Mechanical BE (Physics II) by IMTS INSTITUTE

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IMTS (ISO 9001-2008 Internationally Certified) PHYSICS II

PHYSICS II

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CONTENTS PHYSICS II

Electrical conductivity in metals, Crystal structure and elasticity Unit 1

01-16

Classical Free Electron Theory

Unit 2

17-53

Quantum Free Electron Theory

Unit 3

54-80

Crystal Structure

Unit 4

81-112

Elasticity

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Course Introduction Engineering physicists find employment in a huge variety of areas. Engineering Physics students develop a thorough understanding of fundamentals of physics and the application of this knowledge to practical problems. This background prepares them for careers in engineering, applied science or applied physics with positions in industry, national research laboratories, universities or even as scientific entrepreneurs.

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Subject Introduction In this block we will learn The classical free electron gas, The DrudeLorentz model, The physical model: collisions, Electrical conductivity, Thermal conductivity, Failure of the model and Miller indices, Planes and directions, Cubic structures, Classification, Atomic coordination, Close packing. Unit 1 Objectives, Introduction, The classical free electron gas, The Drude-Lorentz model, The physical model: collisions, Electrical conductivity,

Thermal

conductivity,

Failure

the

model,

Sommerfeld model of the free electron gas, The Eigenstates, Ground state of the quantum free electron gas, The Heat Capacity, Electrical Conductivity,Thermal conductivity, Summary, Keywords, Self Assessment Questions, References.

Unit 2 Objectives, Introduction, Evidence for Quantum Theory , Planck's Quantum Hypothesis , Bohr atom;Electron Wave , Basic Postulates of Quantum Theory , Wave Function , Complex Numbers , Quantum Particle in a Box , Two State System , Quantum Superposition Principle , Heisenberg's Uncertainty Principle , Path Integral

Quantum

Mechanics

Summary,

Keywords,

Self

Assessment Questions, References.

Unit 3 Objectives, Introduction, Miller indices, Planes and directions, Cubic

structures,

Classification,

Atomic

coordination,

packing, Bravais lattices, Point groups, Space groups, , Grain boundaries, Defects and impurities, Prediction of structure, Polymorphism, Physical properties, Classification of Materials, Types of Crystals, X-ray crystallography, X-ray analysis of crystals, Summary, Keywords, Self Assessment Questions, References.

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Unit 4 Objectives, Introduction, Transitions to inelasticity, Stiffness, Relationship to elasticity, Use in engineering,Young's modulus, Usage, Linear versus non-linear, Directional materials, Force exerted by stretched or compressed material, Hooke's law, Shear modulus, Bending stiffness, Atomic properties, Applications, Yield criterion, Isotropic yield criteria, Anisotropic yield criteria, Factors influencing yield stress, Strengthening mechanisms, Implications for structural engineering, Summary, Keywords, Self Assessment Questions, References.

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Unit 1 Classical Free Electron Theory Structure 1.0

Objectives

1.1

Introduction

1.2

The classical free electron gas 1.2.1 The Drude-Lorentz model 1.2.2 The physical model: collisions 1.2.2.1 Electrical conductivity 1.2.2. 2 Thermal conductivity 1.2.2.3 Failure of the model

1.3 Sommerfeld model of the free electron gas 1.3.1 The Eigenstates 1.3.2 Ground state of the quantum free electron gas 1.3.3 The Heat Capacity 1.3.4 Electrical Conductivity 1.3.5 Thermal conductivity 1.4

Summary

1.5

Self Assessment Questions

1.6

References

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1.0

Objectives After studying this unit you will be able to:

1.1



Explain The classical free electron gas



Discuss The Drude-Lorentz model



Describe The physical model: collisions



Elaborate Electrical conductivity



Understand Thermal conductivity

Introduction Let us understand that it is perhaps surprising that a hundred years ago there was virtually no understanding of the physics of solids. The origins of modern condensed matter physics can be traced back to attempts to account for the properties of metals. We will briefly review the classical and semi-classical free electron theories of the metallic state dating from the turn of the century, and we will see both successes and failures. The free electron model provides a meaningful baseline against which the properties of real metals can be judged. More importantly, understanding the successes and failures of the free electron model will highlight the central issues of this course.

1.2

The classical free electron gas 1.2.1

The Drude-Lorentz model

At the turn of the century, Einstein had not yet explained the photoelectric effect, Rutherford had not determined the size of the nucleus, Bohr had not speculated on the discrete nature of electronic \shells" in atoms, and the formulation of quantum mechanics was still decades away. Although the structure of the atom was not known, Thomson had discovered the electron (1897), enabling Drude (1900) and Lorentz (1905) to formulate a model to explain two of the most striking properties of the metallic state, namely the conduction of electricity and heat.

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1.2.2 The physical model: collisions Drude attributed conduction in the metallic state to the most loosely bound electrons in atoms somehow becoming mobile. In fact these \conduction electrons" were assumed to move freely through space apart from collisions, not with each other but rather with the much larger atomic cores, as shown in Fig. 1.1 on the following page. Geometric considerations imply a mean free path between .

collisions of

where R is the core radius and N is the number of atoms per unit volume. Guessing that R is not that much smaller than the atomic radius, Drude deduced ` to be a few A. He assumed that after a collision a conduction electron has random direction and a speed which does not depend on its velocity beforehand, but is determined by the equilibrium distribution function.

At this time, the most natural treatment of free electrons was provided by the kinetic theory of the ideal gas, and so by analogy the collisions were assumed to be instantaneous and physical (billiard ball like). Since each atom contributes of the order of one conduction electron, the electronic density n is of the same order of magnitude as N (~ 10 29 m~3 ). Notice that the free electron gas is a \fake" many body problem. Although they are many in number these are non-interacting electrons, each moving independently and occassionally suffering a collision with an ion core. Within these assumptions we are entitled to treat them one at a time.

So what happens to the electron-electron and electron-nucleus interactions?

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We can postpone answering this question for a little while since the first job of a model is to correctly reproduce observed behaviour. Only then do we have to explain why the model works.

A classical free particle is allowed to have any velocity. By analogy with the ideal gas, the number of conduction electrons per unit volume with velocities in the range v to v + dv_ is n MB dv in the Drude-Lorentz model, where n MB is the Maxwell-Boltzmann (MB) equilibrium velocity distribution function:

where m is the electron mass, and n is the total number of conduction electrons per unit volume, irrespective of their energies. Note that this expression has then form of a normalising constant multiplied by a Boltzmann factor. In \velocity space" (or \k-space", where k = mv=h), the MB distribution is spherically symmetric and centred on the origin.

If we are only interested in the average speed v = \v\, then the distribution function becomes

If the electron gas is subject to some external force F_, then the electrons accelerate. However, in the Drude-Lorentz model an electron emerging from a collision does not \remember" if it had previously been accelerated or not. For a constant applied force the repeated \resetting" of the electron velocities prevents the electrons accelerating indefinitely but results in the establishment of a drift velocity v_d in the direction of the applied field, superimposed on the random thermal motion of the conduction electrons. (In k-space this amounts to shifting the spherically symmetric distribution away from the origin.) Since the collisions are effectively a means of damping, the equation of motion for the drift velocity is

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1.2.2.1

Electrical conductivity

Consider the electron gas in a spatially invariant DC electric field E_. When a steady state is established, dv d =dt = 0 and the electronic drift velocity is constant, corresponding to the current density J_ = nev_d . It follows that the current density is proportional to the applied field J = <0 E_ (i.e. Ohm's law), where the DC conductivity C0 is given by

Using their estimate of the relaxation time r, Drude and Lorentz obtained conductivities which were in quite good agreement with experiment. Notice that without collisions, r > oo and so the conductivity becomes infinite.

In the presence of an AC field E_(t) = E 0 e~i!t , the steady state drift velocity must oscillate at the same frequency as the applied field, although not necessarily in phase with it. It is easily shown that the AC conductivity is then

Playing

around

with

some

basic

equations

from

optics

and

electromagnetic theory we will see in a Problem Sheet that free electron metals should be highly reflective in the visible region of the electromagnetic spectrum, but transparent to ultra-violet. This is indeed the case for many metals.

1.2.2. 2 Thermal conductivity The electrical conductivities of metals and insulators are profoundly different. When it comes to thermal conductivity the distinction is much less dramatic, but metals do tend to conduct heat about one hundred times better than insulators at room temperature. This is an early indication that there is a mechanism for thermal conduction which is not

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related to electrical conduction. But it also suggests that when metals conduct heat the electrons are doing most of the work.

The electron gas model of metals explains thermal conduction as follows. Imagine holding a metal bar in a fire. The electrons in the end that is in the fire tend to be travelling much faster than those in the end which we are holding. But some of the fast ones may happen to be travelling towards the end we are holding. They travel a certain distance before being scattered in some random direction. Thus they carry thermal energy along the bar and before long it becomes too hot to hold. In the Drude-Lorentz model the heat flow is proportional to the tempertaure gradient 1 and the constant of proportionality, i.e. the thermal conductivity K of the free electron gas, turns out to be

where e is the average electron energy and Cy is the heat capacity of the electron gas (the rate at which the energy density of the solid changes with T, keeping the volume constant).

To calculate e we change the variable in Eq. 1.3 from speed to energy, obtaining the equilibrium energy distribution function

also known as the density of occupied energy levels per unit volume, or simply the density of occupied energy levels. The average electron energy is then

(the classical \equipartition" result for three degrees of freedom). Since the electrons do not interact, the total energy of the electron gas is equal to the sum of energies of the individual electrons and it follows that

and so

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Eq. gave thermal conductivities which again were in quite good agreement with experiment.Better still, Above Eqs. gave a \Lorenz number" L, denned by K=(CTT), This explained the empirical observation of Wiedemann and Franz (1853) that L is independent of T and varies very little from metal to metal. The equation also reproduces the observed numerical value of L

1.2.2.3 Failure of the model Hidden behind this great triumph of the Drude-Lorentz model there was a difficulty: it predicts the contribution to C V due to free electrons is 3nk b =2. In fact C V has this approximate magnitude for both metals and insulators. 2 It must be concluded that (i) there is a contribution to the heat capacities of all solids which we haven't yet identified, and (ii) the electronic contribution predicted by the Drude-Lorentz model for metals is not observed. The problem was a serious and stubborn one and cast a shadow over the free electron model for the next twenty years.

In fact there was an older puzzle. The work of Faraday, Ampere, Lenz etc. had shown that a current-carrying wire experiences a force (the Lorenz force) when placed in a magnetic field. In 1879 Hall tried to determine whether this force acted on the wire as a whole or on some substituent responsible for the electric current. He predicted that the force would act upon the substituent (i.e. the electrons) and, since their path length through the wire would be parabolic instead of straight (and therefore longer), an increase in electrical resistance should be observed. Hall failed to observe what we now call magneto-resistance (which is usually very weak), but found that a voltage built up across the wire perpendicular to the direction of the current.

Hall quickly realised he had overlooked something. Initially the electrons are drawn sideways by the magnetic field but then they build up on the

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edge of the wire. This creates an electrostatic repulsion that eventually balances the Lorenz force, preventing the transverse motion of further electrons. Having explained the origin of this transverse voltage, Hall was in for another surprise. The `Hall voltage' observed for Au, Cu, K and Na suggested that these metals have approximately one free electron per atom. This all fitted with clues from chemistry about the \monovalent" nature of these elements. But the Hall voltages for the divalent metals Mg, Cd and Be and the trivalent metal Al were found to have the wrong sign, as if the current were being transported by positively charged particles. This is not in keeping with our picture of conduction by free electrons.

1.3 Sommerfeld model of the free electron gas We observed in the first lecture that there are only three basic ingredients to the free electron model of the metallic state: the electronic states, the statistics of how those states are occupied, and the damping effect caused by the conduction electrons being scattered. The Drude-Lorentz model (based upon classical physics) had some success but could not be reconciled with all the experimental facts. Where did we go wrong? In this section we will start again but this time using a quantum mechanical approach. We'll do more of the maths but don't be put off by that - there is only one important new idea. (The Pauli principle.)

When puzzles like the black-body problem of electromagnetism and the heat capacity problems for solids arose in the nineteenth century, doubts were raised about the understanding of statistical mechanics and thermodynamics. It was clear that the \equipartition of energy" derived from the Boltzmann law 3 was not universal. We now believe that the Boltzmann law does hold, the problem lay with the nature of matter itself.

The 1920's saw the advent of a quantum mechanical description of atoms. The de Broglie wavelength for an electron with energy 13.6 eV (the kinetic

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energy of an electron in a hydrogen atom) is a few A and this is of the same magnitude as the interatomic separations in condensed matter. On this basis one might expect that electrons in solids will demonstrate diffraction and interference. Perhaps we need to take this seriously. 4 Maybe we should also consider the many body nature of the electron gas. In the \one-electron approximation" one replaces the true (unknown) wavefunction of the whole system which we denote

by an

approximate one built up from states which each describe only one electron:

. Maybe we should be aiming to get ^ N

rather than considering each electron separately.

We will return to these two issues when we get a bit more sophisticated, but Sommerfeld brushed them aside by continuing to assume that the conduction electrons in a metal do not interact with each other or with the ions. If they don't feel the ions, they can't be diffracted by them. If they can't feel each other, then the one-electron approximation is exact. Sommerfeld pointed out that there is something more fundamental to grasp: the quantum statistics theorem. This states that the wavefunction of a system of identical spin-half particles must be anti-symmetric with respect to interchange of any two of the substituent particles. Within the one-electron approximation it follows that no two electrons can be in the same quantum state, and we usually call this the Pauli principle. In a rather mysterious way the electrons in a metal would be somehow aware of each other even if there were no Coulomb force between them.

We now need to (i) figure out what are the states (using quantum mechanics), then (ii) make sure we put only one electron in each.

