position, momentum, kinetic energy etc.,of

the particle are obtained by performing suitable mathematical operations on it.

The most important property of ψ is that it gives a measure of the probability of finding a

particle at a particular position. In general ψ is a complex quantity whereas the probability

must be real and positive. Therefore a terrm called probability density is defined. The

probability P(x.t) density is a product of the wave function ψ and its complex conjugate ψ

*

.

ψ is also called the probability amplitude.

Normalization of wave function.

If ψ is a wave function associated with a particle then is the probability of

finding the particle in a small volume dτ . If it is certain that the particle is present in a

volume τ then the total probability in the volume τ is unity i.e . This is the

normalization condition and the wave function ψ is called normalization condition.

In one dimension the normalization condition is

29

Note: When the particle is bound to a limited region the probability of finding the particle at

infinity is zero i.e ., is zero.

Properties of wave function:

The wave function ψ should satisfy the following properties to describe the characteristics of

matter waves.

1. ψ must be a solution of Schrödinger wave equation

2. The wave function ψ should be continuous and single valued everywhere. Because it

is related to the probability of finding a particle at a given position at a given time,

which will have only one value.

3. First derivative of ψ with respect to its variables should be continuous and single

valued everywhere, because ψ is finite.

4. Ψ must be normalized so that ψ must go to 0 as x , so that over all the

space be a finite constant.

Eigen functions and Eigen values

The Schrödinger wave equation is a second order differential equation, it will have many

mathematically possible solutions (ψ). All mathematically possible solutions are not

physically acceptable solutions. The physically acceptable solutions are called well behaved

functions or Eigen functions (ψ). To an acceptable wave function ψ, it has to satisfy the

following criteria.

1. ψ is single valued

2. ψ and its first derivative with respect to its variable are continuous everywhere.

3. ψ is finite everywhere

Eigen functions: Once the Eigen functions are known, they can be used in Schrödinger wave

equation to evaluate the physically measurable quantities like energy, momentum etc.,

these values are called Eigen values. In an operator equation where is an

operator for the physical quantity and ψ is an Eigen function and λ is the Eigen value.

30

31

Applications of Schrodinger’s wave equation

1. For a Particle a in one-dimensional potential well of infinite depth. (or Particle in a box)

Consider a particle of mass ‘m’ moving freely in x- direction in the region from x=0 to x=a. Outside this region potential energy ‘V’ is infinity and within this region V=0.Outside the box Schrodinger’s wave equation is

This equation holds good only if =0 for all points outside the box i.e., , which means that the particle cannot be found at all outside the box.Inside the box v=o, hence the Schrodinger’s equation is given by,

where,

Discussion of the solution The solution of the above equation is given by

where A & B are constants which depending on the boundary conditions of the well.Now apply boundary conditions for this,Condition: I at x =0, = 0. Substituting the condition I In the equation 4, we get A =0and B 0. (If b is also zero for all values of x, ψ is zero . This means that the particle is not present in the well.)Now the equation 3 can be written as Condition: II : at x =a, = 0 Substituting the condition II in equation 5 we get

0= B sin(ka)

where, n = 1,2,3…………….

Substitute the value of k2 in equation (3).

32

x=0 x=a

V=

V=0

x

The equation 6 gives energy values or Eigen value of the particle in the well. When n=0, n

= 0. This means to say that the electron is not present inside the box, which is not true. Hence the lowest value of ‘n’ is 1. The lowest energy corresponds to ‘n’ =1 is called the zero-point energy or Ground state energy.

All the states of n1 are called excited states.To evaluate B in equation (3), one has to perform normalization of wave function.

Normalization of wave function:

Consider the equation,

The integral of the wave function over the entire space in the box must be equal to unity because there is only one particle within the box, the probability of finding the particle is 1.

But

Thus the normalized wave function of a particle in a one-dimensional box is given by,

where n=1,2,3……………

This equation gives the Eigen functions of the particle in the box. The Eigen values are as follows.

33

Since the particle in a box is a quantum mechanical problem we need to evaluate the most probable location of the particle in a box and its energies at different permitted state.

Let us consider first three cases

Case (1) n=1.

This is the ground stat and the particle is normally found in this state.For n=1, the Eigen function is

In the above equation =0 for both x=0 & x=a. but 1 has maximum value 2/a for x=a/2.

Ψ1

A plot of the probability density versus ‘x’ is as shown. From the figure, it indicates the probability of finding the particle at different locations inside the box. =0 at x = 0 and x = a also is maximum 2/a at x = (a/2). This means, in the ground state the particle cannot be found at the walls of the box and the probability of finding the particle is maximum at the central region. The Energy in the ground

state .

Case 2 : n =2 This is the first excited state. The Eigen function for this state is given by

now, 2 =0 for the values

and 2 reaches maximum at

34

1

x=0 x=a/2 x=a x=0 x=a/2 x=a

2

1

0x2

a a4

a

43a

2

these facts are seen in the following plot.

From the figure it can be seen that at x = 0, and at at

It means that in the first excited state the particle cannot be observed either at the walls or at the center. The energy is . Thus the energy in the first excited state is 4 times the zero point energy.

Case 3: n =3. This is the second excited state and the Eigen function for this state is given by

now, 3 =0 for the values

and 2 reaches maximum at

=0 for the values

and reaches maximum 2/a at at which the particle is most likely to be

found. The energy corresponds to second excited state is given by .

UNIT 3. LASER AND OPTICAL FIBERS.

LASERS

LASER is the acronym of Light Amplification by the Stimulated Emission of radiation.

Characteristics of Laser beam

35

0x2

a a4

a2

2

3

0x ax 0x ax

2

3

Directionality: The design of the resonant cavity, especially the orientation of the mirrors to the cavity axis ensures that laser output is limited to only a specific direction.Monochromaticity: The laser beam is characterized by a high degree of monochromaticity (single wavelength or frequency) than any other conventional monochromatic source of light .the bandwidth () of the laser beam is narrow, while ordinary light spreads over a wide range of wavelength.Coherence: The degree of coherence of a laser beam is very high than the other sources. The light from laser source consists of wave trains that are in identical in phase. Further it can be focused to a very small area 0.7m2. High Intensity: The laser beam is highly intense. When laser beam is incident on a square meter of surface, the energy incident is of the order of millions of joules.Focus ability: since laser is highly monochromatic, it can be focused very well by a lens. Yet it is so sharp the diameter of the spot will be close to the wavelength of the focused light. Since even laser is not ideally monochromatic the spot diameter in actual cases will be 100 to 150 times larger than the wavelength.

Basic principles:Radiation interacts with matter under appropriate conditions .The interaction leads to an abrupt transition of the quantum system such as an Atom or a molecule from one energy state to another. If the transition is from a higher state to a lower one, the system gives out a part of its energy and if the transition is in the reverse direction, then it absorbs the incident energy.There are three possible ways through which interaction of radiation and matter can take place, They are

Induced absorption:Induced absorption is the absorption of an incident photon by a system as a result of which the system is elevated from a lower energy state to a higher energy state, where in the difference in energy of two states is precisely the energy of the photon.The rate of induced absorption per unit volume is proportional to the number of atoms in the ground state and to the energy density of the incident photon.Rate of induced absorption = B12UνN1

E2 Incident photon E1 The process can be represented as,Atom +Photon Atom *.

36

Spontaneous Emission:Spontaneous emission is the emission of a photon, when a system transits from higher Energy State to lower Energy State without the aid of any external agency after a time of 10-9s.

Emitted photon

A photon of energy equal to the difference in the energy of the two states is released i.e., The energy of the emitted photon is E2 - E1 .The rate of spontaneous emission per unit volume is proportional to the number of atoms in the excited state.Rate of spontaneous emission = A12N2

The process can be represented as,

Atom * Atom + photon

Stimulated Emission: Stimulated Emission is emission of a photon by a system, under the influence of an incident photon of just the right energy that is equal to the energy difference of the two levels due to which the system transits from a higher energy state to lower Energy State. The photon emitted is called the stimulated photon and will have same phase, energy and direction of movement as that of the incident photon called stimulating photon

E2

Stimulated Photon

Incident photon E1

The rate of induced emission per unit volume is proportional to the number of atoms in the excited state and to the energy density of the incident photon.

Rate of induced absorption = B21UνN2

The process can be represented as: Atom * + photon Atom + (photon + photon)This is the kind of emission, which is responsible for laser action.

To achieve higher probability of stimulated emission we make use of the Phenomenon population inversion.Suppose we have two energy level system E1& E2 and say the number of atoms in E1 is N1and number of atoms in E2 is N2.At any given temperature, under thermal equilibrium conditions Boltzmann relation gives the distribution of atoms in different energy

states.

37

E2

E1

E1

E3

E2

Under normal conditions, the number of atoms in the higher energy state is less than the number of atoms in the lower energy state that is N2<N1, As long as( N2<N1 )the absorption dominates stimulated emission and the medium will absorb light rather than amplify it. In order to achieve higher probability of stimulated emission two conditions has to be satisfied.

a) To obtain a good lasing action, the ratio of stimulated emission to absorption must be higher than unity.

The relation indicates that stimulated emission will be larger than absorption only when N2>N1.The establishment of a situation in which the number of atoms in the higher energy level must be greater than that in the lower energy level state is called population inversion(N2>N1).

b) Under equilibrium condition, the ratio of stimulated emissions to spontaneous emission is

very small. The ratio is

In order to increase stimulated emission it is essential that N2>N1.Even if the population is more in the excited state, there will be a competition between stimulated and spontaneous emission. With metastable states ( the atom resides in that level for an unusually longer time, which is of the order of 10-3 to 10-2 s)the probability of spontaneous can be reduced. The above relation indicates that it is possible to have a higher stimulated emission than spontaneous emission, when photon energy density is high.Three energy level system:Consider three energy levels E1, E2 and E3 of a quantum system of which the level E2 is metastable state. Let the atoms be excited from E1 to E3 state by supply of appropriate energy. And then the atom from the E3 state undergoes downward transition to either E1or E2 states rapidly. Once the atoms undergo downward transitions to level E2 they tend to stay, for a long interval of time, because of which the population of E2 increases rapidly. Transition from E2 to E1 being very slow, in a short period of time the number of atoms in the level E2 is far greater than the level E1. Thus when E2 and E1 are compared the number of atoms in the former is greater than the number of atoms in later. Population inversion has been achieved. The transition from metastable state to ground state is the lasing transition.