1.3.1 The Eigenstates Starting with the time-dependent Schrodinger equation we write the wavefunction of a free electron as the product of a spatial function and a

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temporal function: It is straight forward to show that where e and ip are the eigenenergy and wavefunction in the timeindependent Schrodinger equation. States of this form are termed stationary states since their probability density equals

and is

independent of time. 5 For a free particle the time-idependent Schrodinger equation is simply and one can easily show thatare eigenfunctions with energy Notice that the full wave-function of the free electron has the form exp , the equation for a plane wave with wavevector 6 k. There are infinitely many such solutions of the Schrodinger equation and so we have distinguished them by writing k as a subscript on the wavefunction symbol. is also an eigenfunction of the momentum operator: and so each electron has well-defined momentum given by

Just as in the classical

treatment, we have e = p 2 =2m and any momentum is allowed.

All pieces of metal are finite in size so one might wonder what happens to the electrons as they encounter a surface. In any reasonably sized sample surface effects are negligible and we can largely avoid them by imposing periodic boundary conditions.

Consider a macroscopic piece of metal in the shape of a cube with linear dimension L and volume W = L 3 . Periodic boundary conditions mean that if an electron passes through any particular face of the cube it immediately re-enters it cube through the opposite face. Mathematically this means

Combining Eq. 1.14 and 1.15 we find that the components of k must be of the form

Imposing boundary conditions has introduced an artificial restriction on the allowed k vectors (and hence momenta): they are evenly distributed with a spacing of 2ir=L in the x; y and z directions. But by choosing L to be very

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large we can ensure that the allowed k vectors are arbitrarily close together.

A good thing about having a discrete set of allowed states is that it is easier to keep track when we have to count them. The volume (of kspace) around each allowed k point is

, and the density of allowed

momentum states is therefore

Clearly it is uniform, much the same as for the classical electron gas. The corresponding density of energy levels is

It is conventional at this stage to multiply this result by a factor of 2 to account for spin degeneracy of the states, and to divide by W to give the density of states per unit crystal volume.

1.3.2 Ground state of the quantum free electron gas Now let's consider how the states are occupied, starting with the T > 0 limit which we call the ground state. In the classical electron gas the velocities (and e and k) collapse to zero, but the Pauli principle allows only two electrons (with opposite spin) to have k = 0 in the quantum electron gas. We must place the rest of them in successively higher energy states. The highest occupied level we call the Fermi level, denoting its energy tf and the magnitude of its wavevector k f . In k-space the levels are now uniformly occupied for k < k f and unoccupied outside this Fermi sphere. The Fermi energy is determined simply by the number of conduction electrons per unit volume and the available density of states. For most metals tf is of the order of a few eV (where 1 eV = 1.6 x 10~19 J) and k f ~ 10 11 m_1 = 10 A , corresponding to a Fermi velocity of Vf ~ 10 6 ms-1 . It is vital to realise Vf is the typical electron velocity in the quantum free

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electron gas at T = 0. It should be clear that the properties of the free electron gas are dominated by the Pauli principle and in this sense the free electron gas is fundamentally quantum mechanical.

1.3.3 The Heat Capacity When T > 0 there is a finite chance that the available thermal energy will excite an electron into a state above the Fermi level. You will have seen last year that the probability of a state with energy e being occupied is now given by FD , the Fermi-Dirac factor:

For our purposes we can take the chemical potential the Fermi energy

It is easy to show that

to be equal to and FD

Figure 1.2: Comparison of the density of occupied states at room temperature (solid curves) and T = 500 K (dotted curves) for an electron gas with the density of metallic sodium. MB and FD refer to MaxwellBoltzmann and Fermi-Dirac statistics respectively.

At finite temperature the density of occupied states in the quantum electron gas is where ge is just the density of states. At everyday temperatures (say less than 10 4 K) the Fermi-Dirac distribution is spectacularly different are compared in Fig. from the Maxwell-Boltzmann distribution. 1.2.

Recalculating e with FD statistics, Sommerfeld found that the heat capacity of the quantum free electron gas to be

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Since the Fermi energy is so much larger than usual thermal energies (at room temperature k B T ~ 0:02 eV), it is evident that the Sommerfeld model predicts a much lower heat capacity than its classical counterpart (typically a factor of ~ 50 smaller at room temperature).

What is the physical origin of this result? Increasing the temperature of the classical electron gas moves the whole distribution of occupied states to higher energy, as can be seen in Fig. 1.2. In the quantum gas only those very few electrons within ~ k B T of the Fermi energy can be thermally excited because of the Pauli principle. At low temperature the Sommerfeld model gives a reasonable estimate of the heat capacity of real metals, although the experimental data for potassium in Fig. 1.3 on the following page reveals a contribution to Cy which appears to scale with T 3 . Near room temperature the heat capacity of most solids is of order nk\, and the electronic

Figure 1.3: Temperature dependence of the heat capacity Cy of Potassium. The linear relation of Cy to T predicted by the Sommerfeld model is quite well reproduced by experiment at low temperature. However a cubic contribution (which dominates at higher temperature) can be seen by plotting Cy =T against T.

contribution is completely swamped. Clearly there is another type of effect here, but at least we now understand the electronic contribution to the heat capacity of metals. 1.3.4 Electrical Conductivity It is worth emphasizing that although Sommerfeld solved the Schrodinger equation to calculate the stationary states of the free electron gas, he

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appealed to a collision mechanism to explain electrical resistance just as his predecessors had done. Since the Drude estimate of the collision time seemed to work (it gave the correct room temperature electrical conductivity), this value r ~ 10~14 s was retained by Sommerfeld. But in the Sommerfeld model the average electron speed at this temperature is about 10 times greater than Drude and Lorentz had calculated. We must therefore conclude that A, the average distance traveled between collisions, is ~ 10 times longer than Drude and Lorentz had supposed and this must cast doubt over the collision mechanism they had advanced.

We should abandon the Drude collision mechanism at this stage, but something must scatter electrons, and experimental measurements provide a few small clues. It was well known that the conductivity of metals decreased steadily with T. It was also found that it decreased suddenly upon melting (at constant T). The conductivity of pure metals was found to be reduced when impurities were added. We're not quite ready to digest this information yet, but we'll try to come back to it.

Sommerfeld's retention of a classical description of electron dynamics (Eq. 1.4) requires some justification. How do the wave-like eigenstates of the quantum mechanical free electron gas relate to the particle picture implicit in classical dynamics? We'll tackle this matter in \Solid state physics" (SSP) next term, but in case you are interested: we must identify each electron with a wave packet of free electron waves. These are spatially localized on the scale of the collision length, but are delocalized on the scale of the atom.

1.3.5 Thermal conductivity How did Drude and Lorentz get good values for K when their calculated value of the electronic contribution to Cy was so poor? By looking at Eq. 1.7 we see that Drude and Lorentz were very fortunate. By using MB

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statistics they overestimated Cy but also underestimated the average electron energy by a similar factor. When Sommerfeld corrected Cy and e he obtained more or less the same K as Drude and Lorentz, which had already been found to be in agreement with experiment.

1.4

Summary There are three main pieces to the Sommerfeld theory of the electron gas: Solution of the Schrodinger equation to get the allowed electronic states.The use of Fermi-Dirac statistics to determine the population of the states. Electron scattering: Appeal to some (unknown) scattering mechanism, and classical electron dynamics (between collisions).

After the work of Sommerfeld the free electron model was in much better shape.

1.5

Self Assessment Questions 1. What is(are) the true mechanism(s) for electron scattering? 2. How can the conduction electrons in a metal move seemingly unhindered over such long distances? 3. What is the dominant contribution to the heat capacities of solids for T > 10K? 4. Why does conduction of electricity in some metals appear to take place via the transport of positively charged particles?

1.6

References Hibbeler, R.C., Engineering Mechanics: Dynamics, 7th edition. PrenticeHall, Englewood Cliffs, N.J.,1995. Reif, F., Fundamentals of Statistical and Thermal Physics. McGraw-Hill Inc., 1965. Kittel, C. and Kroemar, H., Thermal Physics, 2nd edition. W.H. Freeman, 1980.

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Demarest, Engineering Electromagnetics. Prentice-Hall. Staelin, D.H., Morgenthaler, A.W. and Kong, J.A., Electromagnetic Waves. Prentice Hall, 1994. Kroemer, H., Quantum Mechanics for Engineering, Materials Science & Applied Physics. Prentice Hall, NJ, 1994.

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Unit 2 Quantum Free Electron Theory Structure 2.0

Objectives

2.1

Introduction

2.2

Evidence for Quantum Theory

2.3

Planck's Quantum Hypothesis

2.4

Bohr atom;Electron Wave

2.5

Basic Postulates of Quantum Theory

2.6

Wave Function 2.6.1 Complex Numbers

2.7

Quantum Particle in a Box

2.8

Two State System

2.9

Quantum Superposition Principle

2.10

Heisenberg's Uncertainty Principle 2.10.1 Path Integral Quantum Mechanics

2.11

Summary

2.12

Keywords

2.13

Self Assessment Questions

2.14

References

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2.0

Objectives After studying this unit you will be able to:

2.1



Describe Evidence for Quantum Theory



Discuss Planck's Quantum Hypothesis



Elaborate Bohr atom;Electron Wave



Give Basic Postulates of Quantum Theory .



Define Wave Function



Explain Quantum Particle in a Box



Understand Quantum Superposition Principle

Introduction Let us understand that Classical (Newtonian) mechanics works perfectly in explaining the world around us, and is accurate enough for even charting the trajectory of probes sent to Jupiter and beyond. So why are we not content with classical physics? Where does the need for quantum theory arise? Quantum theory unveils a new level of reality, the world of intrinsic uncertainty, a world of possibilities, which is totally absent in classical physics. And this bizarre world of quantum physics not only offers us the most compelling explanation of physical phenomena presently known, but is also one of the most prolific source of modern technologies, providing society with a cornucopia of devices and instruments.

2.2

Evidence for Quantum Theory Classical mechanics works very well for large objects that are moving much slower than the velocity of light. Once objects start to move very fast, we need to modify Newton's equations by relativistic equations. On the other hand, for objects that are very small, quantum theory becomes necessary. If one attempts to extend Newton's laws to domains that are far from daily experience, they start to fail and give incorrect results. Historically, at the turn of the nineteenth century, this failure of Newtonian physics became very evident in the studies of the atom. What

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experimental evidence do we have that classical physics is invalid, and that, to date, quantum theory is the most accurate explanation of how nature behaves? Classical physics is what intuitively follows from our five senses, and we have no reason to naively extend the world apprehended by our five senses to microscopic domains of which we have no direct experience. As it turns out, most of the experimental evidence for quantum theory runs counter to one's day to day experiences. One of the fist achievements of quantum theory was the explanation of the structure and stability of atoms, and of the periodic table of elements. Strange phenomena such as super-conductivity, super-fluidity and so on are more macroscopic manifestations of quantum behaviour. Instead of quoting results from experiments far removed from daily life, suffice it to say that most of what goes under the name of high technology is a direct result of the workings of quantum mechanics, and most modern conveniences that we take for granted today would be virtually impossible without it. Observations of radiation from a blackbody and its radiation (measured by spectroscopic lines) provided the first experimental evidence for quantum theory. Every time one sees a neon or sodium light, one is seeing quantum theory in practice. The light from a neon or sodium source is a spectroscopic line. An electric field excites atoms of the neon or sodium atom to a discrete quantum state; the atom then makes a transition by emitting light that is characteristic of the atom, and yields the particular color of light that one sees. Furthermore, semiconductors and electronic chips in general exist due to quantum theory. Electronic devices, from computers, television, to mobile phones are

all based on the

semiconductor, and aeroplanes, ships, cars all use semiconductors in an essential manner. More complex technologies such as MRI (Magnetic Resonance Imaging), lasers, physical chemistry, fabrication of new drugs, modern materials science and so on all draw on the principles of quantum theory. It is no exaggeration to predict that twenty first century technology will largely be based on the principles of quantum physics.

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2.3

Planck's Quantum Hypothesis A black body which is maintained at a constant temperature T steadily loses energy from its surface in the form of electromagnetic radiation. Since the atoms composing the black body are in contact with a heat bath at temperature

, each atom has approximately

amount of energy,

where k = Boltzmann's constant. Since the atoms are jiggling around due to thermal motion, classical electromagnetic theory then predicts that all wavelength's of radiation, in particular upto infinitely short wavelengths, should be emitted by a black body. This classical prediction for the spectrum of radiation that is emitted by such a black-body is contradicted by experiment. Max Planck, a German physicist, correctly explained the experimentally measured black- body spectrum by making the epochmaking conjecture in 1900 that electromagnetic waves are the macroscopic manifestations of packets of wave-energy called photons. Planck further made the quantum hypothesis that the energy of photons is quantized in the sense that the energy of the photons only comes in discrete packets, the smallest packet called a quantum. Photons are massless quantum particles, and all phenomena involving electromagnetic radiation can be fully explained by the quantum theory of photons. The phenomenon of electromagnetic radiation is a classical approximation to the quantum theory of photons. As mentioned in our earlier discussion on radiation in classical electro-magnetism is a valid approximation when the typical energy of the photon is less than the characteristic energy of the instrument with which the experiment is being performed. Photons can have wavelength from zero to infinity. For a wave of frequency equivalently, of wavelength

, the quanta of energy are given by (

, or is the

velocity of light)

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(1)

(2) (3)

We see from the above that the shorter the wavelength of a photon, the greater is its quantum of energy. The constant

is an empirical constant

of nature, required by dimensional analysis, and is called Planck's constant. Its numerical value is given by

(4) (5)

For the sake of convenience, it is customary to work with (6)

(7) (8)

The quantum postulate immediately solves the problem of the black-body spectrum; radiation with increasingly short (ultra-violet) wavelength is incorrectly predicted by classical physics to make an increasingly large contribution to the energy loss. Due to Planck's quantum postulate, to emit even a single quanta of ultra-violet radiation would require a minimum

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energy much larger than the typical thermal energy of about kT that is available for emission, and hence would not be present in the radiated spectrum of a black-body. The black-body spectrum obtained from Planck's postulate is confirmed by experiment, and was the first success of quantum theory.