Metastable state

lasing transition

38

Non-radiative transition

Requisites of a laser system: An active medium is needed which is able to produce laser light. An active medium that supports population inversion. A pumping mechanism is needed to excite the atoms to higher energy levels An optical resonator.

Active medium: A medium in which light gets amplified is called an active medium. The medium may be solid, liquid or gas. In an active medium, only small fractions of atoms of particular species are responsible for stimulated emission and consequent light amplification these atoms are called active centers.

Optical pumping: For realizing and maintaining the condition of population inversion, the atoms has to be raised continuously to the excited state .It requires energy to be supplied to the system. The process by which atoms are raised from the lower level to upper one is called the pumping process.There are number of techniques for excitation, are used in laser devices they are optical pumping, electrical pumping, chemical pumping…etc.

Optical resonator: An optical resonator generally consists of two plane mirrors, with the active material

placed in between them. One of the mirrors is semitransparent while the other one is 100% reflecting. The mirrors are set normal to the optic axis of material.

The optical resonant cavity provides the selectivity of photon states by confining the possible direction of photon propagation .as a result lasing action occurs in this direction.

The distance between the mirrors is an important parameter as it chooses the wavelength of the photons. Suppose a photon is traveling from the direction X to Y.it undergoes reflection at the mirror kept at the other end .the reflected wave superposes on the incident wave and forms stationary wave pattern whose wavelength is given by

λ = 2L/n

Where L is the distance between the mirrors. λ is the wavelength of the photon. n is the integral multiple of half wavelength

Einstein's Coefficients

Consider two atomic states of energies E1 and E2 of a system. Let N1 be the number of atoms in the lower energy state E1, and N2 be the number of atoms in the higher energy state E2 per unit volume of the system. Let radiation with a spectrum of frequencies be incident upon the system .let U d is the energy density incident/unit volume of the system. According to Albert Einstein, atomic transitions can be of three types. They are

39

100%Reflecting mirror Semi transparent

mirror

Active medium

1.Induced or stimulated absorption: In this process atom in the lower state E1 absorbs photon and get excited to higher state E2 .The number of such absorptions per unit time, per unit volume is called rate of absorption.The rate of absorption is proportional to, (a) The number density of lower energy state, i.e., N1, and,

(b) The energy density i.e., Uγ Rate of absorption = B12N1Uγ (1)

Where B12 is the constant of proportionality called Einstein’s coefficient of induced absorption

2.Spontaneous emission: In this process atom in the higher energy state E2 suffer spontaneous transition to the lower energy state E1 by emitting the radiation without the aid of any external agency .The emitted radiation is called spontaneous radiation and the phenomenon is called spontaneous emission .The number of such spontaneous emissions per unit time, per unit volume is called rate of spontaneous emission.

The rate of spontaneous emission is proportional to,(a) The number density of higher energy state i.e., N2

Rate of spontaneous emission = A21N2 (2) Where, A21 is the constant of proportionality called the Einstein's coefficient of spontaneous emission.

3.Induced or stimulated emission: In this process an atom in the higher energy state E2 is induced by the photon of right amount of energy to make a downward transition to the lower energy state E1 .The number of stimulated emission per unit time per unit volume called rate of stimulated emission Rate of stimulated emission, is proportional to,

(a) The number density of the higher energy state i.e., N2, and(b)The energy density i.e., Uγ

Rate of stimulated emission = B21N2Uγ, (3) Where, B21 is the constant of proportionality called Einstein’s coefficient of stimulated emission.

To get a relationship between these coefficients Einstein considered atoms to be in thermal equilibrium with radiation field, which means that the energy density Uγ is constant in spite of the interaction that is taking place between itself and the incident radiation. This is possible only if the number of photons absorbed by the system per sec is equal to the number of photons it emits by both the stimulated and spontaneous emission processes. At thermal equilibrium,

Rate of absorption = Rate of spontaneous emission + Rate of stimulated emissionB12N1Uγ = A21N2 + B21N2Uγ

Or Uγ (B12N1 – B21N2) = A21N2

Or

By rearranging the above equation, we get

40

(4)

But, by Boltzmann's law, we have,

Equation (4) becomes,

(5)

According to Planck's law, the equation for U is,

(6)

Now comparing the equation (5) and (6), term by term on the basis of positional identity, we have,

And,

Or Which implies that the probability of induced absorption is equal to the probability of stimulated emission. Because of the above identity, the subscripts could be dropped, and A21 and B21 can be simply represented as A and B and equation (6) can be rewritten.

At thermal equilibrium the equation for energy density is

SEMICONDUCTOR DIODE LASER

Semiconductor diode laser is a specially fabricated p-n junction device that emits coherent light when it is forward biased. Semiconductor lasers are the smallest (compact) and least

41

P -type

expensive of all the lasers available. Pumping is due to current flowing in the device, makes the semiconductor laser the topmost in efficiency among all the laser devices. Egs,. GaAs, CdSe, all direct band gap semiconductors. In the case of silicon and germanium the energy is released in the form of heat. Hence these materials are not suitable for fabricating laser. In the case of direct band gap semiconductors the band gap energy due to the electron-hole recombination released in the form of photon of light Therefore they are suitable for making lasers.

Construction:

Laser beam

The gallium arsenide laser diode is a single crystal of GaAs, and consists of a heavily doped n and p sections. n- section is derived by doping GaAs with tellurium and the p section is derived by doping GaAs with Zinc. The doping concentration is very high and is of the order of 1017-1019 dopants atoms /cm3.

The diode is extremely small with each of its size of the order 1mm. The p-n junction layer forms the active region. The junction layer (active region) has a

width varying from 1μm to 100μm depending upon the diffusion and temperature conditions at the time of fabrication.

The resonant cavity is required for energy amplification ,is obtained by cleaving the front and back faces of the semiconducting material

The junction lies in a horizontal plane through the center, the top and bottom faces are metallized and ohmic contacts are provided to pass current through the diode.

The front and rear faces are polished or cleaved at right angle to the junction layer, these planes play the role of reflecting mirror.

The other two faces are roughened to trap light inside the crystal.

Working: The GaAs laser diode is subjected to a forward bias by using a d.c source as shown in

the figure .At an absolute zero temperature (T=0K) for a semiconductor the conduction

42

Cleaved faces

Active region

band is completely empty and valance band is completely filled, separated by the energy band gap(Eg). By the process of doping large number of charge carriers are available at room temperature due to thermal excitation.

When p-n junction is forward biased with a large current, it raises the electrons from

the valance band to the conduction band, because this is an unstable state within a short time .the electrons in the conduction band drop to the lowest level in that band. At the same time, the electrons near the top of the valence band will drop to the lowest unoccupied levels, leaving behind holes. The lowest level within the conduction band is with a large concentration of electrons and the top of the valence band is full of holes. This is the state of population inversion. The narrow region where the state of population inversion is achieved is called active region.

At some instant, one of the excited electrons from the conduction band falls back to the valence band and recombines with a hole, and the energy associated with this recombination is emitted as a photon of light. This photon as it moves, stimulates the recombination of another excited electron with a hole and releases another photon.

These two photons are in phase with each other and have the same wavelength, thus travel together, get reflected at the end face. As they travel back, they stimulate more electron hole recombination with the release of additional photons of same wavelength coherence in nature. This beam gets resonated by travelling back and forth, and finally comes out through the partially reflecting face. As the current increase continuously, more electrons excited to the conduction band and generate holes in the valance band. This situation maintains population inversion.

The stimulated electron hole recombination’s cause emission of coherent radiation of very narrow bandwidth. Semiconductor diode laser emits light at a wavelength lies in IR and visible region.

At low forward current level, the electron-hole recombination cause spontaneous emission of photons and junction acts as an LED.

These lasers are used in communications laser printers CD players etc.

43

Nd:YAG LASER : Flash lamp Nd:YAG Rod Ellipsoidal reflector

M1 Trigger pulse Capacitor bank

^^^^^ Power supply Resistor

Flash lamp Nd:YAG Rod

Ellipsoidal reflector

CONSTRUCTION: Nd:YAG laser is one of the most popular solid state laser. Yittrium aluminiun garnet

Y3Al5O12 ,commonly called YAG is an optically isotropic crystal.

Some of the Y3+ ions in the crystal are replaced by neodymium ions, Nd3+. The crystal atoms do not participate in the lasing action but serve as host lattice in which the active centers, namely Nd3+ ions reside.

It consists of an elliptically cylindrically reflector housing the laser rod along one of its focus lines and a flash lamp along the other focus line.

The two ends of the laser rod are polished and silvered and constitute the optical resonator.

The light leaving one focus of the ellipse will pass through the other focus after reflection from the silvered surface of the reflector. Thus the entire radiation is focused on the laser rod.

44

E2

E1

Working:

Energy level diagram

The pumping of the Nd3+ ions to upper state is done by a krypton arc lamp.

The optical pumping with light of wavelength range 5000 to 8000A⁰excites ground state Nd3+ ions to higher states.

The metastable state E3 is the upper laser level and E2 forms the lower laser level.

The upper laser level E3 will be rapidly populated, as the excited Nd3+ ions quickly make a downward transition from the upper energy bands.

The population inversion is achieved between E3 and E2 level.