2.4

Bohr atom;Electron Wave Various experiments towards the end of the nineteenth century indicated that atoms are made out of a positively charged nucleus with negatively charged particles moving around it. Atoms must interact with light. A simple manifestation of this interaction is the phenomenon of sight; light is 'reflected' off material bodies composed out of atoms. The reason an object has a color, say green, is similar to our previous analysis of a neon light. The atoms of a green object absorb only the green frequency of light, and then re-radiate green color light, thus making it appear green to our eyes. By analyzing the light emitted by atoms, one reaches the remarkable conclusion that the energy that the atom can absorb or emit only takes certain discrete values. This in turn implies that the atom's energy is quantized, and that the atom can have only a set of discrete energies. Figure 1 shows the discrete energy levels of a hydrogen atom.

It was initially thought that the atom was a microscopic version of the solar system. The idea is mistaken for the following reason. An electron moving in a closed orbit has to classically keep on accelerating since (the direction of) its velocity is constantly changing. Since it is known from electromagnetic theory that an accelerating charge always radiates, the electron would continuously lose energy, and consequently would soon spiral into the nucleus, and in effect the atom would collapse. Hence, according to Newtonian physics, atoms are inherently unstable and cannot exist. This classical ``prediction'' contradicted the entire body of knowledge that chemistry had developed based on the idea of atoms, and

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that there is a different kind of an atom for each of the different elements. The question which confronted physicists was the following: are atoms real, are they made out of material particles? If so, why don't they obey the laws of classical mechanics and of electromagnetism? A complete explanation of the atom only emerged in 1926. In the absence of an alternative to Newton's law, pioneers of quantum theory had to reason intuitively and metaphorically. By the early 1900's pioneers like Niels Bohr, Max Born, Arnold Sommerfeld and Victor deBroglie had developed ad hoc rules to explain the existence of the atom, inspired by the quantum postulate of Planck. Recall Planck's conjecture that light wave consists of photons which cannot have continuous energy - unlike the case for classical radiation - but instead, that the energy of the photon is quantized and can only have discrete quanta of energy. Bohr and deBroglie made a number of postulates regarding the atom, and we only discuss those that were vindicated by later developments. The ideas of Bohr and others were conjectures, since there was no underlying theory from which these could be deduced. An electron inside an atom moves in the Coulomb potential due to the positively charged nucleus, leading to an attractive force and resulting in the electron being in a bound state. Figure 2 schematically shows the hydrogen atom, which consists of a nucleus made out a proton ( and zero neutron to be consistent with later discussions), and an electron ``circulating'' it. More precisely, a hydrogen atom consists of an electron of mass charge velocity

and charge

which is bound to a nucleus (proton) having

. The energy of an electron in a Hydrogen atom, moving with , is then given by

(9)

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where

is given in eq. Bohr made the ad hoc conjecture in 1913 that inside an

atom, the electron can exist only in certain special allowed states. If the electron is in such a state, it will not radiate even though it is constantly accelerating. This conjecture immediately leads to a discrete set of energies for the atom. Bohr explained the atoms' absorption and emission of radiation by the electron making ``quantum transitions'' from one its allowed states to another one. Bohr further conjectured that there is a lowest energy state for the electron, called the ground state, and once the electron is in this ground state, it will no longer radiate. Bohr conjectured that the size of an atom should be determined by the fundamental constants involved in binding an electron to the nucleus, namely the electron mass and charge quantum

and

phenomenon,

respectively, and since the existence of the atom is a Planck's

constant

should

also

appear.

By merely doing dimensional analysis, it can easily be seen that the combination has

the

dimension

length,

and

has

value

near

This is a remarkable coincidence, since this is the typical size of an atom, and shows the power of dimensional analysis. A more careful analysis shows that theradius of the hydrogen atom is approximately given by

m. Question: Why does the speed of light

not appear in the estimate for the

size of the atom? What are the special allowed states of the electron? To get the correct energy levels, Bohr was further led to the (correct) conjecture that the angular momentum

of the electron in the atom is quantized such that

(10)

Bohr further made the incorrect conjecture that the electron inside the hydrogen atom moves in an exactly circular orbit. (11)

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and

effect

the

radii

for

electron

motion

quantized.

Furthermore, for a particle moving in an exact circle, we have that the attractive force of the Coulomb potential be exactly balanced by the centrifugal force, and yields

and which yields from eq. that the allowed (discrete) energies of the hydrogen atom as given by

(13)

(14)

The energy of the hydrogen atom comes out to be negative, as expected since the electrons are in a bound state with the nucleus. The energy level of the hydrogen given by the Bohr is correct, although Bohr made a number of correct and incorrect assumptions and conjectures to come up with this result. The structure of the hydrogen atom is shown in Figure 2. Why should the electrons inside an atom have only a discrete set of allowed states? Bohr had no explanation for this experimental fact, and it was only later, in 1923, that deBroglie offered the following explanation. Just like the photon is a particle that manifests as electromagnetic waves, one can conversely conjecture that a particle, say an electron,

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also has a wave associated with it. For a particle of mass velocity

moving at

, deBroglie conjectured that there is an associated wave with a

wavelength given by

(15)

Figure 3: DeBroglie Wave of an Electron

How does the idea of deBroglie explain the behaviour of the electron inside the atom? At the scale of the atom, which is of the order of m, deBroglie conjectured that the idea of an electron being a classical particle having a definite position and velocity is no longer valid. Furthermore, reasoning by analogy with the concept of resonance in waves for which only certain frequencies are allowed, deBrolie conjectured that the only allowed waves for the electron are resonant waves. To see how Bohr's conjectures follow from the idea of an electron wave, for a circular orbit of radius

, a state of the electron with

complete wavelengths yields, similar to eq for the case of resonance for a circular object, the following.

(16)

(17)

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The restriction of the states of the electron to be only resonant waves in turn provides an explanation of the discrete energy levels observed for atoms. The discrete quantized energies of the atom correspond to the allowed resonant frequencies. The electron can make ``quantum transitions'' from a higher energy state to a lower energy state by radiating photons, and vice versa by absorbing photons. Furthermore, the fact that there is a lowest frequency for a resonant wave explains why an electron can be in a bound state with the nucleus without radiating, since there is no lower state into which it can make a transition.

In summary, an electron can have a definite energy in an atom and be in a stable stationary state, but the price that we must pay is that we no know its exact position. This is the only way we can avoid the classical result that an accelerating charge must radiate. Note the inability to know the position and velocity of an electron in an atom is not like our ignorance in statistical mechanics; rather, the ignorance in quantum phenomena is an inherent limitation that is placed by nature on what can in principle be known, in this case about the electron's position and velocity. This quantum postulates of deBroglie yields a stable atom. But there is the following paradox inherent in deBroglie's postulate of an electron wave. Each and every time an electron is observed in an experiment, it is seen to be a point-like particle; on the other hand, a wave is spread over space. So this is the paradox: how can the electron be a point-particle and at the same time be a ``wave''? This is the famous ``wave-particle'' duality that permeates quantum physics. What is the nature of the wave that deBroglie postulated? Is the electron wave a physical wave, like a sound wave or an electromagnetic wave? For five years, the electron wave that deBroglie postulated was simply an interesting metaphor, without any sound theoretical or mathematical foundation. In 1926 it was finally understood that the electron wave of Bohr is not a physical wave, but

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instead, is a probability wave. What do we mean by a probability wave, and how does this relate to the behaviour of an electron? We address this question in the section below.

2.5

Basic Postulates of Quantum Theory The modern formulation of quantum theory rests primarily on the ideas of Erwin Schr

dinger, Werner Heisenberg and P.A.M. Dirac. In the period

from 1926-29 they laid the mathematical foundations for quantum mechanics, and this theory has successfully stood the test of innumerable experiments over the last seventy years. At present, there is not a single experimental result which cannot be explained by the principles of quantum theory. Unlike Einstein's theory of relativity which reinterpret's the meaning of classical concepts such as time, position, velocity, mass and so on, quantum theory introduces brand new and radical ideas which have no pre-existing counterpart in classical physics. To understand the counter-intuitive and paradoxical ideas that are essential for the understanding of quantum mechanics, we develop it in contrast to what one would expect from classical physics, and from intuition based on our perceptions of the macroscopic world. Recall that a classical system is fully described by Newton's laws. In particular, if we specify the position and velocity of a particle at some instant, its future evolution is fully determined by Newton's second law. In quantum mechanics, the behaviour of a quantum particle is radically different from a classical particle. The essence of the difference lies in the concept of measurement, which results in an observation of the state of the system. A classical particle, whether it is observed or unobserved, is in the same state. By contrast, a quantum particle has two completely different modes of existence, something like Dr Jekyll and Mr Hyde. When a quantum particle is observed it appears to be a classical particle having say a definite position or momentum, and is said to be in a physical state. However, when it is not observed, it exists in a counter-intuitive state,

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called a virtual, or a probabilistic, state. To illustrate the difference between a classical and quantum particle, let us study the behaviour of a classical and quantum particle confined inside a potential well of infinite depth. Consider a particle of mass

, confined to a one-dimensional box,

with perfectly reflecting sides due to the infinite potential, of length

Suppose the particle has a velocity

, and hence momentum

Let us study what classical and quantum physics have to say about the particle confined to a box. Classical Description The classical (Newtonian) description of a particle is that the particle travels along a well-defined path, with a velocity

. Since the box has

perfectly reflecting boundaries, every time the particle hits the wall, its velocity is reversed from

, and it continues to travel until it hits the other wall and

bounces back and so on. We hence have

(18) (19)

The point to note is that the position and velocity of the classical particle are determined at every instant, regardless of whether it is being observed or not. Quantum Description A particle inside a potential well is similar to an electron inside an atom, and hence is be described by a resonant wave. The reason we choose the example is because it has all the features of the

-atom, but is much simpler. The specific features of an

electron inside an atom discussed earlier reflect the general principle of

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quantum theory which states that, if the momentum of the particle is fully known, we then have correspondingly no knowledge of its position. The precise relation between the uncertainty in position and momentum is given by the Heisenberg Uncertainty Principle discussed. Note we can interchange the role of momentum with position, and a similar analysis follows. Hence, similar to the case of the Bohr atom, the electron in the potential well is in a bound state with a definite energy, but at the same time it no longer has a definite position. In summary, the particle inside the potential well has a definite momentum (and hence has definite energy), but its position inside the well is a random variable. When it is not observed, the quantum particle exists in a random state, which in physics is called a virtual state; in particular, the position of the particle can be anywhere within the interval

What happens if we perform a measurement to actually ``see'' what is the position occupied by the particle? The measurement will find the electron to be always at some definite point; the act of measurement causes the electron to make a quantum transition from its virtual state to an actual physical state. In summary, the quantum particle has two forms of existence: a virtual state when it is not being observed, and a physical state which is observed when a measurement is performed on the particle. In this sense, the physical state of the quantum particle is Dr Jekyll, and its hidden virtual state is Mr Hyde.

2.6

Wave Function Recall that in our discussion of the atom, we discussed deBroglie's conjecture the electron is described by a (probability) wave.The resolution of the paradoxes posed by Bohr, deBroglie and others is the following: the electron wave that deBroglie postulated is not a classical wave, but rather is a probability wave. The fundamental quantity that specifies quantum probabilities is a wave-like entity called a probability amplitude, also

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called the Schr

dinger wave function, and is denoted by

be emphasized that the probability wave

. It should

of a quantum particle is

not an ordinary classical wave; rather, the only thing it has in common with a classical wave is that it is spread over space. The non-classical nature of

can seen from the following fact: the moment the quantum

particle is observed (measured), say to be at a definite position

, the

wavefunction of the quantum particle instantaneously goes to zero ('collapses') everywhere else in space, since once we find the particle at position x there is zero likelihood of finding it at any other point! This process of the collapse of the wavefunction causes an irreversible change in the system and is called decoherence. The wave function

in general

is a complex number, and is the central quantity in quantum mechanics, replacing the role played by the position and velocity of a particle in Newtonian mechanics. In quantum mechanics one gives up any attempt to know what the object intrinsically is, what in philosophy is called the object in-itself. The wave function

contains all the possible information

that can be extracted from the object by a process of repeated measurements. Quantum probabilities, denoted by wave function

, are related to the

by the following relation.

(20)

The equation above is the great discovery of quantum theory, namely, that behind what we observe lies the hidden world of probability, which in turn is fully described by the wave function determines how

The Schr

dinger equation

changes with time; given the initial wavefunction

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of the particle at t=0, i.e. Schr

, the future behaviour is then fixed by the

dinger equation. In summary, we see that the classical particle

which is pointlike has now been replaced by a quantum particle which is described by the wavefunction

. The quantum particle is

inherently random when it is not being observed, and in the sense of probability exists everywhere in space and hence is wave-like; when a measurement is performed to ascertain the position of the quantum particle, it is always found to be pointlike and hence the particle-like behaviour of the quantum particle. This is the wave- particle duality referred to earlier. The non-local collapse of

has puzzled

physicists since it apparently needs the information that the particle has been observed at position x to be communicated at infinite speed to the rest of space for the instantaneous 'collapse' to take place and would seem to violate the special theory of relativity. However, detailed analysis has shown that quantum measurement theory is consistent with the special theory of relativity; and more importantly, the non-local nature of the wavefunction

is consistent with all the experiments that have

been devised to test this aspect of quantum mechanics.

2.6.1 Complex Numbers A complex number

, is represented, for real numbers a and b, by

(21) (22) Complex numbers form a system of arithmetic similar to real numbers. For the more mathematically minded, it can be shown that to solve for the roots of an

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arbitrary n-th-order polynomial equation, it is necessary and sufficient to extend the real numbers to complex numbers.

2.7

Quantum Particle in a Box We examine more carefully the behaviour of a quantum particle inside a box; in particular, what it means to observe and to not to observe the particle. For the quantum particle confined to a box of length, we know that the probability of finding the particle outside the interval must be zero, and hence we need

to vanish outside the interval. Let

inside the interval; in other words by

lies inside the interval

. Following the reasoning of deBroglie, we take

be a point , denoted to be a

resonant wave we have the following.