The laser emission occurs in infrared region at a wavelength 10,600A .⁰

Nd:YAG lasers find applications like welding, hole drilling….etc.

45

1.06μm

E3

Pump Bands

Upper laser level

Lower lasing level

Ground level

Applications of Lasers Lasers find applications in various fields such as medicine, material processing, communications, energy resources, and photography…. Etc. Laser welding, cutting and drilling

Welding:

the

The laser is focused on to a spot to be welded. Due to heat generation, the material melts over a tiny area on which beam focussed. An impurity such as oxides floats upon the surface. On cooling the material within becomes homogenous solid structure which makes weld

strong No destruction occurs in the regions around the welded portion Since it is a contact less process no foreign material comes in contact in to the welded

joint. Since the heat-affected zone is small, laser welding is ideal in areas such as

microelectronics.CO2 lasers are the most popular ones in this particular application.

Cutting:

Oxygen

Laser cutting is generally done assisted by gas blowing. The combustion of the gas burns the metal thus reducing the laser power requirement for

cutting.

46

Laser light

Welding spot

Laser light

When laser melts the local region the gas blows away the tiny splinters. The blowing action increases the depth and speed of the cutting process.

The quality of cutting is very high. Complicated 3D cutting profiles is possible. There will be no thermal damage and chemical change when cutting is done to regions

around cut area.

Drilling: Drilling of holes is achieved by subjecting materials to laser pulses of duration 10-4to 10-3

second. The intense heat generated over a short duration by the pulses evaporates the material

locally, thus leaving a hole. ND-YAG laser is used to drill holes in metals, where as co2 is used in both metallic and

non -metallic materials. In conventional methods tools wear out while drilling whereas the problem doesn't exist

with laser setup. Drilling can be achieved at any angle. Very fine holes of diameter 0.2 to 0.5mm could be drilled with a laser beam Very hard materials or brittle materials could be subjected to laser drilling since there is

no mechanical stress with a laser beam.

Pulsed Laser Deposition (PLD):PLD is a new unconventional evaporation technique. It is one of the newer techniques for depositing thin films, making use of interaction of laser beams with material surfaces. The PLD process is schematically depicted as shown in the figure

47

In its configuration a high power laser situated outside the vacuum deposition chamber is focused by means of external lenses onto the target surface, which serves as evaporation source. Lasers that have been most widely used for PLD center around Nd:YAG laser (wavelength= 1.064µm) and gas excimer laser.Irrespective of the laser used, absorbed beam energy is converted into thermal, chemical and mechanical energy, causing electronic excitation of target atoms and exfoliation of the surface. The evaporants form a plume above the target consisting of a collection of energetic neutral atoms, molecules, ions, electrons, atom clusters, micron sized particulates and molten droplets. They are propelled to the substrate where they condense to form a film.One of the drawbacks of PL technique is that plume is highly directional which makes it difficult to uniformly deposit films over large substrate areas. Another drawback is splashing of macroscopic particles during laser- induced evaporation. Splashing of macroscopic particles include rapid expansion of gas trapped beneath target surface, rough target surface morphology whose mechanically weak projections are prone to fracturing during pulsed thermal shocks etc. To avoid splashing, a rapidly spinning pin-wheel like shutter between target and substrate is introduced. Slower moving particulates can be batted back, allowing more mobile atoms, ions and molecules to penetrate this mechanical mass filter.

OPTICAL FIBERS.

Construction

Optical fibers are the light guides used in optical communications as wave-guides. They are transparent dielectrics and able to guide visible and infrared light over long distance. They are made up of two parts. Core the inner cylindrical material made of glass or plastic and the cladding material the other part which envelops the inner core. The cladding is also made of similar material but of lesser refractive index. Both the core and the cladding materials are enclosed in a polyurethane jacket (as shown in fig. 1), which safeguards the fiber against chemical reaction with surroundings, and also against abrasion and crushing.

Core

Wave guides (propagation mechanism)

A wave guide is a tubular structure through which energy of some sort could be guided in the form of waves. In OF light waves can be guided through a fiber, it is called light guide.

48

Cladding

Polyurethane

The guiding mechanism

The cladding in an OF always has a lower refractive index (RI) than that of the core. The light signal which enters into the core can strike the interface of the core and the cladding only at large angles of incidence because of the ray geometry. The light signal undergoes reflection after reflection within the fiber core. Since each reflection is a total internal reflection (when the angle of incidence is greater than the critical angle, the ray undergoes total internal reflection), the signal sustains its strength and also confines itself completely within the core during propagation. Thus, the optical fiber functions as a wave guide.

Numerical Aperture and Ray Propagation in the Fiber

Let us consider the special case of ray which suffers critical incident at the core cladding interface. The ray, to begin with travels along AO entering into the core at an angle of 0

To the fiber axis. Let it be refracted along OB at an angle 1 in the core and further proceed to fall at critical angle of incidence (= 90-1) at B on the interface between core and cladding. Since it is critical angle of incidence, the refracted ray grazes along BC.

1 B

0

0 Core (n1) A

Cladding (n2)

Incident ray

Now it is clear from the figure that any ray enters at an angle of incidence less than 0 at O, will have to be incident at an angle greater than the critical angle at the interface, and gets total internal reflection in the core material. Let OA is rotated around the fiber axis keeping 0

same, then it describes a conical surface. We can say that it a beam converges at a wide angle into the core, then those rays which are funneled into the fiber within this cone will only be totally internally reflected, and thus confined within for propagation. Rest of the rays emerges from the sides of the fiber.

The angle 0 is called the wave guide acceptance angle, or the acceptance cone half-angle, and sin 0 is called the numerical aperture (N.A.) of the fiber. The numerical aperture represents the light-gathering capability of the optical fiber.

Condition for Propagation:

Let n0, n1and n2 be the RI of surrounding medium, core and cladding respectively.

49

Acceptance cone

Now, for refraction at the point of entry of the ray AO into the core, by applying the Snell’s law that,

N1 Sin 0 = N2 Sin 1 -----1)At the point B on the interface,The angle of incidence = 90 - 1 Again applying Snell’s law, we have,n1 sin (90-1) = n2 sin 90.Or, n1 cos 1 = n2 Or, Cos 1 = n2/n1 ----2)Rewriting Eq (1), we have,

Sine 0 = n1/n0 sin 1 =

Substituting for Cos 1 from Eq. (2), we have,

If the medium surrounding the fiber is air, then n0 = 1,Or,

If I is the angle of incidence of an incident ray, then the ray will be able to propagate, If, I < 0

Or if Sin I < Sin 0

Or, Sin I < N.A.

Fraction Index Changes ():The fractional index change is the ratio of the refractive index difference between the core and the cladding to the refractive index of core of an optical fiber.

Therefore,

= ------3)

Relation between N.A. and

From Equ. 3) , (n1-n2) = n1 ---4)

We have,

Since n1~ n2 , (n1+ n2) = 2n1

50

Therefore,

Though an increase in the value of increase N.A., and thus enhances the light gathering capacity of the fiver, we cannot increase to a very large value, since it leads to what is called “intermodal dispersion” which causes signal distortion.

Types of Optical fibers.In any optical fiber, the whole material of the cladding has a uniform RI value. But the RI of the core material may either remain constant or subjected to variation in a particular way. The curve which represents the variation of RI with respect to the radial distance from the axis of the fiber is called the refractive index profile.

The optical fibers are classified under 3 categories, namely,a)Single mode fiber, b) step index multimode fiber and c)graded index multimode fiber.

This classification is done depending on the refractive index profile, and the number of modes that the fiber can guide.

a) Single Mode fiber (step index)A single mode fiber has a core material of uniform RI value. Similarly cladding also has a material of uniform RI but of lesser value. This results in a sudden increase in the value of RI from cladding to core. Thus its RI profile takes the shape of a step. The diameter value of the core is about 8 to 10 m and external diameter of cladding is 60 to 70 m. Because of its narrow core, it can guide just a single mode as shown in Fig. 3. Hence it is called single mode fiber. Single mode fibers are most extensively used ones and they constitute 80% of all the fibers that are manufactured in the world today. They need lasers as the source of light. Though less expensive, it is very difficult to splice them. They find particular application in submarine cable system.

b) Step –index Multimode fiber:

51

60 to 70 m8 to 10 m

Refractive index profile

Ray Propagation

Light ray

Cladding

Core

The geometry of a step-index multimode fiber is as shown in Fig.4.Its construction is similar to that of a single mode fiber but for the difference that, its core has a much larger diameter by the virtue of which it will be able to support propagation of large number of modes as shown in the figure. Its refractive index profile is also similar to that of a single mode fiber but with a larger plane regions for the core.

The step-index multimode fiber can accept either a laser or an LED as source of light. It is the least expensive of all. Its typical application is in data links which has lower bandwidth requirements.

c) Graded-Index Multimode Fiber

Graded index multimode fiber is also denoted as GRIN. The geometry of the GRIN multimode fiber is same as that of step index multimode fiber. Its core material has a special feature that its refractive index value decreases in the radially outward direction from the axis, and becomes equal to that of the cladding at the interface. But the RI of the cladding remains uniform. It RI profile is also shown in Fib.5 Either a laser or LED can be the source for the GRIN multimode fiber. It is most expensive of all. Its splicing could be done with some difficult. Its typical application is in the telephone trunk between central offices.

52

100 to 250 m50 to 100 m

RI Profile

Core

Cladding

Fig. 4. Step index multimode fiber

Attenuation in optical fibers

The total power loss offered by the total length of the fiber in the transmission of light is called attenuation.

The important factors contributing to the attenuation in OF are

1) Absorption loss, ii) Scattering loss iii) Bending loss iv) Intermodal dispersion loss and v)coupling loss.

The total losses in the fiber are due to the contribution of losses due to absorption, scattering radiation and coupling. All these losses are wavelength dependent and it can be minimized by selecting the operating wavelength.