[C is a constant which we will fix later.] The constant

that has been introduced

in the equation above is purely on dimensional grounds. The argument of a sine function is an angle, which is dimensionless. Since and

has dimensions of mass

dimensionful quantity

has dimensions of length,

velocity, we have to divide out by a a

which has the dimensions of

. We will soon see that

this constant is none other than the famous Planck's constant, whose numerical value is given in eq.(6). We need to smoothly match outside the interval and

, and hence

must vanish at the boundary points

. Note that from above that at ; however, to achieve

for the regions inside and

we have as expected

, we need to constrain (quantize) the

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possible values of

. From the properties of the sine function we know that

for any integer

. Hence for the momentum

of the quantum

particle we have the following. We see from eq. that the momentum of a quantum particle is not like that of a classical particle. A classical particle inside a box can have any momentum

; however for the quantum particle, since the wave function

has to vanish outside

the momentum of the quantum particle is

quantized, and can only have a discrete set of values, and consequently its energy is quantized as well. This is a general feature of a quantum system, and as one can imagine, quantum theory derives its name from this phenomenon. In summary, for a quantum particle, its energy is given by

(36)

(37)

Note that both momentum

and energy

come in discrete amounts measured by

have been quantized, that is and

respectively. Now it is

known from classical physics that the energy for example of a string tied at two ends also becomes discrete. But what is unique about the quantization of momentum and energy is that it is measured in terms of a universal constant

. For a quantum particle, what can be physically

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measured is not its position, but rather,the probability of the particle being at different positions inside the box. The probability is given by

In Figure's 10 and 11 the probability of the particle being at various positions in the interval

is plotted. Note the salient point that unlike the

classical particle which passes through each point within the interval

the quantum particle, say in the first excited state, will never be found at the midway point of the interval. In general, the probability distribution of the quantum particle shows it to be anything but a point particle when it is in its virtual state. The graph of

describes the result of a number of experiments. Since

we are looking at a particle, say an electron, with a definite momentum, we prepare such electrons in the following manner. Heat a filament, which causes it to eject electrons with a wide range of momenta. We then subject the electrons to a fixed magnetic field, and the electrons will curve around with different radii depending on their momenta. We position our box at a particular radii, and, let the electrons with a definite momentum allowed by the box to enter into the box, and give it time to 'settle down'.

We then measure the position of the particle inside the box. The way this is done is to measure the position of the particle by say shining light on it; suppose we find that its position is having a definite momentum and with a definite position

. So we end up with an electron

as well,which we had prepared carefully, due to the measurement that we performed.

One may object to the statement that the particle is at a definite position , since did we not assume that the position of the electron inside the box is random? The answer is yes, the position is random. What we have done is to obtain one possible position of the electron. In other words, when the

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position of the particle in not measured, it is in a (random) virtual state. By the act of measurement, we caused the particle to make a quantum transition from its virtual state to an actual physical state. We have to repeat the experiment again and again, and every time sending in an electron with the same momentum, and then measuring its position. We will soon discover that the position of the (identical) electrons entering into the box varies, and the electron is found at all points inside the box. We repeat the experiment

times, and record the number of times,

that the electron is found at the position

. We can then calculate the

probability that the electron is at the different positions, given by For the given momentum chosen, say

, we have, in accordance with the

general formula the following

An an estimate, we have

. The result of the

experiment will yield, for the second excited state, the distribution of the positions as shown in Figure 11. The probabilities computed from quantum theory behave the same way as that of classical probability. What separates classical and quantum probabilities is the existence of the wave function, and we explore these differences in the next section.

2.8

Two State System In quantum theory, a particle is described by specifying all the possible states it can have. To simplify our discussion, consider a particle that can have only two possible states. Such a system is the simplest possible one for a quantum particle, and is also called, for obvious reasons as a twostate system. An example of a two-state system is the spin of an electron. In addition to moving around in space, the electron has an intrinsic angular momentum called spin. The spin of the electron can either point

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up or down, and hence forms a two-state system. A quantum two-state system is described by determining its amplitude to be in the two states. How should we mathematically describe our two-state system? The two different states should be "orthogonal" to each other, in the sense that being in one state is completely different from being in the other state. The simplest way to realize this expected orthogonality of the two states is to represent them by two-dimensional vectors, and the idea of orthogonality translates exactly into the concept of vectors being perpendicular. Hence, we will represent the wavefunction for a two-state system by twodimensional vectors. One should note that the two-dimensional vector space has got nothing to do with a physical two-dimensional space, but rather, should be viewed as a mathematical construction for describing the spin of an electron.

To precisely discuss the spin of an electron, we first have to choose a coordinate system for the electron. Consider an external magnetic field pointing along the z-axis. Consider two special cases for the spin of the electron. Case (a) The spin points along the z-axis, which we denote as the spin is pointing "up". Case (b) The spin points towards the negative zaxis,which we denote as the spin is pointing "down". The wavefunction for these two special cases are the following. (41) (42)

diagram spin pointing up, down and aribitrary So far we could have been discussing classical physics, since the spin pointing up or down with 100% certainty is a classical concept. A quantum mechanical spin is more subtle, since we can superpose two states and obtain a state that points up or down along the z-axis with only a certain likelihood. For such a FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621

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quantum mechanical state, obtained by superposing a quantum spin pointing up with a one pointing down, we have its wavefunction

given

(43)

with the following physical interpretation (44) (45)

The fact that a quantum particle can be in two states simultaneously is highly counter-intuitive and paradoxical. Since there is nothing special about spin, one can replace spin up and spin down by any two independent states. To illustrate the paradox, Schr

dinger proposed the

following experiment. Suppose a cat is inside a sealed and opaque box, with a radioactive substance inside the box as well. The radioactive material randomly emits alpha particles, and if it emits a strong burst of alpha particles, it will trigger a container to release a poisonous gas, causing the cat to die. The question Schr

dinger asked is the following:

As long as we do not open the box (technically speaking: perform a measurement), we do not know what has transpired, and there is some likelihood that the cat is either dead or alive. Hence the cat's wavefunction will be

(46)

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This famous cat, called Schr

dinger's cat, illustrates the counter-intuitive

and bizarre world of quantum mechanics, that the cat can be alive and dead at the time! Schr

dinger felt this was an absurd situation, since -

regardless of whether a measurement is performed - the cat should either be dead or alive, since how can the cat be dead and alive at the same time? The paradox that Schr

dinger's cat brings out is the need to

understand what is the physical meaning of the entity that we obtain by superposing two, or, for that matter, many states. To understand the superposition principle, we study the famous two-slit experiment.

2.9

Quantum Superposition Principle The most counter-intuitive aspect of quantum mechanics is the essential role that measurement plays in determining the behaviour of physical reality. We already have encountered something strange and bizarre in Schr

dinger's cat, namely, how can a system simultaneously be in two

orthogonal

states?

fully

appreciate

the

counter-intuitive

and

paradoxical nature of quantum mechanics we study the two-slit experiment in some detail. The heart of quantum mechanics is tied down to the wave-particle duality of elementary particles. An elementary particle can be localized (captured) as if it were a point like particle; on the other hand it can exist everywhere just like a wave field which has an extended structure. In this section we shall illustrate the wave-attribute and particleattribute of an elementary particle. In the macro-world, the concept of a particle is easy to comprehend. One starts with a piece of matter and keep on breaking it until one reaches the smallest constituent of matter. This smallest constituent is a particle. For example, the powder of a chalk can approximately be regarded as particles. In this way a particle is a point-like object. In geometry, a point has no size. However in physics one needs to measure 'size', that is, one needs a very powerful microscope in order to

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determine microscope size. The microscope in this scale is just the high energy accelerator and detector and at present energy the smallest size that we can measure is up to the order of

m. A wave is also easy to

visualize in the macro-world, it is the motion of a disturbance and a simple wave is an extended entity with a periodic structure. Water wave, sound wave are just energy propagating in continuous media: water and air. Originally it was thought that light wave also needs the presence of a medium, the ether. This ether is ruled out by the constancy of the speed of light. In micro-world we cannot directly 'see' the particles of waves, so how do we extend the concept of particle and wave from macro-world to the micro-world? This is done by examining the behaviour of a physical system under interference experiments. In macro-world, bullets can be taken as particles. In the experiment as illustrated in Figure14, a bullet from the firing gun can only go through either slit 1 or slit 2 and it is detected by the movable detector at the backstop. The experiment is first done by covering the slit 2 so that the bullet can go through only slit 1. After firing for a suitable time interval detected is plotted along the

, the distribution of the bullets

-direction and the distribution curve

is obtained. The experiment is repeated with the same time interval with the slit 1 being covered instead of slit 2. The result is the distribution curve

When both slits 1 and 2 are open, the combined distribution curve is obtained, and it is found that is in fact a sum if and

. In this way we identify particle-like behavior for a physical

system in the microwrold by the distribution curves

and

. There is no interference in the sense that the probability

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that a bullet will reach a point

is by taking a path eitherthrough slit 1 or

through slit 2, and this explains the final result being a sum of

and

. A similar experiment is performed for water wave as shown in Figure 15. The detector can only measure the intensity which is proportional to the square of the height

of the wave,

of the wave. Let

and

, respectively, be the amplitudes (heights) of the waves arriving at the detector when slit 2 and slit 1 are closed; we then have

and

are the intensity distributions of the water wave when slit 2 and slit 1 are closed respectively. When slits 1 and 2 are open, the resultant intensity is, from the superposition of waves, given by

(47) (48)

Recall the reason that the intensity of the interference pattern, namely , is not the sum of the individual amplitude is due to constructive

and

destructive

interference,

and

which

gives

the

characteristic minima and maxima of interference. Interference, as originally used by Young for light, is the best indication for a phenomenon being wave-like.

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Figure 15: Waves

To ascertain whether an elementary particle such as an electron behaves like a wave or particle, we carry out the interference experiment similar to the one we have considered for water waves. What we need to state at the outset that the interference patterns

and

can both be

obtained for the electron depending on how we perform the experiment. The experimental arrangement consists of an electron gun which sends identical electrons through a screen which has two slits to a wall where an apparatus keeps track of the point at which the electron stops. The electron gun produces the electrons one by one, so that at any given time there is only one electron traveling from the electron gun to the wall. We consider two different experiments with this arrangement, namely, one experiment in which a measurement is carried out to determine which slit the electron went through, and a second experiment in which no measurement is made to determine which slit the electron goes through. In both cases a large number, say

, electrons are sent in, one by one,

and the distribution of the positions at which the electrons hit the wall is measured. Experiment with Detection

We perform the experiment as given in Figure 16 with both slits 1 and 2, open and with the additional requirement that we determine which slit FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621

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the electron actually passes through. This can be arranged by fixing two detectors, say a light source, at the back of the slits as shown in Figure 16. Since we know which slit the electron goes through we can plot three distribution curves.

and

are the distribution curves for electrons go

through slit 1 and slit 2 respectively. Similar to the result obtained for bullets, the probability of the electron arriving at a point on the wall when both slits are open, denoted by

is given by

(49)

is the distribution curve for electrons that passes through either slit 1 or 2. We consequently have the result that when the electron's path is measured, it has a particle-like behavior. Experiment without Detection Consider now the same experiment as before, but with the detectors removed, as shown in Figure 17. In other words, we do not make any measurement to determine which slit the electron goes through. The interference pattern

is exactly like

as obtained for water waves.

This suggests that electrons behave like waves and we have to introduce a probability amplitude amplitude

for electrons when slit 2 is closed and an

for electrons when slit 1 is closed. We then have in analogy

with waves

(50) (51)

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When both slits 1 and 2 are open, and no measurement is made, the resultant distribution

is the square of modulus of the sum of

and

. The probability amplitudes obey the superposition principle when the different paths are not known, and yield

(52)

It is the superposed amplitude

that determines the outcome when no

measurement is performed. Hence

(53) (54)

The superposition principle is the unique feature of quantum mechanics, and shows graphically that, under some circumstances, particles behave as probability waves.

Note from Figure17 that the points of minima, say

, of the interference

pattern indicate that no electrons will be detected at those points. This is a remarkable, since if say only one slit was open there is a finite likelihood of an electron arriving at

, but with both slits open, unlike the case for bullets, no

electron can arrive there. This result is counter-intuitive since one would expect, as in the case of bullets, that for both slits open the electron would have two FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621

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ways of arriving at point

. In sum, when we do not observe which path the

electron takes, it behaves like a wave. An actual interference experiment for atoms instead of electrons is given in Figure 18, and leads to the conclusion that the electron distribution curve can either be wave-like or particle-like depending on whether we require the information as to which slit the electron passes through.

Knowing which slit the electron passes through, the electron exhibits particle-like behaviour and result is obtained; not knowing which slit the electron goes through, the electron exhibits wave-like behaviour and result follows. This is the famous wave-particle duality of quantum mechanics.

2.10 Heisenberg's Uncertainty Principle Planck's quantum postulate has radical and counter-intuitive implications for what can be experimentally measured. This aspect was elucidated only in 1927 by Werner Heinsenburg, another German physicist, and goes by the name of Heisenberg's Uncertainty Principle. Suppose one wants to measure the position of a particle; one can shine light on it and locate it by observing the light that is reflected by the particle. Suppose one wants to know the position of the particle to a very high degree of precision; then, since light of a given wavelength

cannot resolve distances less than

we will have to shine light on the particle with smaller and smaller wavelength to determine more and more precisely what is its position. And this is where we run into the quantum postulate: the minimum amount of light that we can shine on the particle has to have at least one quanta of

energy, which is

; as we make

smaller and smaller, the energy of

the minimum quanta becomes larger and larger.

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Hence to make a very precise measurement of the position of the particle, we are forced by the quantum principle to impart a high amount of energy to the particle and results in increasing the kinetic energy to the particle. This kinetic energy imparted to the particle changes the velocity of the particle in an uncontrollable and irreversible manner, and we end up with a final velocity of the particle which is different from the value it had before we made the measurement.

What is the uncertainty that results from the fact that light only comes in quanta with a minimum energy? For the case shown in Figures 19 and 20, the uncertainty in position of the particle whose position we are determining approximately equal to the wavelength

of light that we are

shining on it, since any distance much smaller than

cannot be resolved.

Hence we have

(55)

In the process of measurement, we have to scatter off the particle, a

photon which has at least to one quantum of light with wavelength

amount of energy, corresponding . If we knew for certain that the

particle would absorb one quantum of light, we could always account for it and there would be no uncertainty. However, and here is where the random and unpredictable aspect of quantum measurement comes in, during the process of measurement, the particle has a finite probability of

absorbing any amount of energy from

. One might

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object that even in a classical measurement, a certain amount of energy has to be imparted to the particle being observed. Although this is true, in classical physics, in principle, the energy imparted in the process of measurement can be made arbitrarily small, whereas in quantum physics, a precision of say

will necessarily involve a minimum energy of

to carry out the measurement. This in essence is the dividing line between classical and quantum measurement theory. Consider measuring the position of a quantum particle which is moving with an initial velocity

; after the measurement process it will have a

final velocity given by

. The energy of a free quantum particle is not

quantized (it is not in a bound state), the particle can absorb any amount of energy upto a maximum of energy less than

. The way the particle absorbs

is to first absorb the photon of wavelength

and then with a finite probability spontaneously re-emit another photon of wavelength

, and with energy

; clearly we must have

.In effect, the particle absorbs energy equal to

. Figure 21

symbolically shows a particle absorbing and re-emiting a photons in the process of its position being measured. If one repeats the experiment with initial velocity

, the particle will have

a final velocity which will not have a fixed value, but rather will vary over a range of velocities, denoted by

The variation in the final velocity is

due to the varying amounts of energy that the particle absorbs in the process of measurement. The fluctuation in the energy effectively absorbed by the particle is the inherent randomness in the process of a FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621

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quantum measurement, and which induces a quantum transition from a virtual to a physical state. Hence, no matter what was energy of the particle before the measurement, after the measurement, it has an uncertainty in its energy due to having absorbed energy anywhere between energy

to energy

. Hence, the uncertainty in the

particle's energy after the measurement process is

and is given by (56)

(57)

Recall the energy of a free particle after the measurement is given by

, and hence the uncertainty in the energy of the particle translates into uncertainty in the particles final momentum

. Since all

the quantities from now on refer to only the particle that is being observed, we drop the subscript of particle. We hence have

From special relativity we always have that

; hence

To recapitulate, we started by trying to precisely measure the position of the particle with no desire to disturb its velocity. But we discovered that, due to the quantum principle, the more precisely we measured the position of the particle the more we uncontrollably disturbed the velocity of the particle. Hence we ended up with a precise measurement of the

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position of the particle, and due to this very measurement we lost information on the precise value of the particle's velocity. Heisenberg postulated that a measurement made by any means (not necessarily by using light) will, due to the quantum postulate, introduce uncontrollable disturbances in the object being observed. If say the position of the particle is measured to only to a precision

, then the momentum

can be known only to a precision of satisfy

the

celebrated

Heisenberg

Uncertainty

which relation.

Heisenberg's Uncertainty Principle states that any measurement made will satisfy the uncertainty relation, and be of only a limited precision; the classical concept of having an arbitrarily precise knowledge of both

and

does not hold in the micro-world. Heisenberg's uncertainty principle has stunning implications. If we fully know the position of a particle, that is , then eq. implies that

, and visa versa! In other words,

position and momentum are mutually exclusive, in that complete knowledge of one necessarily means giving up all knowledge of the other. But this is not the end; position and momentum, even though being mutually exclusive, are nevertheless related by eq. in that one can have partial knowledge of both. What happens when we don't make any measurement, for example to determine the position or the momentum of the particle? Does it have a definite position or a definite momentum? The answer is no; the particle is in a probabilistic state in which both its position and momentum have a likelihood of having a whole range of values. The probabilistic state of a quantum particle implies the counterintuitive result that the outcome of an observation depends on what we decide to measure! For example if we decide to make a very precise measurement of the position of the particle we will end up with a large uncertainty in its momentum, whereas if we decide to make a very precise

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measurement of the momentum of that same particle we will end up with a large uncertainty in its position! To get a concrete idea of the Uncertainty Principle, consider a hydrogen atom in which an electron is in a bound state with a proton due to their mutual electrical attraction. Since there is an attractive force one may ask why doesn't the electron fall into the nucleus (proton) and by doing this minimize the potential energy of the atom (and which in fact is the incorrect prediction of classical physics)? The reason is the Uncertainty Principle. If the electron were to fall into the nucleus, its position would be determined fairly precisely and this would mean that

would become very small; the uncertainty principle would

then imply that

is very large, and this would give the electron a very

large amount of kinetic energy resulting in the electron flying far away from the proton and in effect breaking-up the atom. The atom reaches a compromise by letting the electron move around in a finite volume whose size is fixed so as to minimize the kinetic energy due to

and while at

the same time lowering as much as possible the electron's potential energy. It is this trade-off between the uncertainty of the position and momentum of the electrons inside an atom which is responsible for the finite size of atoms. The actual size of the atom of course depends on the charge

and mass

quantum effects due to to be about

of the electron, together with the extent that are operational. This yields the size of an atom

m, and is the actual size of a typical atom.

2.10.1 Path Integral Quantum Mechanics Recall in classical physics to determine the future evolution of a particle using Newton's second law, we need to specify its exact position as well as its exact velocity at the start of the particle's motion. However, we learnt from Heisenberg's uncertainty relation that we cannot, even in principle, determine simultaneously the exact position and velocity of a particle. The FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621

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best we can do is to specify the initial coordinate

of the particle, and the

future behaviour of a quantum particle is then given by the Schr

dinger

equation. We explore the physical implications of the fact that we no longer know the initial velocity of the particle.

To start with, how are we to describe a quantum particle? Recall in the two-slit experiment we saw that if a quantum particle is not observed as going through a particular slit, then, in the sense of probability, a single particle can be thought of as going through both slits simultaneously. Paths taken by the quantum particle in the sense of probability (unobserved) are called virtual paths to distinguish them from experimentally observed paths called physical paths. Now consider making more and more slits with smaller and smaller widths until we have infinitely many slits with zero width as in Figure 22 - in other words no slits at all! We now see that the quantum particle, as it evolves from its observed initial position to its final position simultaneously takes all possible virtual paths from its starting to its finishing point. This is the tremendous

generalization

quantum

mechanics

over

classical

mechanics: in the latter the classical particle takes only one definite physical path in evolving from its initial to its final position, whereas in quantum mechanics the 'particle' propagates probabilistically and takes all possible virtual paths from it's observed initial to its final position. The probability for the quantum particle to take a particular path is given essentially by the potential that is acting on the particle as well as its kinetic energy. Path Integral quantum mechanics was formulated by Feynman in 1949. Path integral quantum mechanics starts directly from the virtual paths that a quantum particle takes, and derives the results of Schr

dinger and Heisenberg. Fundamental to path integral quantum

mechanics is the probability amplitude for a particle to go from initial position and time

to the final position and time

. To evaluate

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the probability amplitude we need to define the concept of the Lagrangian. Recall energy is given by

(64)

2.11 Summary The modern formulation of quantum theory rests primarily on the ideas of Erwin Schr

dinger, Werner Heisenberg and P.A.M. Dirac. In the period

2.12 Keywords Heisenberg's Uncertainty Principle : Suppose one wants to measure the position of a particle; one can shine light on it and locate it by observing the light that is reflected by the particle. Suppose one wants to know the position of the particle to a very high degree of precision; then, since light of a given wavelength

cannot resolve distances less than

we will have to shine light on the particle with smaller and smaller wavelength to determine more and more precisely what is its position.

2.13 Self Assessment Questions 1. Describe Evidence for Quantum Theory . 2. Discuss Planck's Quantum Hypothesis.

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3. Elaborate Bohr atom;Electron Wave . 4. Give Basic Postulates of Quantum Theory.. 5. Define Wave Function. 6. Explain Quantum Particle in a Box.

2.14 References Hibbeler, R.C., Engineering Mechanics: Dynamics, 7th edition. PrenticeHall, Englewood Cliffs, N.J.,1995. Reif, F., Fundamentals of Statistical and Thermal Physics. McGraw-Hill Inc., 1965. Kittel, C. and Kroemar, H., Thermal Physics, 2nd edition. W.H. Freeman, 1980. Demarest, Engineering Electromagnetics. Prentice-Hall. Staelin, D.H., Morgenthaler, A.W. and Kong, J.A., Electromagnetic Waves. Prentice Hall, 1994. Kroemer, H., Quantum Mechanics for Engineering, Materials Science & Applied Physics. Prentice Hall, NJ, 1994.

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Unit 3 Crystal Structure Structure 3.0

Objectives

3.1

Introduction

3.2

Miller indices

3.3

Planes and directions

3.4

Cubic structures

3.5

Classification 3.5.1 Atomic coordination 3.5.2 Close packing 3.5.3 Bravais lattices 3.5.4 Point groups 3.5.5 Space groups

3.6

Grain boundaries

3.7

Defects and impurities

3.7

Prediction of structure

3.9

Polymorphism

3.10

Physical properties

3.11

Classification of Materials

3.12

Types of Crystals

3.13

X-ray crystallography

3.14

X-ray analysis of crystals

3.15

Summary

3.16

Keywords

3.17

Self Assessment Questions

3.18

References

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3.0

Objectives After studying this unit you will be able to:

3.1



Explain Miller indices



Define Cubic structures



Discuss Atomic coordination



Define Bravais lattices



Understand Point groups



Elaborate Space groups



Describe Grain boundaries

Introduction Let us understand that in mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry. Patterns are located upon the points of a lattice, which is an array of points repeating periodically in three dimensions. The points can be thought of as forming identical tiny boxes, called unit cells, that fill the space of the lattice. The lengths of the edges of a unit cell and the angles between them are called the lattice parameters. The symmetry properties of the crystal are embodied in its space group.

A crystal's structure and symmetry play a role in determining many of its physical properties, such as cleavage, electronic band structure, and optical transparency.

The crystal structure of a material or the arrangement of atoms within a given type of crystal structure can be described in terms of its unit cell. The unit cell is a tiny box containing one or more atoms, a spatial arrangement of atoms. The unit cells stacked in three-dimensional space describe the bulk arrangement of atoms of the crystal. The crystal FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621

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structure has a three dimensional shape. The unit cell is given by its lattice parameters, the length of the cell edges and the angles between them, while the positions of the atoms inside the unit cell are described by the set of atomic positions (xi , yi , zi) measured from a lattice point.

Simple cubic (P)

3.2

Body-centered cubic (I) Face-centered cubic (F)

Miller indices Planes with different Miller indices in cubic crystals

Vectors and atomic planes in a crystal lattice can be described using a three-value Miller index notation (ℓmn). The ℓ, m and n directional indices are separated by 90, and are thus orthogonal. In fact, the ℓ component is mutually perpendicular to the m and n indices. By definition, (ℓmn) denotes a plane that intercepts the three points a1/ℓ, a2/m, and a3/n, or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell (in the basis of the lattice vectors). If one or more of the indices is zero, it simply means that the planes do not intersect that axis (i.e. the intercept is "at infinity").

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Considering only (â&#x201E;&#x201C;mn) planes intersecting one or more lattice points (the lattice planes), the perpendicular distance d between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula:

3.3

Planes and directions The crystallographic directions are fictitious lines linking nodes (atoms, ions or molecules) of a crystal. Similarly, the crystallographic planes are fictitious planes linking nodes. Some directions and planes have a higher density of nodes. These high density planes have an influence on the behavior of the crystal as follows:

Optical properties: Refractive index is directly related to density (or periodic density fluctuations).

Adsorption and reactivity: Physical adsorption and chemical reactions occur at or near surface atoms or molecules. These phenomena are thus sensitive to the density of nodes.

Surface tension: The condensation of a material means that the atoms, ions or molecules are more stable if they are surrounded by other similar species. The surface tension of an interface thus varies according to the density on the surface.

Dense crystallographic planes Microstructural defects: Pores and crystallites tend to have straight grain boundaries following higher density planes.

Cleavage: This typically occurs preferentially parallel to higher density planes.

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Plastic deformation: Dislocation glide occurs preferentially parallel to higher density planes. The perturbation carried by the dislocation (Burgers vector) is along a dense direction. The shift of one node in a more dense direction requires a lesser distortion of the crystal lattice.

In the rhombohedral, hexagonal, and tetragonal systems, the basal plane is the plane perpendicular to the principal axis.

3.4

Cubic structures For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted a); similarly for the reciprocal lattice. So, in this common case, the Miller indices (ℓmn) and [ℓmn] both simply denote normals/directions in Cartesian coordinates. For cubic crystals with lattice constant a, the spacing d between adjacent (ℓmn) lattice planes is (from above):

Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes:

Coordinates in angle brackets such as <100> denote a family of directions which are equivalent due to symmetry operations, such as [100], [010], [001] or the negative of any of those directions.

Coordinates in curly brackets or braces such as {100} denote a family of plane normals which are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.

For face-centered cubic (fcc) and body-centered cubic (bcc) lattices, the primitive lattice vectors are not orthogonal. However, in these cases the

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Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions.

3.5

Classification The defining property of a crystal is its inherent symmetry, by which we mean that under certain 'operations' the crystal remains unchanged. For example, rotating the crystal 180 about a certain axis may result in an atomic configuration which is identical to the original configuration. The crystal is then said to have a twofold rotational symmetry about this axis. In addition to rotational symmetries like this, a crystal may have symmetries in the form of mirror planes and translational symmetries, and also the so-called "compound symmetries" which are a combination of translation and rotation/mirror symmetries. A full classification of a crystal is achieved when all of these inherent symmetries of the crystal are identified.[1]

3.5.1 Atomic coordination By considering the arrangement of atoms relative to each other, their coordination numbers (or number of nearest neighbors), interatomic distances, types of bonding, etc., it is possible to form a general view of the structures and alternative ways of visualizing them.

HCP lattice (left) and the fcc lattice (right).

3.5.2 Close packing The principles involved can be understood by considering the most efficient way of packing together equal-sized spheres and stacking closepacked atomic planes in three dimensions. For example, if plane A lies beneath plane B, there are two possible ways of placing an additional

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atom on top of layer B. If an additional layer was placed directly over plane A, this would give rise to the following series :

...ABABABAB....