53

Fig. 5a; Graded index multimode fiber

100 to 250 m50 to 100 m

RI Profile

Attenuation in fiber is defined as the ratio of power input and power output. It is denoted by symbol . Mathematically attenuation of the fiber is given by,

, dB/km

Where Pout and Pin are the power output and power input respectively, and L is the length of the fiber in km.

1. Absorption loss:

There are two type of absorption, one is Absorption by impurities and the other Intrinsic absorption. In the case of first type, the type of impurities is generally transition metal ions such as iron, chromium, cobalt and copper. During signal propagation when photons interact with these impurities, the electron absorbs the photons and get excited to higher energy level. Later these electrons give up their absorbed energy either as heat energy or light energy. The reemission of light energy is of no use since it will usually be in a different wavelength or at least in different phase with respect to the signal. The other impurity which would cause significant absorption loss is the OH (Hydroxyl) ion, which enters into the fiber constitution at the time of fiber fabrication. In the second type i.e., intrinsic absorption, the fiber itself as a material, has a tendency to absorb light energy however small it may be. Absorption in a fiber is the absorption that takes place in the material assuming that there are no impurities and the material is free of all inhomogeneities, and hence it is called intrinsic absorption which sets the lowest limit on absorption for a given material.

2. Scattering loss:

The power loss occurs due to the scattering of light energy due to the obstructions caused by imperfections and defects, which are of molecular size, present in the body of the fiber itself. The scattering of light by the obstructions is inversely proportional to the fourth power of the wavelength of the light transmitted through the fiber. Such a scattering is called Rayleigh scattering. The loss due to the scattering can be minimized by using the optical source of large wavelength.

3. Bending losses

Bending losses occur due to the presence of macro bends and micro bends that are caused while manufacturing and as well as due to the applied stress on the fiber. At the point of bend the light will escape to the surrounding medium due to the fact that the angle of incidence at that point becomes lesser than the critical angle. Hence it will not undergo total internal reflection. In order to avoid this type of losses, the optical fiber has to be laid straight for

54

Defects

long distances and they should be freed from the external stresses by providing mechanical strength by external encasements.

4. Coupling losses

Coupling losses occur when the ends of the fibers are connected. At the junction of coupling, air film may exist or joint may be inclined or may be mismatched and then can be minimized by following the technique called splicing.

Applications

Optical fibers find their applications in the fields of communication, medicine, industry and domestic.

1. Communication applications: (P to P communication)

The block diagram of a typical fiber optic communication system is as shown in the figure. Voice or any information is converted into electrical signals using the coder (photo multiplier) and is connected to the optical transmitter. The light signals from the optical transmitter are connected to the photo detector through optical fiber. The electrical signals from the photo-detector are decoded by the decoder to the original state, which may be the voice or information. In communication electrical signal is converted into optical signal, propagated through optical fibers and converted back into electrical signal at the other end.

55

Back scattered loss

Bending loss

Micro bend

Macro bend

Optical fibers find very important, applications in communication because of the following advantages they possess.

a. Transmission loss is lowb. Fiber is lighter and compact than equivalent copper cables.c. Fibers have large data rate compared to equivalent copper cablesd. There is no interference in the transmission of light from electromagnetic waves

generated by electrical applications.e. Fibers are free from corrosion effect caused by salt, pollution and radiation. Hence,

they are more reliable.f. Tapping information from fiber is impossible. Hence, the transmission is more

secured.g. The cost of fiber optic communication system is lower than that of an equivalent cable

communication system..

Optical fibers find few disadvantages in the communication systems. They are:

a. Fiber loss is more at the joints if the joints do not match (the joining of the two ends of the separate fibers are called splicing)

b. Attenuation loss is large as the length of the fiber increases.c. Repeaters are required at regular interval of lengths to amplify the weak signal in long

distance communication.d. Bends will increase the loss of the fiber. Hence, the fiber should be laid straight.

Point to Point haul communication system is employed in telephone trunk lines. This system of communication covers the distances, 10 km and more. Long-haul communication has been employed in telephone connection in the large cities of New York and Los Angeles. The use

56

CoderOptical Transmitter

Photo detectorDecoder

Voice or Any information

Electrical

Signal

Optical fiber

High propagation

Voice or

Information

Schematic diagram of P to P Communication using Optical Fiber

of single mode optical fibers has reduced the cost of installation of telephone lines and maintenance, and increased the data rate.

Local Area Network (LAN) Communication system uses optical fibers to link the computer-oriented communication within a range of 1 or 2 km

Community Antenna Television (CATV) makes use of optical fibers for distribution of signal to the local users by receiving a multichannel signal from a common antenna.

2. Medical application

Endoscopes are used in the medical field for image processing and retrieving the image to find out the damaged part of the internal organs of the human system. It consists of bundle of optical fibers of large core diameter, whose ends are arranged in the same sequence. Endoscope is inserted to the inaccessible damaged part of the human system. When light is passed through the optical bundle, the reflected light received by the optical fibers forms the image of the inaccessible part on the monitor. Hence, the damage caused at that part can be estimated and also it can be treated.

3. Industrial

Optical fibers are used in the design of Boroscopes, which re used to inspect the inaccessible machinery parts. The working principle of boroscope is same as that of endoscopes.Domestic.

Optical fiber bundles are used to illuminate the interior places where the sunlight has no access to reach. It can also be used to illuminate the interior of the house with the sunlight or the incandescent bulb by properly coupling the fiber bundles and the source of light. They are also used in interior decorating articles.

57

UNIT 4 CRYSTAL STRUCTURE, FERRO AND PIEZO ELECTRICS.

A CRYSTAL is a 3D periodic array of atoms or molecules. An ideal CRYSTAL is constructed by the infinite repetition of identical units in space. In the simplest crystals the structural unit is a single atom as in Cu, Ag, Au, Fe, Al and alkali metals.

All crystals can be described in terms of a lattice with a group of atoms attached to every lattice point. This group of atoms is called as motif or basis. Therefore the lattice can be considered to the skeleton of the crystal structure.

What is a lattice?It is a periodic 3D array of points in space such that the environment about any point is the same as the environment about any other point.

What is a motif or basis?It is an atom or a molecule (simple or complex) or groups of unrelated atoms that is repeated at every point of the crystal lattice forming the crystal structure.

Hence we sum up by saying that CRYSTAL STRUCTURE = LATTICE + MOTIF (OR BASIS)

LATTICE MOTIF OR BASIS

CRYSTAL STRUCTURE

A lattice is defined by three fundamental translation vectors a, b and c such that the atomic arrangement looks the same in every respect when viewed from the point r as when viewed

58

from the point r' = r + ua + vb + wc where u, v and w are arbitrary integers. The lattice and the translational vectors a, b and c are said to be primitive if any two points r and r' from which the atomic arrangement looks the same always satisfies the relation r' = r + ua + vb + wc with a suitable choice of integers u, v and w. With this definition of primitive translational vectors there is no cell of smaller volume that can serve as building block for crystal structure.

The primitive translational vectors are often used to define the crystal axis. However, non-primitive crystal vectors are often used when they have a simpler relation to the symmetry of the structure. The crystal axis a, b and c from the edges of a parallelopiped. If there are lattice points only at the corners then it is called as a primitive parallelopiped.

A lattice translational operation is defined as the displacement of a crystal by a crystal translational vector T = ua + vb + wc. Any two lattice points are connected by a vector of this form.

More than one lattice is always possible for a given crystal structure and once a lattice is chosen more than one set of axes is always possible. The motif is identified once these choices have been made.

Unit cell

The unit cell is the basic building block of the crystal lattice. By repeated translation of the building blocks along the crystal axes, we can construct the entire lattice.

59

The unit cell can be formally defined as " a geometrical structure that forms the smallest unit of the space lattice that when repeated along the direction of the crystallographic axis generates the space lattice".

A unit cell can be primitive or non primitive. A primitive cell is one where there are lattice points only at the corners. A non-primitive cell is one that has, in addition to the lattice points at the corners, other lattice points at the center of the cell faces or in the interior.primitive Non-primitive

A unit cell is defined by three translational vectors and the angles between them.

There is an unlimited number of possible lattices because there is no natural restriction on the lengths of the lattice vectors or on the values of the angles between them. However crystal obey symmetry laws. The lattice to be chosen must adhere to the symmetry of the crystal. This requires restriction on the relative lengths of the translational vectors and on the angles between them. When such restrictions are applied then there are only 14 types of space lattices possible with one being a general lattice and the other 13 special lattices. These 14 space lattices are called as Bravais lattices.

60

a

b

c

The crystal systems

Depending on the basic symmetry requirements, crystals can be grouped into the following systems. For each system the relative lengths and angles have been given.

System Relation between a, b and c

Relation between , and

Examples

CUBIC a = b = c = = = 90 NaCl, CaF2

TETRAGONAL a = b c = = = 90 NiSO4, SnO2

ORTHORHOMBIC a b c = = = 90 KNO3, BaSO4

HEXAGONAL a = b c = = 90, = 120 SiO2, AgIRHOMBOHEDRAL a = b = c = = 90 CaCO3, CaSO4

MONOCLINIC a b c = = 90, 90 FeSO4, Na2SO4

TRICLINIC a b c 90 K2Cr2O7, CuSO4.5H2O

The fourteen Bravais Lattices are distributed among the 7 crystal systems as follows

CUBIC

TETRAGONAL

ORHTHORHOMBIC

HEXAGONAL

61

RHOMBOHEDRAL

MONOCLINIC

TRICLINIC

All the Bravais' lattices follow two golden rules

1. The environment about any point is identical to the environment about any other point2. The lattice obeys the symmetry laws of the crystal system

Directions and planes in a latticeOnce the lattice for the structure is identified and the axes fixed, the coordinates of

every lattice point is known. Any lattice point can serve as the origin and once the origin is fixed a direction within the lattice can be identified. Normally a line joining any two lattice points can represent a direction. This line can be moved parallel to itself until it passes through the origin and it still represents the same direction. A direction within the lattice is denoted by a set of three integers, u v and w enclosed within square brackets i.e. [uvw]. To find out the values of u, v and w of a line consider the following diagram.

c

b 0

a

The line drawn with the unit cell till it reaches the boundary of the unit cell represents the direction

Treat this line as a vector and find out the length of its projections along the a-, b- and c-directions

Write down the coefficients and convert them into a set of three smallest integers. These integers enclosed within square brackets represent the direction.