This type of crystal structure is known as hexagonal close packing (hcp). If however, all three planes are staggered relative to each other and it is not until the fourth layer is positioned directly over plane A that the sequence is repeated, then the following sequence arises:

...ABCABCABC...

This type of crystal structure is known as cubic close packing (ccp)

The unit cell of the ccp arrangement is the face-centered cubic (fcc) unit cell. This is not immediately obvious as the closely packed layers are parallel to the {111} planes of the fcc unit cell. There are four different orientations of the close-packed layers.

The packing efficiency could be worked out by calculating the total volume of the spheres and dividing that by the volume of the cell as follows:

The 74% packing efficiency is the maximum density possible in unit cells constructed of spheres of only one size. Most crystalline forms of metallic elements are hcp, ccp or bcc (body-centered cubic). The coordination number of hcp and fcc is 12 and its atomic packing factor (APF) is the number mentioned above, 0.74.

3.5.3 Bravais lattices When the crystal systems are combined with the various possible lattice centerings, we arrive at the Bravais lattices. They describe the geometric

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arrangement of the lattice points, and thereby the translational symmetry of the crystal. In three dimensions, there are 14 unique Bravais lattices which are distinct from one another in the translational symmetry they contain. All crystalline materials recognized until now (not including quasicrystals) fit in one of these arrangements. The fourteen threedimensional lattices, classified by crystal system, are shown above. The Bravais lattices are sometimes referred to as space lattices.

The crystal structure consists of the same group of atoms, the basis, positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the 14 Bravais lattices. The characteristic rotation and mirror symmetries of the group of atoms, or unit cell, is described by its crystallographic point group.

3.5.4 Point groups The crystallographic point group or crystal class is the mathematical group comprising the symmetry operations that leave at least one point unmoved and that leave the appearance of the crystal structure unchanged. These symmetry operations include 

Reflection, which reflects the structure across a reflection plane



Rotation, which rotates the structure a specified portion of a circle about a rotation axis



Inversion, which changes the sign of the coordinate of each point with respect to a center of symmetry or inversion point



Improper rotation, which consists of a rotation about an axis followed by an inversion.

Rotation axes (proper and improper), reflection planes, and centers of symmetry are collectively called symmetry elements. There are 32

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possible crystal classes. Each one can be classified into one of the seven crystal systems.

3.5.5 Space groups The space group of the crystal structure is composed of the translational symmetry operations in addition to the operations of the point group. These include: 

Pure translations which move a point along a vector



Screw axes, which rotate a point around an axis while translating parallel to the axis



Glide planes, which reflect a point through a plane while translating it parallel to the plane.

3.6

Grain boundaries SEM micrograph of surface of a colloidal crystal. Structure and morphology consists of ordered crystallites, grains or domains of particles as well as interdomain lattice defects in the form of grain boundaries.

Highlighted image of surface of a colloidal crystal. Emphasis on microstructural defects to illustrate the defect/domain morphology typical of an elemental crystal.

Grain boundaries are interfaces where crystals of different orientations meet. A grain boundary is a single-phase interface, with crystals on each side of the boundary being identical except in orientation. The term "crystallite boundary" is sometimes, though rarely, used. Grain boundary areas contain those atoms that have been perturbed from their original lattice sites, dislocations, and impurities that have migrated to the lower energy grain boundary.

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Treating a grain boundary geometrically as an interface of a single crystal cut into two parts, one of which is rotated, we see that there are five variables required to define a grain boundary. The first two numbers come from the unit vector that specifies a rotation axis. The third number designates the angle of rotation of the grain. The final two numbers specify the plane of the grain boundary (or a unit vector that is normal to this plane).

Grain boundaries disrupt the motion of dislocations through a material, so reducing crystallite size is a common way to improve strength, as described by the Hall-Petch relationship. Since grain boundaries are defects in the crystal structure they tend to decrease the electrical and thermal conductivity of the material. The high interfacial energy and relatively weak bonding in most grain boundaries often makes them preferred sites for the onset of corrosion and for the precipitation of new phases from the solid. They are also important to many of the mechanisms of creep.

Grain boundaries are generally only a few nanometers wide. In common materials, crystallites are large enough that grain boundaries account for a small fraction of the material. However, very small grain sizes are achievable. In nanocrystalline solids, grain boundaries become a significant volume fraction of the material, with profound effects on such properties as diffusion and plasticity. In the limit of small crystallites, as the volume fraction of grain boundaries approaches 100%, the material ceases to have any crystalline character, and thus becomes an amorphous solid.

3.7

Defects and impurities Real crystals feature defects or irregularities in the ideal arrangements described above and it is these defects that critically determine many of

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the electrical and mechanical properties of real materials. When one atom substitutes for one of the principal atomic components within the crystal structure, alteration in the electrical and thermal properties of the material may ensue.[2] Impurities may also manifest as spin impurities in certain materials. Research on magnetic impurities demonstrates that substantial alteration of certain properties such as specific heat may be affected by small concentrations of an impurity, as for example impurities in semiconducting ferromagnetic alloys may lead to different properties as first predicted in the late 1960s.[3][4] Dislocations in the crystal lattice allow shear at lower stress than that needed for a perfect crystal structure.[5]

3.8

Prediction of structure The difficulty of predicting stable crystal structures based on the knowledge of only the chemical composition has long been a stumbling block on the way to fully computational materials design. Now, with more powerful algorithms and high-performance computing, structures of medium complexity can be predicted using such approaches as evolutionary algorithms, random sampling, or metadynamics.

The crystal structures of simple ionic solids (e.g. NaCl or table salt) have long been rationalized in terms of Pauling's rules, first set out in 1929 by Linus Pauling, referred to by many since as the "father of the chemical bond".[6] Pauling also considered the nature of the interatomic forces in metals, and concluded that about half of the five d-orbitals in the transition metals are involved in bonding, with the remaining nonbonding d-orbitals being responsible for the magnetic properties. He therefore was able to correlate the number of d-orbitals in bond formation with the bond length as well as many of the physical properties of the substance. He subsequently introduced the metallic orbital, an extra orbital necessary to

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permit uninhibited resonance of valence bonds among various electronic structures.[7]

In the resonating valence bond theory, the factors that determine the choice of one from among alternative crystal structures of a metal or intermetallic compound revolve around the energy of resonance of bonds among interatomic positions. It is clear that some modes of resonance would make larger contributions (be more mechanically stable than others), and that in particular a simple ratio of number of bonds to number of positions would be exceptional. The resulting principle is that a special stability is associated with the simplest ratios or "bond numbers": 1/2, 1/3, 2/3, 1/4, 3/4, etc. The choice of structure and the value of the axial ratio (which determines the relative bond lengths) are thus a result of the effort of an atom to use its valency in the formation of stable bonds with simple fractional bond numbers.[8][9]

After postulating a direct correlation between electron concentration and crystal structure in beta-phase alloys, Hume-Rothery analyzed the trends in melting points, compressibilities and bond lengths as a function of group number in the periodic table in order to establish a system of valencies of the transition elements in the metallic state. This treatment thus emphasized the increasing bond strength as a function of group number.[10] The operation of directional forces were emphasized in one article on the relation between bond hybrids and the metallic structures. The resulting correlation between electronic and crystalline structures is summarized by a single parameter, the weight of the d-electrons per hybridized metallic orbital. The d-weight calculates out to 0.5, 0.7 and 0.9 for the fcc, hcp and bcc structures respectively. The relationship between d-electrons and crystal structure thus becomes apparent.[11]

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3.9

Polymorphism Quartz is one of the several thermodynamically stable crystalline forms of silica, SiO2. The most important forms of silica include: Îą-quartz, Î˛-quartz, tridymite, cristobalite, coesite, and stishovite.

Polymorphism refers to the ability of a solid to exist in more than one crystalline form or structure. According to Gibbs' rules of phase equilibria, these unique crystalline phases will be dependent on intensive variables such as pressure and temperature. Polymorphism can potentially be found in many crystalline materials including polymers, minerals, and metals, and is related to allotropy, which refers to elemental solids. The complete morphology of a material is described by polymorphism and other variables such as crystal habit, amorphous fraction or crystallographic defects. Polymorphs have different stabilities and may spontaneously convert from a metastable form (or thermodynamically unstable form) to the stable form at a particular temperature. They also exhibit different melting points, solubilities, and X-ray diffraction patterns.

One good example of this is the quartz form of silicon dioxide, or SiO2. In the vast majority of silicates, the Si atom shows tetrahedral coordination by 4 oxygens. All but one of the crystalline forms involve tetrahedral SiO4 units linked together by shared vertices in different arrangements. In different minerals the tetrahedra show different degrees of networking and polymerization. For example, they occur singly, joined together in pairs, in larger finite clusters including rings, in chains, double chains, sheets, and three-dimensional frameworks. The minerals are classified into groups based on these structures. In each of its 7 thermodynamically stable crystalline forms or polymorphs of crystalline quartz, only 2 out of 4 of each the edges of the SiO4 tetrahedra are shared with others, yielding the net chemical formula for silica: SiO2.

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Another example is elemental tin (Sn), which is malleable near ambient temperatures but is brittle when cooled. This change in mechanical properties due to existence of its two major allotropes, α- and β-tin. The two allotropes that are encountered at normal pressure and temperature, α-tin and β-tin, are more commonly known as gray tin and white tin respectively. Two more allotropes, γ and σ, exist at temperatures above 161 C and pressures above several GPa.[12] White tin is metallic, and is the stable crystalline form at or above room temperature. Below 13.2 C, tin exists in the gray form, which has a diamond cubic crystal structure, similar to diamond, silicon or germanium. Gray tin has no metallic properties at all, is a dull-gray powdery material, and has few uses, other than a few specialized semiconductor applications.[13] Although the α-β transformation temperature of tin is nominally 13.2 C, impurities (e.g. Al, Zn, etc.) lower the transition temperature well below 0 C, and upon addition of Sb or Bi the transformation may not occur at all.[14]

3.10 Physical properties Twenty of the 32 crystal classes are so-called piezoelectric, and crystals belonging to one of these classes (point groups) display piezoelectricity. All piezoelectric classes lack a center of symmetry. Any material develops a dielectric polarization when an electric field is applied, but a substance which has such a natural charge separation even in the absence of a field is called a polar material. Whether or not a material is polar is determined solely by its crystal structure. Only 10 of the 32 point groups are polar. All polar crystals are pyroelectric, so the 10 polar crystal classes are sometimes referred to as the pyroelectric classes.

There are a few crystal structures, notably the perovskite structure, which exhibit ferroelectric behavior. This is analogous to ferromagnetism, in that, in the absence of an electric field during production, the ferroelectric

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crystal does not exhibit a polarization. Upon the application of an electric field of sufficient magnitude, the crystal becomes permanently polarized. This polarization can be reversed by a sufficiently large counter-charge, in the same way that a ferromagnet can be reversed. However, it is important to note that, although they are called ferroelectrics, the effect is due to the crystal structure (not the presence of a ferrous metal).

3.11 Classification of Materials Different materials have difference electrical characteristics and this leads to their classification in any of the three categories which are as follows.

Insulators are materials which do not conduct electricity. But how do we explain this in terms of the theory we have already learned in the lessons mentioned above. Well stated along those lines, an insulator simply means a material which does not have any free electrons available for carriage of current. This is because the valence band electrons of these materials are bound very tightly to their parent nuclei and require vast amounts of energy to break that bond. In other words these materials have a large forbidden energy gap. Of course this effect could be overcome to some degree when energy is supplied to these tightly bound electrons in the form of say heat. This means that at higher temperatures there might be a slight conduction available in them.

Conductors are materials which lie on the other end of the spectrum and the forbidden energy gap is literally zero. This in effect means that the valence and conduction bands of these atoms are literally overlapping with each other so there is hardly any restriction of electrons to jump to and from these bands.

Semiconductors are midway between insulators and conductors. In these materials the energy gap is neither very large like insulators nor negligible like conductors. There is a moderate energy gap which does not require

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much energy for the electrons to jump from valence to conduction band. The conduction band of these materials is nearly empty if not fully empty. At a temperature of zero degrees Kelvin these materials act as insulators since there are no electrons in the conduction band but at room temperature the forbidden energy gap is reduced to a minimal. Infact the typical values of the energy gap in a semiconductor is of the order of 1 eV.

Another thing that separates these materials from conductors is that while the current in a conductor is solely due to the flow of electrons; the current in a semiconductor is not only due to flow of electrons but also due to flow of holes about which we learnt previously. The electrons flow across the conduction band whilst the holes flow via the valence band only.

The typical properties of the above mentioned category of materials are due to their unique structure which is known as the lattice structure and basically it consists of a repeated arrangement of atoms in a particular geometric pattern. There are several different types of crystal structure such as the Simple Cubic structure, Base Centered Cubic structure, Face Centered Cubic structure and Diamond structure.

In SC structure the lattice is in the form of a cube having an atom at each of its vertices. The only material which exists in SC form is Podium.

In BCC the atoms are at the vertices of the cube plus one atom at the center of cube, hence each atom has 8 neighbours. E.g. Sodium In FCC there are atoms at the corners of the cube as well as the centers of each of the faces thus giving a figure of 12 atoms in the neighbourhood of each atom.

Diamond lattice - the structure of semiconductors is slightly different and the semiconductor materials such as Silicon and Germanium have a

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structure known as the diamond lattice wherein there are four equidistant neighbours of each atom in the form of a tetrahedron.

We will learn more about the types of semiconductor materials and the role of the crystal structure in determining properties such as atomic binding in separate articles.

3.12

Types of Crystals Shapes and Structures

Copper sulfate has a triclinic crystal structure.

There are seven crystal lattice systems. You can view examples of each type by following one of the 'Elsewhere on the Web' links I have provided.

Cubic or Isometric - not always cube shaped! You'll also find octahedrons (eight faces) and dodecahedrons (10 faces).

Tetragonal - similar to cubic crystals, but longer along one axis than the other, forming double pyramids and prisms.