For the line shown, the lengths of the projections are 0a, ½b and 1c.

The coefficients are 0, ½,1

62

The smallest integers are [012]

Equivalent directions

Two non-parallel directions can be related by lattice symmetry. Such directions are called equivalent directions. For example, (refer to the figure) if the unit cell is given a rotation of /4 about the axis shown, then

[001]

[010]

[100]

[100] direction will coincide with [010] direction. A further rotation of /4 will make it coincide with the [100] direction. In this way by making suitable rotations the directions [100], [010], [001], [100], [010] and [001] can be made to be coincident with each other. These directions are called equivalent directions and they are represented by enclosing any member of this family with angular brackets i.e. 100.

63

Lattice planes and Miller Indices

Bragg treated the phenomena of diffraction of X-rays from crystals as if X-rays were specularly reflected from imaginary planes. The atoms of a crystal are supposed to lie on these imaginary planes. A notation has been developed to represent these planes. The planes are represented by a set of triple integers known as Miller indices.

Miller Indices

Indexing procedure to find the Miller Indices.

a)Find the intercept and express the intercept it terms of the basis vector a,b and cOA = ua, OB =vb and OC = wc

b) Divide the intercept by the magnitude of the Basis vector

OA /a = u, OB/b = v and OC /c = Wc)Take the reciprocal of the intercept

1/u 1/v 1/x

d)Simplifying by multiplying the denominator of LCM

(h k l) (hkl) represents the Miller indices of the required plane ABC

Definition: Miller indices are the reciprocals of the intercept made by the plane on crystallographic axis when reduced to its three small possible integer (h k l).

Note: 1. (hkl) may represent a single plane or a complete family of planes2. If a plane is paralle to a given axis then the corresponding Miller index is zero

64

C

B O

A

Consider a place ABC is intercepting X axis at A Y axis at B and Z axis at C

Let OA, OB and OC are the intercepts made by the plane on Crystallographic axis XY and Z respectively.

3. If the plane cuts the negative axis then the corresponding Miller index is negative and is written with a bar on top of it.

Some examples of Miller indices

(100)Z

Y

X

(010)

Z

Y

X

65

(111)

Z

Y

X

(232)

Z

Y

X(020)

Z

Y

X

Expression for inter-planar spacing

66

Inter-planar spacing, denoted by d, is the perpendicular distance between two consecutive planes. The value of d depends on the Miller indices of the plane and on the crystal system. Consider a set of planes (hkl) in a cubic unit cell.

Miller indices of the plane (hkl) Intercepts on the a-, b- and c-axis

are OA, OB and OC respectively ON is the normal from the origin O

to the plane

The first plane of the member always passes through the origin. Therefore since O lies on the first plane, the perpendicular from the origin to the plane represents the inter-planar distance, d. Therefore ON = d

Let the normal ON make angles 1, 2 and 3 with respect to the a-, b- and c-axes. Then Cos(1), Cos(2) and Cos(3) represents the direction cosines of the line ON. Direction cosines satisfy a well known property Cos2(1) + Cos2(2) + Cos2(3) = 1.

Considering the triangle ONA Cos(1) = ON/OA. Similarly considering triangles ONB and ONC we get Cos(2) = ON/OB and Cos(3) = ON/OC.

ON = d From the definitions of h, k and l we get OA = a/h, OB = b/k and OC = c/l.

Substituting for ON, OA, OB and OC we get

Since the system is cubic a = b = c

Therefore

Introduction to X – ray diffraction:

Diffraction is bending of light across an obstacle whose dimensions are comparable with the wavelength of light. But in the case of X – rays, their wavelengths are of order of 10 -

8cm. An optical grating will be ineffective to cause diffraction of X – rays, since the concerned dimensions in the grating exceeds that of the X – rays by many orders. But we know that the dimensions of atoms are of the order of 10-8cm,and the atoms are distributed perfectly uniformly in a crystal as a 3 – dimensional array. Both because of atomic dimensions, and because of the regularity in the arrangement of atoms, a crystal provides an

67

Z N

C

O B Y A

X

excellent facility to diffract the X – rays. In this sense, a crystal can be thought of as a 3 – dimensional grating for X – rays.

Bragg’s Diffraction:

In 1913, Sir William Lawrence Bragg and his son Sir William Henry Bragg, put forward a different approach for studying X – ray diffraction by crystals. They considered crystal in terms of planes in which regularity in the distribution of atoms could be identified. It is possible to identify different families of such planes. The interplanar separation remains constant for a given family of planes. Each plane family will be inclined at an angle to the other. The component planes of a plane family are called Bragg planes. Further, the two scientists Bragg and Bragg, considered an X – ray beam of wavelength incident in a direction at an angle to a family of Bragg planes with interplanar spacing d, and showed that constructive interference takes place between the rays scattered by the atoms in different Bragg planes when the condition,

n = 2 d sin

where n is an integer, is obeyed.

The above condition is known as Bragg’s law.

In order to carry out X – ray studies based on Bragg’s law, a single crystal of the given specimen is mounted such that, the X – ray beam is inclined upon the crystal at an angle , and a detector scans through various angles for the diffracted X – rays. The detector shows peaks for those angles at which constructive interference occurs.

Bragg’s Law:

The following figure shows a crystal in which the dashed lined indicate atomic planes, or Bragg planes, and the dots indicate the position of the atoms. Let a monochromatic parallel beam of X – rays of wavelength be incident upon the crystals at an angle as shown. The ray along AB, which is part of the incident beam, is scattered by the atom B in the first plane.

Another ray along DE which is part of the same incident beam is scattered by the atom E in the next adjacent parallel plane.

Each of the atoms in the crystal, scatters the incident rays in all directions. Among the rays scattered from B and E, those which are parallel to each other, and have a path difference of n, give rise to constructive interference.

Consider the rays BC and EF, scattered from B and E. Let BP and BQ be drawn perpendicular to the rays DE and EF respectively.

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X – ray scattering by crystal

Now, we can say that, up to BP, the path covered is same during incidence. From BQ onwards, the parallel rays BC and QF will travel keeping step with each other. Hence the excess path traveled by the ray along DEF over that along ABC can be written as,

= PE + EQ,

= BE sin + BE sin,

= 2BE sin.

But BE is the interplanar spacing, i.e., the spacing between the atomic planes. Let it be denoted as d.

= 2d sin.

Constructive interference takes place if = n, where n = 1,2,3,…

i.e., if 2d sin = n.

the above equation is called Bragg’s law, and is called glancing angle.

Here the rays AB and DE are incident at an angle to the Bragg plane. Also we see that, the corresponding scattered rays BC and EF undergo constructive interference only when they are inclined at the same angle to the Bragg plane.

Thus in essence, the constructive interference conditions match the law of reflection in geometrical optics. This happens irrespective of thee wavelength of the X – ray beam. Hence, it is referred to as Bragg reflection.

69

d

AC

B

E

P Q

FD

Ferroelectric

paraelectric

FERRO AND PIEZO ELECTRICS.

Ferro electricity

Ferro electrics constitute an important group of dielectrics. They are anisotropic crystals which exhibit spontaneous polarization by acquiring electric displacement below certain temperature. By analogy with magnetism these materials are called ferroelectrics. The spontaneous polarization occurs in the ferroelectrics by the action of internal process and without the influence of an external electric field.

Examples for ferroelectrics are

1. Crystals exhibiting Rochelle salt structure , viz, Rochelle salt (Seignette salt) it is the Sodium potassium tartaric acid (NaK (C4H4O6) 4H2O ).

2. Peroskite group structure consisting mainly of titanates and niobotes . examples for this crystals are BaTiO3 (Barium Titanium Oxide)

3. The dihydrogen phosphate and arsenates : examples are KH2PO4 (KDP)

Properties of Ferroelectrics:

1. Anomalous dependence of dielectric constant on temperature: Usually in ordinary dielectrics, dielectric constant does not change much with temperature. But in case of ferroelectric crystal, dielectric constant exhibits one or more sharp maxima when dielectric constant varies with a value of several thousands. The temperature corresponding to sharp maxima is called curie temperature. It is also defined as the critical temperature at which it looses its ferroelectric property and enters into paraelectric phase.

The static dielectric constant also changes with temperature T as per the relation

for T>Tc

Where C is a constant, and θ is very close to the curies temperature Tc.

Above relation is called the Curie-Weiss law.

70

εr

Tc T

2. Nonlinear dependence of its polarization on the external electric field: In ordinary dielectrics, the polarization P varies linearly with dielectric constant. (relation between P, χ and E). Therefore they are called linear dielectrics. But in case of ferroelectrics the relation between P and E is complex in nature and they are called as non-linear dielectrics. Therefore one can say that dielectric constant is not a constant in case of ferroelectrics but depends on the applied electric field E.

3. Ferroelectrics exhibit hysteresis: When a ferroelectric crystal is subjected to an alternating field, the polarization P Vs electric field E describes a closed loop called the hysteresis loop. The polarization will rise non linearly and reach saturation at certain value of Ps. The polarization will not change even if E is increased. If the field is decreased, the polarization versus E will not follow the same path as that obtained for increasing E. When the external electric field is switched off the value of P does not return to zero and ferroelectric retains a residual polarization Pr. To bring back the polarization to zero value, a field Ec known as coercive field must be applied in the opposite direction.