Orthorhombic - like tetragonal crystals except not square in cross section (when viewing the crystal on end), forming rhombic prisms or dipyramids (two pyramids stuck together).

Hexagonal - six-sided prisms. When you look at the crystal on-end, the cross section is a hexagon.

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Trigonal - possess a single 3-fold axis of rotation instead of the 6-fold axis of the hexagonal division.

Triclinic - usually not symmetrical from one side to the other, which can lead to some fairly strange shapes.

Monoclinic - like skewed tetragonal crystals, often forming prisms and double pyramids.

This is a very simplified view of crystal structures. In addition, the lattices can be primitive (only one lattice point per unit cell) or non-primitive (more than one lattice point per unit cell). Combining the 7 crystal systems with the 2 lattice types yields the 14 Bravais Lattices (named after Auguste Bravais, who worked out lattice structures in 1850). The structure of real crystals is pretty complicated! You can read about crystallography and mineral structures here and here.

Crystals Grouped by Properties There are four main categories of crystals, as grouped by their chemical and physical properties:

Covalent Crystals A covalent crystals has true covalent bonds between all of the atoms in the crystal. You can think of a covalent crystal as one big molecule. Many covalent crystals have extremely high melting points. Examples of covalent crystals include diamond and zinc sulfide crystals.

Metallic Crystals Individual metal atoms of metallic crystals sit on lattice sites. This leaves the outer electrons of these atoms free to float around the lattice. Metallic crystals tend to be very dense and have high melting points.

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Ionic Crystals The atoms of ionic crystals are held together by electrostatic forces (ionic bonds). Ionic crystals are hard and have relatively high melting points. Table salt (NaCl) is an example of this type of crystal.

Molecular Crystals These crystals contain recognizable molecules within their structures. A molecular crystal is held together by non-covalent interactions, like van der Waals forces or hydrogen bonding. Molecular crystals tend to be soft with relatively low melting points. Rock candy, the crystalline form of table sugar or sucrose, is an example of a molecular crystal.

As with the lattice classification system, this system isn't completely cutand-dried. Sometimes it's hard to categorize crystals as belonging to one class as opposed to another. However, these broad groupings will provide you with some understanding of structures. I'll test your knowledge by referring to these crystal shapes in crystal-growing tutorials!

3.13 X-ray crystallography

X-ray

crystallography

can

locate

every

atom

zeolite,

aluminosilicate with many important applications, such as water purification.

X-ray crystallography is a method of determining the arrangement of atoms within a crystal, in which a beam of X-rays strikes a crystal and diffracts into many specific directions. From the angles and intensities of these diffracted beams, a crystallographer can produce a threedimensional picture of the density of electrons within the crystal. From this

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electron density, the mean positions of the atoms in the crystal can be determined, as well as their chemical bonds, their disorder and various other information.

Since many materials can form crystals such as salts, metals, minerals, semiconductors, as well as various inorganic, organic and biological molecules

X-ray

crystallography

has

been

fundamental

the

development of many scientific fields. In its first decades of use, this method determined the size of atoms, the lengths and types of chemical bonds, and the atomic-scale differences among various materials, especially minerals and alloys. The method also revealed the structure and functioning of many biological molecules, including vitamins, drugs, proteins and nucleic acids such as DNA. X-ray crystallography is still the chief method for characterizing the atomic structure of new materials and in discerning materials that appear similar by other experiments. X-ray crystal structures can also account for unusual electronic or elastic properties of a material, shed light on chemical interactions and processes, or serve as the basis for designing pharmaceuticals against diseases.

In an X-ray diffraction measurement, a crystal is mounted on a goniometer and gradually rotated while being bombarded with X-rays, producing a diffraction pattern of regularly spaced spots known as reflections. The twodimensional images taken at different rotations are converted into a threedimensional model of the density of electrons within the crystal using the mathematical method of Fourier transforms, combined with chemical data known for the sample. Poor resolution (fuzziness) or even errors may result if the crystals are too small, or not uniform enough in their internal makeup.

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X-ray crystallography is related to several other methods for determining atomic structures. Similar diffraction patterns can be produced by scattering electrons or neutrons, which are likewise interpreted as a Fourier transform. If single crystals of sufficient size cannot be obtained, various other X-ray methods can be applied to obtain less detailed information; such methods include fiber diffraction, powder diffraction and small-angle X-ray scattering (SAXS). If the material under investigation is only available in the form of nanocrystalline powders or suffers from poor crystallinity, the methods of electron crystallography can be applied for determining the atomic structure.

For all above mentioned X-ray diffraction methods, the scattering is elastic; the scattered X-rays have the same wavelength as the incoming X-ray. By contrast, inelastic X-ray scattering methods are useful in studying excitations of the sample, rather than the distribution of its atoms

Early scientific history of crystals and X-rays

Drawing of square (Figure A, above) and hexagonal (Figure B, below) packing from Kepler's work, Strena seu de Nive Sexangula.

Crystals have long been admired for their regularity and symmetry, but they were not investigated scientifically until the 17th century.

As shown by X-ray crystallography, the hexagonal symmetry of snowflakes results from the tetrahedral arrangement of hydrogen bonds about each water molecule. The water molecules are arranged similarly to the silicon atoms in the tridymite polymorph of SiO2. The resulting crystal structure has hexagonal symmetry when viewed along a principal axis.

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Crystal symmetry was first investigated experimentally by Nicolas Steno (1669), who showed that the angles between the faces are the same in every exemplar of a particular type of crystal,[2] and by Ren Just Hay (1784), who discovered that every face of a crystal can be described by simple stacking patterns of blocks of the same shape and size. Hence, William Hallowes Miller in 1839 was able to give each face a unique label of three small integers, the Miller indices which are still used today for identifying crystal faces. Hay's study led to the correct idea that crystals are a regular three-dimensional array (a Bravais lattice) of atoms and molecules; a single unit cell is repeated indefinitely along three principal directions that are not necessarily perpendicular. In the 19th century, a complete catalog of the possible symmetries of a crystal was worked out by Johann Hessel,[3] Auguste Bravais,[4] Yevgraf Fyodorov,[5] Arthur Schnflies[6] and (belatedly) William Barlow. From the available data and physical reasoning, Barlow proposed several crystal structures in the 1880s that were validated later by X-ray crystallography;[7] however, the available data were too scarce in the 1880s to accept his models as conclusive.

X-ray crystallography shows the arrangement of water molecules in ice, revealing the hydrogen bonds that hold the solid together. Few other methods can determine the structure of matter with such sub-atomic precision (resolution).

X-rays were discovered by Wilhelm Conrad Rntgen in 1895, just as the studies of crystal symmetry were being concluded. Physicists were initially uncertain of the nature of X-rays, although it was soon suspected (correctly) that they were waves of electromagnetic radiation, in other words, another form of light. At that time, the wave model of light

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specifically, the Maxwell theory of electromagnetic radiation was well accepted among scientists, and experiments by Charles Glover Barkla showed that X-rays exhibited phenomena associated with electromagnetic waves, including transverse polarization and spectral lines akin to those observed in the visible wavelengths. Single-slit experiments in the laboratory of Arnold Sommerfeld suggested the wavelength of X-rays was about 1 Angstrm. However, X-rays are composed of photons, and thus are not only waves of electromagnetic radiation but also exhibit particle-like properties. The photon concept was introduced by Albert Einstein in 1905,[8] but it was not broadly accepted until 1922,[9][10] when Arthur Compton confirmed it by the scattering of X-rays from electrons.[11] Therefore, these particle-like properties of X-rays, such as their ionization of gases, caused William Henry Bragg to argue in 1907 that X-rays were not electromagnetic radiation.[12][13][14][15] Nevertheless, Bragg's view was not broadly accepted and the observation of X-ray diffraction in 1912[16] confirmed for most scientists that X-rays were a form of electromagnetic radiation.

3.14 X-ray analysis of crystals

The incoming beam (coming from upper left) causes each scatterer to reradiate a small portion of its intensity as a spherical wave. If scatterers are arranged symmetrically with a separation d, these spherical waves will be in synch (add constructively) only in directions where their path-length difference 2d sin θ equals an integer multiple of the wavelength λ. In that case, part of the incoming beam is deflected by an angle 2θ, producing a reflection spot in the diffraction pattern.

Crystals are regular arrays of atoms, and X-rays can be considered waves of electromagnetic radiation. Atoms scatter X-ray waves, primarily through

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the atoms' electrons. Just as an ocean wave striking a lighthouse produces secondary circular waves emanating from the lighthouse, so an X-ray striking an electron produces secondary spherical waves emanating from the electron. This phenomenon is known as elastic scattering, and the electron (or lighthouse) is known as the scatterer. A regular array of scatterers produces a regular array of spherical waves. Although these waves cancel one another out in most directions through destructive interference, they add constructively in a few specific directions, determined by Bragg's law:

Here d is the spacing between diffracting planes, 胃 is the incident angle, n is any integer, and 位 is the wavelength of the beam. These specific directions appear as spots on the diffraction pattern called reflections. Thus, X-ray diffraction results from an electromagnetic wave (the X-ray) impinging on a regular array of scatterers (the repeating arrangement of atoms within the crystal).

X-rays are used to produce the diffraction pattern because their wavelength 位 is typically the same order of magnitude (1-100 ngstrms) as the spacing d between planes in the crystal. In principle, any wave impinging on a regular array of scatterers produces diffraction, as predicted first by Francesco Maria Grimaldi in 1665. To produce significant diffraction, the spacing between the scatterers and the wavelength of the impinging wave should be similar in size. For illustration, the diffraction of sunlight through a bird's feather was first reported by James Gregory in the later 17th century. The first artificial diffraction gratings for visible light were constructed by David Rittenhouse in 1787, and Joseph von Fraunhofer in 1821. However, visible light has too long a wavelength (typically, 5500 ngstrms) to observe diffraction from crystals. Prior to the

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first X-ray diffraction experiments, the spacings between lattice planes in a crystal were not known with certainty.

The idea that crystals could be used as a diffraction grating for X-rays arose in 1912 in a conversation between Paul Peter Ewald and Max von Laue in the English Garden in Munich. Ewald had proposed a resonator model of crystals for his thesis, but this model could not be validated using visible light, since the wavelength was much larger than the spacing between the resonators. Von Laue realized that electromagnetic radiation of a shorter wavelength was needed to observe such small spacings, and suggested that X-rays might have a wavelength comparable to the unitcell spacing in crystals. Von Laue worked with two technicians, Walter Friedrich and his assistant Paul Knipping, to shine a beam of X-rays through a copper sulfate crystal and record its diffraction on a photographic plate. After being developed, the plate showed a large number of well-defined spots arranged in a pattern of intersecting circles around the spot produced by the central beam.[16][17] Von Laue developed a law that connects the scattering angles and the size and orientation of the unit-cell spacings in the crystal, for which he was awarded the Nobel Prize in Physics in 1914.[18]

As described in the mathematical derivation below, the X-ray scattering is determined by the density of electrons within the crystal. Since the energy of an X-ray is much greater than that of a valence electron, the scattering may be modeled as Thomson scattering, the interaction of an electromagnetic ray with a free electron. This model is generally adopted to describe the polarization of the scattered radiation. The intensity of Thomson scattering declines as 1/m with the mass m of the charged particle that is scattering the radiation; hence, the atomic nuclei, which are thousands of times heavier than an electron, contribute negligibly to the scattered X-rays.

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3.15 Summary The crystal structure of a material or the arrangement of atoms within a given type of crystal structure can be described in terms of its unit cell. The unit cell is a tiny box containing one or more atoms, a spatial arrangement of atoms. The unit cells stacked in three-dimensional space describe the bulk arrangement of atoms of the crystal. The crystal structure has a three dimensional shape.

3.16 Keywords Cleavage: This typically occurs preferentially parallel to higher density planes.

Plastic deformation: Dislocation glide occurs preferentially parallel to higher density planes. The perturbation carried by the dislocation (Burgers vector) is along a dense direction.

3.17 Self Assessment Questions 1. Explain Miller indices. 2. Define Cubic structures. 3. Discuss Atomic coordination. 4. Define Bravais lattices. 5. Understand Point groups. 6. Elaborate Space groups. 7. Describe Grain boundaries.

3.18 References Hibbeler, R.C., Engineering Mechanics: Dynamics, 7th edition. PrenticeHall, Englewood Cliffs, N.J.,1995. Reif, F., Fundamentals of Statistical and Thermal Physics. McGraw-Hill Inc., 1965.

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Kittel, C. and Kroemar, H., Thermal Physics, 2nd edition. W.H. Freeman, 1980. Demarest, Engineering Electromagnetics. Prentice-Hall. Staelin, D.H., Morgenthaler, A.W. and Kong, J.A., Electromagnetic Waves. Prentice Hall, 1994. Kroemer, H., Quantum Mechanics for Engineering, Materials Science & Applied Physics. Prentice Hall, NJ, 1994.

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Unit 4 Elasticity Structure 4.0

Objectives

4.1

Introduction

4.2

Transitions to inelasticity

4.3

Stiffness

4.4

Relationship to elasticity 4.4.1 Use in engineering

4.5

Young's modulus 4.5.1 Usage

4.6

Linear versus non-linear

4.7

Directional materials

4.8

Force exerted by stretched or compressed material

4.10

Hooke's law

4.10

Shear modulus

4.12

Bending stiffness

4.13

Atomic properties 4.13.1 Applications

4.14

Yield criterion

4.15

Isotropic yield criteria

4.16

Anisotropic yield criteria

4.17

Factors influencing yield stress

4.18

Strengthening mechanisms

4.19

Implications for structural engineering

4.19

Summary

4.20

Keywords

4.21

Self Assessment Questions

4.22

References

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4.0

Objectives After studying this unit you will be to:

4.1



Explain Transitions to inelasticity



Describe Stiffness



Discuss Relationship to elasticity



Define Young's modulus



Explain Directional materials



Discuss Force exerted by stretched or compressed material



Define Hooke's law

Introduction Modelling elasticity Let us understand that the elastic regime is characterized by a linear relationship between stress and strain, denoted linear elasticity. The classic example is a metal spring. This idea was first stated[1] by Robert Hooke in 1675 as a Latin anagram "ceiiinossssttuu" whose solution he published in 1678 as "Ut tensio, sic vis" which means "As the extension, so the force."