It is clear that there is a close resemblance between the electric properties of non-linear dielectrics and the magnetic properties of ferromagnetic materials. It is why the non-linear dielectrics are called ferroelectrics.

Classification of Ferroelectrics.

Depending upon their chemical composition and structure ferroelectrics can be classified as 1. Tartrate group2. Dihydrogen phosphate and arsenates of alkali metals3. Oxygen octahedron group

1. Tartrate group

Example is Rochelle salt which is Sodium-Potassium salt of tartaric acid (NaK (C4H4O6) 4H2O. They are the first solid known to exhibit ferroelectric properties. The unique properties of these materials are that they exhibit ferroelectric properties between -18 ºC to +23 ºC only. The graph of spontaneous polarization Vs temperature shows two curie temperature (transition temperature).

2. Dihydrogen phosphate and arsenates of alkali metals.

The typical example for this type is KH2PO4 . The spontaneous polarization Vs Temperature curve of this material shows only one curie temperature at 123 K.

3. Oxygen octahedron group.

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The best example under this type is BaTiO3 (Barium Titanium Oxide). It has curie temperature at 120 ºC, 5 ºC and at -80 ºC.

They exhibit hysteresis below its curie temperature. Above curie temperature BaTiO3

corresponds to the cubic structure. In cubic structure Ba2+ ions occupy the corners of cube. The centre of the cubic face are occupied by O2- ions. The O2- ions forms an octahedron structure and at the centre of which the small Ti4+ ion is located.

When the temperature is lowered below 120 ºC (i.e., below critical temperature), the material becomes spontaneously polarized and at the same time its structure changes. The direction of spontaneous polarization may lie along any of the cube edges giving a total 6 possible direction for the spontaneous polarization. Along the direction of spontaneous polarization of a given domain, the material expands, where as perpendicular to the direction of polarization it contracts. Thus material is no longer cubic but corresponds to tetragonal.

The two more transition temperature of BaTiO3 are at 5 ºC and -80 ºC, At 5 ºC, the spontaneous polarization changes its direction from one of the cubic edges to a direction corresponding to a face diagonal.

At 80 ºC, the spontaneous polarization changes from a direction corresponding to a face diagonal to one along a body diagonal. The three transition temperature reflects the dielectric constant and its spontaneous polarization of a material.

Piezo Electricity

The French Physicist Pierre Curie and Paul-Jean Curie discovered the Piezo electric effect in the year 1880. Some of the dielectrics, when compressed in a certain direction become polarized and polarization charges appear on its surface. This polarization of a dielectric as a result of mechanical deformation is called direct piezoelectric effect. If the crystals are subjected to tension, the polarity of the polarization charges is reversed. The amount of polarization charges appearing on the crystal depends on the crystal lattice type and lattice direction along which the deformation occurs.

The Piezoelectric effect occurs only in such crystal in which the structure does not possess a centre of symmetry. Crystals that exhibit the piezoelectric effect are called piezoelectrics..

Examples: Ammonium Phosphate, Quartz, PZT (Lead Zirconate Titanate). They are also ferroelectrics.

If the mechanical pressure on the piezoelectric crystal is altered, a varying voltage, which is related to the applied pressure, is produced by the crystal. The voltage can be as small as a fraction of a volt or as large as several thousand volts depending on the material and pressure.

This piezoelectric effect is reversible. i.e., if a piezoelectric crystal is placed in an electric field it will be polarized and deformed. If the direction of the electric field is changed the deformation sign changes, say from contraction to extension. This mechanical deformation of dielectrics caused by an external electric field is known as the reverse piezoelectric effect (also called as electrostriction). If an alternating voltage is applied between the two opposite faces of the crystal, it vibrates with the frequency of the field.

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+ +

O

Electric dipoles in a piezoelectric crystal

Crystals with center of symmetry

Consider the crystal lattice such as the one shown below. The lattice ions are located at certain regular intervals. In the lattice a unit cell is considered.

+ - + - +

- + - + -

+ - + - +

- + - + -

+ - + - +

The unit cell is said to have a center of symmetry at O because from O, if we draw a vector to any lattice ion, say A (A is a negative ion) and then draw the reverse vector from O, we see that it meets the same type of lattice ion ( that is the negative ion) at B. Crystals in which such a symmetry exists in the arrangement of lattice ions are said to have a center of symmetry.

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+ + + + + + + + +

+

-

+

-

+

-

+

-

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

+ + + + + + + + +

+

-

+

-

+

-

+

-

- - - - - - - - - -

+

+

+

--

-

Crystals with No center of symmetry

Consider a crystal with hexagonal unit cell. The position of its positive and negative ions could be identified as shown in figure. In the unit cell we can identify a point O about which geometrically the unit cell is symmetric. Now from O, let us then draw the reverse vector from O. The reverse vector meets the opposite type of lattice ion( that is with negative charge) at B. Such crystals are said to be with no center of symmetry and the unit cell is said to be noncentrosymmetric. All piezoelectric crystals possess noncentrosymmetric unit cells.

State of polarization when no stress is acting:

We consider a crystal with hexagonal unit cell. The unit cell will be comprising of three positive and three negative charges that are alternated in a frame with six sides as shown in the figure. The centre of mass of the negative charges will be at O the centre of the unit cell. But it coincides with the centre of mass of the positive charge due to the three positive charges. Hence the dipole moment of the unit cell is zero. Because of this reason there can’t be any net polarization for the crystal in the free state.

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-

-

O

Development of polarization in piezoelectric crystals:

When pressure is applied on the crystal, it undergoes physical compression following which the lattice ion shifts their positions. Accordingly the charges in the unit cell will be displaced as shown in the figure. The displacement results in the centres of mass of the negative and positive charges being shifted in opposite directions. Effectively we will be having a pair of closely lying opposite charges i.e, a dipole created in each unit cell while the crystal is being stressed. Thus the stress lends a dipole moment for each unit cell. After allowing for the charge neutralization within the crystal, the whole process results in the appearance of charges on the faces perpendicular to the direction of stress, thus accounting for polarization of the piezoelectric material.

Mechanism of Piezoelectric effect in Quartz crystal

Natural quartz has the shape of a hexagonal prism with a pyramid attaches to each end.It has one three fold symmetry axis called optic axis and three two fold axes called the electrical axis. The line joining the apex point of pyramids is the 3 fold axis.

The three lines which pass through the opposite corners of the crystal constitutes its three x- axes or electrical axes. Similarly the 3 lines which are perpendicular to the sides of the hexagonal form the 3-y axis which are known as mechanical axes. Thin plates of quartz crystal cut perpendicular to one of its x-axis are known as x-cut plates. Similarly, thin plates of the crystal cut perpendicular to one of its y-axis are known as y-cut plates.

Let us consider X cut crystal plane. Let its thickness be t and length l (along optic axis). If an alternating voltage is applied to metal electrodes fixed to opposite faces of the crystal plate, alternating stress and strain are set up both in its thickness and length. The frequency of the thickness vibrations is given by

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+

+

+

-

-

-

- +

And the frequency of the length vibrations is given by

Where m=1,2,3, ….etc., stand for fundamental, first overtone, second overtone…respectively. Y is Young’s modulus along the appropriate direction and rho is the density of the crystal plate.

Applications of Piezoelectric materials in Smart structures:

Smart structure is considered as a system of input-output functions comprising of many components performing sensing, actuating and information processing. Smart structure is defined as one that can sense a change in its environment and respond back in a useful or controlled fashion.

Components /parts of smart structure:

Input

a) Sensor: It receives the information from the external environment. These are functional materials that sense the intensity of stimulus due to the applied stress, strain, electrical, thermal, optical or other phenomenon. These can be embedded into the structure itself or bonded externally to the structure.

Material used for the design of sensors are Piezoelectric crystals. It acts as sensor by developing an electric potential when it is mechanically stressed across its faces. Examples are quartz, lead zirconate titanate (PZT), barium titanate, polyvinylidene fluoride(PVDF) etc.

b) Control unit: This is the important part of a smart structure, which acts like brain. Its function is to receive the sensed information from the sensor, to process it and give suitable control command to the actuator to perform as per the requirement of the system. These are mostly microprocessors and using automatic control functions and signal processing to instruct the actuators.

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c) Actuators: This is the performing part of the smart structure, which responds to the sensed input. It receives information/control action from the control unit and actuates to control the required performance of the structure.

Piezoelectric materials are used as actuators as they undergo mechanical deformation when an electric field is applied. Example

Applications of Piezoelectric materials:

Piezoelectric materials find extensive use in electronics industry. Their main use is in frequency control of oscillators. A piezoelectric crystal is placed between the plates of a capacitor of a circuit whose frequency is same as the natural frequency of mechanical vibration of the crystal. The circuit acts as tuned circuit of very high Q-value, and possesses excellent frequency stability. They are also used as electro-acoustic tranducers (to convert electrical enery into mechanical and vice-versa). The transducer action in employed in producing ultrasonics which find use in SONAR (sound navigation and ranging), nondestructing testing of materials, MEMS. Lead Zirconate titanate is used in accelerometers, earphones, gas lighters.

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UNIT 5. PHYSICS OF THIN FILMS.

PHYSICS OF THIN FILMS

GENERAL INTRODUCTION:

The study of deposition of thin films has made a tremendous impact on the level of technology we utilize in our daily lives. Thin film coatings provide enhanced optical performance on items ranging from camera lenses to sunglasses. Architectural glass is often coated to reduce the heat load in large office buildings, and provide significant cost savings by reducing air conditioning requirements.

Many of the components of plumbing fixtures used in jewelry industry to decorate jewels are manufactured by depositing thin films of chromium onto injection molded plastic parts. The useful life of tool bits has also been increased by the application of thin films that are chemical compounds. Steel cutting tools used in lathes and mills are often coated with the chemical compound titanium nitride to reduce wear of the cutting edges

The deposition of thin films composed of chemical compounds may be performed in several ways. Co-deposition is a technique in which vapors of two different materials are generated simultaneously. These two vapors condense together, forming an alloy or compound. Other techniques for deposition of compounds include thermal evaporation of the compound (as is performed for salt coatings), sputtering of the compound, and reactive sputtering or evaporation. In the reactive processes, atoms of the evaporant (typically a metal) chemically react with gas species which are intentionally injected into the process chamber. Each of these processes will be described in detail.

HISTORY OF THIN FILM AND ITS DEPOSTION TECHNIQUE

Early references to the science of thin film deposition include the research conducted by Michael Faraday in 1857. In his series of experiments, Faraday created thin metallic films by exploding metal wires in a vacuum vessel. Historically, the techniques for thin film deposition have evolved in approximately in the following order starting from thermally induced evaporation (by electrical resistance heating, induction heating, and electron beam heating), followed by sputtering (diode, triode, magnetron, ion beam), arc processes, and the most recently, laser ablation techniques.

The deposition of thin film (physical vacuum depositonin-PVD) ivolves three steps viz., 1) creation of an evaporant from the source material 2) transport of the evaporant from the source to the substrate (item to be coated) and 3) condensation of the evaporant onto the substrate to form the thin film deposit. The thin film deposition process demands high vacuum condition. There are two reasons why this process is best conducted under vacuum: 1). the process of evaporation involves significant

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amounts of heat, if oxygen were present, any reactive metal would form oxides; and 2). collisions with gas molecules during the transport of evaporant from source to substrate would reduce the net deposition rate significantly, and would also prevent growth of dense films.

When an evaporated material arrived at the surface of the substrate, evaporated material condenses on the substrate in a complex sequence of events that determine many of the physical properties of the deposited film. The steps in the growth of thin films are generally referred to as nucleation and growth. In nucleation, the atoms and molecules which are arriving (called ad atoms) at the surface lose thermal energy to the surface, and the surface absorbs that energy. Depending on the amount of thermal energy the ad atoms and the surface have, the ad atoms move about on the surface until they lose the thermal energy required to move about the surface (referred to as Adam mobility). As nuclei continue to form, the film grows into a continuous sheet covering the substrate. Chemical interactions between the ad atoms and the surface determine the strength of the bond between the film and substrate.

In the deposition process the strength of the bond between the film and the substrate place a major role. For example it is very difficult to deposit a thin gold film on a silicon substrate. This is because Gold does not form a chemical bond with silicon dioxide, and therefore, the adherence of gold films on glass is very weak. Improvement of this adhesion may be made by first depositing a thin (500Å thick) "Binder" layer of chromium or niobium, then depositing the gold over the binder layer. Chromium and niobium do form chemical bonds with the silicon dioxide in glass, and also form metallic bonds with the following gold layer. Once a few

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Evaporant

Source

Substrat

monolayers of evaporant have condensed on the substrate, the film continues to grow in thickness as if the entire substrate were made of the material being deposited. During film growth the microstructure of the deposited film will be developed. This microstructure may be described in terms of grain size, orientation, porosity, impurity content, and entrained gases. Normally, vacuum deposition processes are selected over other processes (electrochemical deposition, flame spraying, etc.) to achieve the following desirable properties:

1) High chemical purity.2) Good adhesion between the thin film and substrate.3) Control over mechanical stress in the film.4) Deposition of very thin layers, and multiple layers of different materials.5) Low gas entrapment.

For each of the vacuum deposition process, keep in mind the ultimate goal is to provide a means for depositing a thin film having the required physical and chemical properties. The parameters one can control to achieve the specified goals are:

1) Kinetic energy of the ad atoms.2) Substrate temperature.3) Deposition rate of the thin film.4) Augmented energy applied to the film during growth.5) Gas scattering during transport of the evaporant.

By varying these parameters one can generate thin films of a given material that have different mechanical strength, adhesion, optical reflectivity, electrical resistivity, magnetic properties and density.

Thermally Induced Evaporation / Thermal evaporation

In this process, heat is input into the source material (often called the charge) to create a plume of vapor which travels in straight-line paths to the substrate. Upon arrival at the substrate, the atoms, molecules, and clusters of molecules condense from the vapor phase to form a solid film. The heat of condensation is absorbed by the substrate. On a microscopic scale the localized heating from this process can be enormous. It is common, in the development of metal coating techniques for thin cross-section plastic parts, to melt substrates during the initial deposition runs. With experience, one can select source-to-substrate distances (h) and deposition rates which will allow coating of temperature sensitive substrates without melting. Following figure shows an illustration of thermal evaporation technique.

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There are several methods by which heat can be delivered to the charge to cause vaporization: electric resistance heating, flash evaporation, arc evaporation, exploding wire technique, laser evaporation (PLD- Laser ablation) and electron beam evaporation. In this various techniques of thermal evaporation, the major principle is heating the source/ charge material by various means.

Introduction

Thermal evaporation may be achieved directly or indirectly by a variety of physical methods.

1. Resistive heating: This method consists of heating the material with a resistively heat filament or boat, generally made of refractory metals such as W, Mo, Ta and Nb, with or without ceramic coatings. Crucibles of quartz, graphite, alumina, beryllia, and ziroconia are used with indicret heating. The choice of the support material/ target material is primarily determined by the evaporation temperature and resistance to alloying and /or chemical reaction with the evaporant. With the exception of highly reactive materials like Si, Al, Co, Fe and Ni, most materials present no problem with evaporation from suitable supports.

a) Sublimation: If a material has a sufficiently high vapor pressure before melting occurs, it will sublimate, and the condensed vapors form a film. Since the rates of sublimation for most materials are small, this method does not find widespread applications.

2. Flash evaporation: A rapid evaporation of a multicomponent alloy or compound, which tends to distill fractionally, may be obtained by continuously dropping fine particles of the material onto a hot surface so that numerous discrete evaporations occur. Alternatively, a mixture of the components in powder form may be fed into the evaporator. The arrangement is shown below. Fine powder (100 – 200 mesh) is fed into a heated Ta or Ir boat by mechanically or ultrasonically agitating the feed chute. This system has been used for preparing films of III – V group compounds of the

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periodic table (Be2, Te3 etc.,) . The Ta boat is held at 1300 to 1400 degree C for III – V compound, and the substrate is heated to 200 – 550 degree C for obtaining single phase compounds..

The draw back of this technique is some times the flash evaporated films show considerable deviation from the original composition. Further, since the large amount of rapidly released gas produces spattering of particles from the evaporant, this method is not easily controllable.

3. Arc Evaporation: By striking an arc between two electrodes of a conducting material, sufficiently high temperatures can be generated to evaporate refractory materials such as Nb and Ta. This method, widely used for evaporation of carbon for electron microscope specimens, employs a standard dc arc-welding generator connected to the electrodes with a capacitor across the electrodes. The arc, initiated by bringing the electrodes together, may last ~ 1/10 sec. Deposition rates ~50 A°/sec or higher can be obtained for refractory metals, but the process is not easily reproducible.

4. Exploding-wire Technique: This technique consists of exploding a wire by a sudden resistive heating of the wire with a transient high current density approaching 106

A/cm2. This is achieved by discharging a bank of condensers (~10 to 100 micro farad), charged to a voltage ~1 to 10 KV, through a metallic wire. Thus, a catastrophic destruction and vaporization of the wire at some region takes place. The draw back of this technique is the thin film so formed showed regions of defects due to the condensation of micro particles spattered during the explosion.

5. Laser evaporation: This method is also called laser ablation technique. Nd:YAG laser can be used for this method (see Pulsed Laser Deposition in Laser chapter). The enormous intensity of a laser may be used to heat and vaporize materials by keeping the laser source outside the vacuum system and focusing the beam onto the surface of the material to be evaporated.

High energy density pulsed laser beams have been used to deposit thin films of a variety of elements, alloys and compounds. In this process, a laser source, external to the vacuum vessel generates a beam which is focused, passed through a viewport and impinges on a target within the vacuum vessel. Sufficient energy is generated to blast (ablate) material from the surface of the target. This ablated material consists of neutral atoms, ions, clusters of atoms and macro particles. The amount of material depositedper laser pulse is very consistent, allowing one to accurately deposit films of a specified thickness. The deposition rate is low compared to other techniques (electron beam evaporation and sputtering, for example).

Laser ablation, as a deposition technique is currently limited to research and

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development laboratories due to the low deposition rate, the additional safety issues involved with the use of lasers and the expense of the equipment. Some of the thin films that have been deposited using laser ablation include super conducting thin films, ceramic coatings, and amorphous metallic layers.

6. Electron beam evaporation: The simple resistive heating of an evaporation source suffers from the disadvantages of possible contamination from the support material and the limitations of the input power, which make it difficult to evaporate high-melting point materials. These drawbacks may be overcome by an efficient source of heating by electron bombardment of the material. In practice, this type of source is capable of evaporating any material at rates ranging from fractions of an angstrom to micron per second. Thermal decomposition and structural changes of some chemical compounds may occur because of the intense heat and / or energetic electron bombardment.

The simplest electron-bombardment arrangement consists of a heated tungsten (W) filament to supply electrons which are accelerated by applying a positive potential tot the material for evaporation. The electron lose their energy in the material vary rapidly, their range being determined by their energy and the atomic number of the material. Thus the surface of the material becomes a molten drop and evaporates. There are two types of electron beam evaporation techniques that are discussed in detail in later section.

Vacuum deposition Technique

Resistive heating:

This process is probably the most widely used deposition process. Both conducting metals and resistive films can be deposited with this method.

Principle:The principle of vacuum deposition is that evaporant charge is activated thermally and the molecules and atoms are transferred from the surface into an extremely low pressure gas phase. The source material may be liquid or solid. In the solid case, the evaporation process is known as sublimation.

Instrumentation:

The bell jar is made up of glass or metal. The jar is highly evacuated usually 10 -5 to 10-6 Torr. High vacuum is necessary to make the mean free path of the evaporated molecules many times the diameter of the bell jar. The schematic diagram is shown below.

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Working:

The substrate and source material to be evaporated are placed in the bell jar. The system is pumped down to the appropriate pressure and the source material is heated by an electrical unit until it vapourised. When the source material’s vapour pressure exceeds that existing in the bell jar, the material vapourizes rapidly. Under high vacuum condition, the mean free path of the vapourized atoms or molecules is greater than the distance from the source to the substrate. The vapourised atoms radiate in all directions and condense on all lower temperature surfaces with which they collide, including the substrate. The substrate is usually heated during evaporation to improve film adhesion.

The film produced by vacuum evaporation exhibits a fine grain structure. The uniformity and the repeatability of the films are improved, if the angle of incidence of the radiating vapour on the substrate is made steeper. Therefore limits the number of substrate that may be processed at any one time. The heating filment such as tungsten is refractory metal having high melting point and low volatibility. The heaters used for evaporation have different shapes such as cone, spiral, boat, dimple boat, crucible etc.,.

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metal foil boat filament Baffled boat

Some of the requirements for a good filament material are:

1) High melting point.2) Low solubility for the charge materials.3) Filament should be wettable by the charge materials.4) Filament should withstand thermal shocks well.

Some of the inherent disadvantages of resistance heated thermal evaporation that should be kept in mind when selecting a deposition technique:1) The source may generate impurities which may co-deposit in the condensing thin film.2) Accurate control of the deposition rate is difficult.3) The composition of alloy thin films deposited may differ from that of the charge material (especially if the elements in the alloy have markedly different vapor pressures).4) The amount of material which may be evaporated per run is limited.5) The substrate will experience heating due to radiant energy from the source.

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SPUTTERING TECHNIQUE

Before the development of the thermal evaporation technique, the method of sputtering was widely used for the preparation of the films for certain metals eg. Platinum and molybdenum for which the thermal evaporation method is difficult on account of their high melting point.

Principle:

The process involves maintaining a discharge in a gas, perfectly innert at pressure of about 10-3 mm of Hg. The cathode being made of the metal to be sputtered. The mechanism is essentially one local boiling of the cathode surface resulting from intese local heating arising from bormbordmetjn by the positive ions in the discharge.

The plasma contains equal numbers of positive argon ions and electrons and as well as neutral argon atoms. Ar+ ions are accelerated to the negatively charged cathode striking that electrode and sputtering off atoms. These travel through plasma and deposit on anode. Rate of sputtering depends on the sputtering yield which is defined as the number of atoms or molecules ejected from the target per incident ion.

For DC sputtering target electrode is conducting. To sputter dielectric materials RF power source is used.

Comparison of evaporation and sputtering

EVAPORATION SPUTTERING

high vacuum path

few collisions line of sight deposition

little gas in film

low vacuum, plasma path

many collisions less line of sight deposition

gas in film

larger grain size smaller grain size

fewer grain orientations many grain orientations

poorer adhesion better adhesion

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ELECTRON BEAM EVAPORATION TECHNIQUE

Electron Beam Heating Evaporation

This technique is based in the heat produced by high energy electron beam

bombardment on the material to be deposited. The electron beam is generated by an

electron gun, which uses the thermoionic emission of electrons produced by an

incandescent filament (cathode). Emitted electrons are accelerated towards an anode

by a high difference of potential (kilovolts). The crucible itself or a near perforated

disc can act as the anode. A magnetic field is often applied to bend the electron

trajectory, allowing the electron gun to be positioned below the evaporation line (see

diagram)

Cooling the crucible avoids contamination problems from heating and degasification.

There are three types of electron beam gunsa) Work accelerated gun b) self accelerated gun and c) bent beam electron gun

a) Work accelerated gun: (Figure a)

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In this case the electron beam is coming out of the loop type of a filament. It is accelerated directly towards the charge or through an appropriate shield thus concentrating the beam. Due to intense heat the charge melts.

Self accelerated gun: (Figure b)In this case the electrons are emitted from hair pin type of filament and focused through a walnut cylinder on the material. Depending on the energy of the electron beam, the charge gets vapourised and the vapour condense on the substrate. Since the electron beam travels in straight line, the whole arrangement requires more space.

Bent beam electron gun: (Figure C)In this method the electron beam is bent by an appropriate magnetic field and then focused on the charge. i.e., the electron beam takes a curved path to minimize the space. The maximum current that can be applied to the filament is calculated by I = 2.3 x 10-6 V3/2 /d A/cm2

“d” separation between anode and cathode. To obtain high current density one has to use higher accelerating voltage and smaller cathode to anode distance.

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1. Scanning Electron Microscope

SEM is one of the widely utilized instrument for material characterization. It is capable of producing high resolution images of a sample surface. SEM images have a characteristic three dimensional appearance and are useful for judging the surface structure of the sample. The main advantage of SEM is its ability to study the heterogeneity of the materials, to visualize various mineral components in their distinct growth forms and their relation in terms of overall micro fabric and texture.

Principle of SEM:When high velocity electrons strike the surface of a solid object in vacuum, electrons are either reflected or emitted from the surface of the sample. Most widely used signals are Secondary electrons, back scattered electrons and characteristic X-rays. The signals are generated with interaction between incident electrons and atoms in a specimen.

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A schematic of typical SEM is shown in the above figure. Electrons thermionically emitted from a tungsten filament (cathode) are drawn to an anode and focussed by two successive condenser lenses into a beam with a very fine spot size that is typically 10Å in diameter. Pairs of scanning coils located at the objective lens deflect the beam either linearly or in raster fashion over a rectangular area of the specimen surface. Electron beams having energies ranging from a few keV to 50keV, with 30keV a common value, are utilized. Upon impinging on the specimen, the primaary electrons decelerate and in losing energy, transfer it inelastically to other atomic electrons and to the lattice. Through continous random scattering events the primary beam effectively spreads and fills a teardrop-shaped interaction volume with a multitude of electronic excitations. The result is a distribution of electrons which manage to leave the specimen with an energy spectrum shown schematically in figure. In addition target X-rays are emitted and other signals such as light, heat and specimen current are produced, the sources of their origin can be imaged with approriate detectors. Various SEM techniques are differentiated on the basis of what is subsequently detected and imaged.

1. Secondary electrons

The most common imaging mode relies on detection of this very lowest portion of the emitted energy distribution. Their very low energy means they originate from a subsurface depth of no larger than several angstroms. The signal is captured by a detector consisting of scintillator/ photomultiplier . Secondary electron image gives surface morphology.

2. Backscattered electrons

These are the high-energy electrons which are elestically scattered and essentially possess the same energy as the incident electrons. The phenomenon which makes use of backscattered electrons is electron backscatter diffraction (EBSD). As the finely focussed electron beam of an SEM penetrates a crystalline specimen, i.e., a grain, electrons are inelastically scattered and lose angular correlation with the primary beam. In this process, a point source of electrons effectively forms and it Bragg diffracts from the sample lattice. Computer analysis of these geometrically complex EBSD patterns enables the crystallographic orientation of individual grains

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to be extracted. As a result a microtexture of polycrystalline films is obtained.

3. X-rays

An SEM is like a large X-ray vacuum tube used in conventional X-ray diffraction systems. Electrons emitted from the filament are accelerated to high enrgies where they strike the specimen target. In the process, X-rays characteristic of atoms in the irradiated area are emitted. By analyzing their energies the atoms can be identified and by coounting the number of X-rays emitted, the concentration of atoms in the specimen can be determined. Instrumentation for this major X-ray spectroscopy technique, known as X-ray energy dispersive analysis (EDX) , is practically always attached to the SEM column.

ApplicationsTopography

The surface features of an object or how it look, its texture and relation between these features and material properties (hardness, reflectivity etc.)

Morphology The shape and size of the particles making up the object, direct relation between these structures and material properties (ductility, strength reactivity etc.)

Composition The elements and compounds that the object is composed and the relative amount of them, relationship between composition and materials properties (melting point, reactivity, hardness, etc.)Crystallographic information Arrangement of atoms in the object, direct relation between these arrangements and material properties (conductivity, electrical properties, strength, etc.)

Other applications include

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Material evaluation with respect to grain size, surface roughness, porosity, particle size distribution and material homogeneity

2.Transmission Electron Microscopy (TEM)TEM is indispensable for the structural imaging of nanometer-sized features. In comparison, SEM is about an order of magnitude poorer. As the name implies, the TEM is used to obtain structural information from specimens thin enough to transmit electrons. Thin films are, therefore ideal for study but they must be removed from electron impenetrable substrates prior to insertion in the TEM.

3. Scanning Tunneling Microscope (STM) All conductors are imageable in principle and there is less danger of distorting and altering surface features in high electric fields. STM enables direct imaging of atoms, unlike other indirect techniques for deteriminig surface crystallography that depend on the amplitude and phases of scattered electron, photon or ion waves.

4. Atomic force Microscope(AFM) Shortly after STM appeared, the atomic force microscope was conceived as a response to the question: if surfaces could be imaged by a current, why not by a force? Amajor advantage of detecting forces rather than current is that all kinds of material surfaces including metals, semiconductors, and insulators are imageable. In order to mechanically sense atomic-scale surface topography, the sharp tip should be mounted at the end of a soft cantilever spring. Presently, AFM cantilevers are micromachined from silicon, silicon dioxide and silicon nitride. There is a mechanical raster-scanning system (usually piezoelectric), a method to sense the cantilever deflection (usually optical, eg. Interferometry), a feedback system to monitor and control the cantilever force, and a display to convert the force-position data into an image.

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