This linear relationship is called Hooke's law. The classic model of linear elasticity is the perfect spring. Although the general proportionality constant between stress and strain in three dimensions is a 4th order tensor, when considering simple situations of higher symmetry such as a rod in one dimensional loading, the relationship may often be reduced to applications of Hooke's law.

Because most materials are elastic only under relatively small deformations, several assumptions are used to linearize the theory. Most importantly, higher order terms are generally discarded based on the small deformation assumption. In certain special cases, such as when considering a rubbery material, these assumptions may not be FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621

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permissible. However, in general, elasticity refers to the linearized theory of the continuum stresses and strains.

4.2

Transitions to inelasticity Above a certain stress known as the elastic limit or the yield strength of an elastic material, the relationship between stress and strain becomes nonlinear. Beyond this limit, the solid may deform irreversibly, exhibiting plasticity. A stress-strain curve is one tool for visualizing this transition.

Furthermore, not only solids exhibit elasticity. Some non-Newtonian fluids, such as viscoelastic fluids, will also exhibit elasticity in certain conditions. In response to a small, rapidly applied and removed strain, these fluids may deform and then return to their original shape. Under larger strains, or strains applied for longer periods of time, these fluids may start to flow like a liquid, with some viscosity.

4.3

Stiffness For pain and/or loss of range of motion of a joint, see joint stiffness. For the term regarding the stability of a differential equation, see stiff equation.Stiffness is the resistance of an elastic body to deformation by an applied force along a given degree of freedom (DOF) when a set of loading points and boundary conditions are prescribed on the elastic body. It is an extensive material property. Calculations

The stiffness, k, of a body is a measure of the resistance offered by an elastic body to deformation. For an elastic body with a single Degree of Freedom (for example, stretching or compression of a rod), the stiffness is defined as

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where P is the force applied on the body Î´ is the displacement produced by the force along the same degree of freedom (for instance, the change in length of a stretched spring)

In the International System of Units, stiffness is typically measured in newtons per metre. In English Units, stiffness is typically measured in pound force (lbf) per inch.

Generally speaking, deflections (or motions) of an infinitesimal element (which is viewed as a point) in an elastic body can occur along multiple Degrees of Freedom (maximum of six Degrees of Freedom at a point). For example, a point on a horizontal beam can undergo both a vertical displacement and a rotation relative to its undeformed axis. When the Degrees of Freedom is M, for example, a M x M matrix must be used to describe the stiffness at the point. The diagonal terms in the matrix are the direct-related stiffnesses (or simply stiffnesses) along the same degree of freedom and the off-diagonal terms are the coupling stiffnesses between two different degrees of freedom (either at the same or different points) or the same degree of freedom at two different points. In industry, the term influence coefficient is sometimes used to refer to the coupling stiffness.

It is noted that for a body with multiple Degrees of Freedom, Equation (1) generally does not apply since the applied force generates not only the deflection along its own direction (or degree of freedom), but also those along other directions (or Degrees of Freedom). For example, for a cantilevered beam, the stiffness at its free end is 12*E*I/L^3 rather than 3*E*I/L^3 if calculated with Equation (1).

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For a body with multiple Degrees of Freedom, to calculate a particular direct-related stiffness (the diagonal terms), the corresponding Degree of Freedom is left free while the remaining Degrees of Freedom should be constrained. Under such a condition, Equation (1) can be used to obtain the direct-related stiffness for the degree of freedom which is unconstrained. The ratios between the reaction forces (or moments) and the produced deflection are the coupling stiffnesses.

The inverse of stiffness is compliance, typically measured in units of metres per newton. In rheology it may be defined as the ratio of strain to stress [1], and so take the units of reciprocal stress, e.g. 1/Pa. Rotational stiffness

A body may also have a rotational stiffness, k, given by

where M is the applied moment θ is the rotation

In the SI system, rotational stiffness is typically measured in newtonmetres per radian.

In the SAE system, rotational stiffness is typically measured in inchpounds per degree.

Further measures of stiffness are derived on a similar basis, including: 

shear stiffness - ratio of applied shear force to shear deformation



torsional stiffness - ratio of applied torsion moment to angle of twist

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4.4

Relationship to elasticity In general, elastic modulus is not the same as stiffness. Elastic modulus is a property of the constituent material; stiffness is a property of a structure. That is, the modulus is an intensive property of the material; stiffness, on the other hand, is an extensive property of the solid body dependent on the material and the shape and boundary conditions. For example, for an element in tension or compression, the axial stiffness is

where A is the cross-sectional area, E is the (tensile) elastic modulus (or Young's modulus), L is the length of the element.

For the special case of unconstrained uniaxial tension or compression, Young's modulus can be thought of as a measure of the stiffness of a material. 4.4.1 Use in engineering The stiffness of a structure is of principal importance in many engineering applications, so the modulus of elasticity is often one of the primary properties considered when selecting a material. A high modulus of elasticity is sought when deflection is undesirable, while a low modulus of elasticity is required when flexibility is needed

An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically (i.e., non-permanently) when a force is applied to it. The elastic modulus of an object is defined as the slope of its stress-strain curve in the elastic deformation region:

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where lambda (位) is the elastic modulus; stress is the force causing the deformation divided by the area to which the force is applied; and strain is the ratio of the change caused by the stress to the original state of the object. If stress is measured in pascals, since strain is a unitless ratio, then the units of 位 are pascals as well.

Since the denominator becomes unity if length is doubled, the elastic modulus becomes the stress needed to cause a sample of the material to double in length. While this endpoint is not realistic because most materials will fail before reaching it, it is practical, in that small fractions of the defining load will operate in exactly the same ratio. Thus for steel with an elastic modulus of 30 million pounds per square inch, a 30 thousand psi load will elongate a 1 inch bar by one thousandth of an inch, and similarly for metric units, where a thousandth of the modulus in Gigapascals will change a meter by a millimeter.

Specifying how stress and strain are to be measured, including directions, allows for many types of elastic moduli to be defined. The three primary ones are:

Young's modulus (E) describes tensile elasticity, or the tendency of an object to deform along an axis when opposing forces are applied along that axis; it is defined as the ratio of tensile stress to tensile strain. It is often referred to simply as the elastic modulus.

The shear modulus or modulus of rigidity (G or

) describes an

object's tendency to shear (the deformation of shape at constant volume) when acted upon by opposing forces; it is defined as shear stress over shear strain. The shear modulus is part of the derivation of viscosity.

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The bulk modulus (K) describes volumetric elasticity, or the tendency of an object to deform in all directions when uniformly loaded in all directions; it is defined as volumetric stress over volumetric strain, and is the inverse of compressibility. The bulk modulus is an extension of Young's modulus to three dimensions.

Three other elastic moduli are Poisson's ratio, Lam's first parameter, and P-wave modulus.

Homogeneous and isotropic (similar in all directions) materials (solids) have their (linear) elastic properties fully described by two elastic moduli, and one may choose any pair. Given a pair of elastic moduli, all other elastic moduli can be calculated according to formulas in the table below at the end of page.

Inviscid fluids are special in that they cannot support shear stress, meaning that the shear modulus is always zero. This also implies that Young's modulus is always zero.

4.5

Young's modulus In solid mechanics, Young's modulus, also known as the tensile modulus, is a measure of the stiffness of an isotropic elastic material. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. This can be experimentally determined from the slope of a stress-strain curve created during tensile tests conducted on a sample of the material.

It is also commonly, but incorrectly, called the elastic modulus or modulus of elasticity, because Young's modulus is the most common elastic

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modulus used, however there are other elastic moduli, such as the bulk modulus and the shear modulus.

Young's modulus is named after Thomas Young, the 19th century British scientist. However, the concept was developed in 1727 by Leonhard Euler, and the first experiments that used the concept of Young's modulus in its current form were performed by the Italian scientist Giordano Riccati in 1782 predating Young's work by 25 years

Young's modulus is the ratio of stress, which has units of pressure, to strain, which is dimensionless; therefore Young's modulus itself has units of pressure.

The SI unit of modulus of elasticity (E, or less commonly Y) is the pascal (Pa or N/m); the practical units are megapascals (MPa or N/mm) or gigapascals (GPa or kN/mm). In United States customary units, it is expressed as pounds (force) per square inch (psi). 4.5.1 Usage The Young's modulus allows the behavior of a bar made of an isotropic elastic material to be calculated under tensile or compressive loads. For instance, it can be used to predict the amount a wire will extend under tension or buckle under compression. Some calculations also require the use of other material properties, such as the shear modulus, density, or Poisson's ratio.

4.6

Linear versus non-linear For many materials, Young's modulus is essentially constant over a range of strains. Such materials are called linear, and are said to obey Hooke's law. Examples of linear materials include steel, carbon fiber and glass.

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Non-linear materials include: rubber and soils (except at very small strains).

4.7

Directional materials Young's modulus is not always the same in all orientations of a material. Most metals and ceramics, along with many other materials, are isotropic: Their mechanical properties are the same in all orientations. However, metals and ceramics can be treated with certain impurities, and metals can be mechanically worked to make their grain structures directional. These materials then become anisotropic, and Young's modulus will change depending on the direction from which the force is applied. Anisotropy can be seen in many composites as well. For example, carbon fiber has a much higher Young's modulus (is much stiffer) when force is loaded parallel to the fibers (along the grain). Other such materials include wood and reinforced concrete. Engineers can use this directional phenomenon to their advantage in creating structures. Calculation Young's modulus, E, can be calculated by dividing the tensile stress by the tensile strain:

where E is the Young's modulus (modulus of elasticity) F is the force applied to the object; A0 is the original cross-sectional area through which the force is applied; Î&#x201D;L is the amount by which the length of the object changes; L0 is the original length of the object.

4.8

Force exerted by stretched or compressed material The Young's modulus of a material can be used to calculate the force it exerts under a specific strain.

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where F is the force exerted by the material when compressed or stretched by Î&#x201D;L.

Hooke's law can be derived from this formula, which describes the stiffness of an ideal spring:

where

4.9

Elastic potential energy The elastic potential energy stored is given by the integral of this expression with respect to L:

where Ue is the elastic potential energy. The elastic potential energy per unit volume is given by:

where

is the strain in the material.

This formula can also be expressed as the integral of Hooke's law:

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Relation among elastic constants

For homogeneous isotropic materials simple relations exist between elastic constants (Young's modulus E, shear modulus G, bulk modulus K, and Poisson's ratio Î˝) that allow calculating them all as long as two are known:

Approximate values

Influences of selected glass component additions on Young's modulus of a specific base glass

Young's modulus can vary somewhat due to differences in sample composition and test method. The rate of deformation has the greatest impact on the data collected, especially in polymers. The values here are approximate and only meant for relative comparisons. Approximate Young's modulus for various materials[3] Material

GPa

lbf/in (psi)

Rubber (small strain)

0.01-0.1

1,500-15,000

ZnO NWs[citation needed]

21-37

3,045,792-5,366,396

PTFE (Teflon)[citation needed]

0.5

75,000

Low density polyethylene[citation needed]

0.2

30,000

HDPE

0.8

Polypropylene

1.5-2

217,000-290,000

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Bacteriophage capsids

1-3

Polyethylene terephthalate

2-2.7

Polystyrene

3-3.5

435,000-505,000

Nylon

2-4

290,000-580,000

Diatom frustules (largely silicic acid)

0.35-2.77

50,000-400,000

Medium-density fibreboard

580,000

Pine wood (along grain)

8.963

1,300,000

Oak wood (along grain)

1,600,000

(under 30

4,350,000

Magnesium metal (Mg)

6,500,000

Aluminium

10,000,000

Glass (see chart)

50-90

Kevlar[7]

70.5-112.4

High-strength

concrete

150,000-435,000

compression)

Mother-of-pearl (nacre, largely calcium 70

10,000,000

carbonate) Tooth

enamel

(largely

calcium 83

12,000,000

phosphate) Brass and bronze

100-125

Titanium (Ti)

17,000,000 16,000,000

Titanium alloys

105-120

15,000,000-17,500,000

Copper (Cu)

117

17,000,000

Glass fiber reinforced plastic (70/30 by 40-45

5,800,000-6,500,000

weight fibre/matrix, unidirectional, along grain) Carbon fiber reinforced plastic (50/50 125-150

18,000,000-22,000,000

fibre/matrix, unidirectional, along grain) Silicon[10]

185

Wrought iron

190210

Steel

200

29,000,000

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polycrystalline Yttrium iron garnet (YIG)

193

28,000,000

single-crystal Yttrium iron garnet (YIG)

200

30,000,000

Beryllium (Be)

287

42,000,000

Molybdenum (Mo)

329

Tungsten (W)

400-410

58,000,000-59,500,000

Sapphire (Al2O3) along C-axis

435

63,000,000

Silicon carbide (SiC)

450

65,000,000

Osmium (Os)

550

79,800,000

Tungsten carbide (WC)

450-650

65,000,000-94,000,000

Single-walled carbon nanotube

1,000+

145,000,000+

Diamond (C)[14]

1220

150,000,000175,000,000

4.10

Hooke's law

Hooke's law accurately models the physical properties of common mechanical springs for small changes in length

Hooke's law describes how far the spring will stretch with a specific force

In mechanics, and physics, Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load added to it as long as this load does not exceed the elastic limit. Materials for which Hooke's law is a useful approximation are known as linear-elastic or "Hookean" materials. Hooke's law in simple terms says that strain is directly proportional to stress.

Mathematically, Hooke's law states that where

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x is the displacement of the end of the spring from its equilibrium position (in SI units: "m"); F is the restoring force exerted by the material (in SI units: "N" or kgms -2); and k is the force constant (or spring constant) (in SI units: "Nm-1" or "kgs-2").

When this holds, the behavior is said to be linear. If shown on a graph, the line should show a direct variation. There is a negative sign on the right hand side of the equation because the restoring force always acts in the opposite direction of the displacement (for example, when a spring is stretched to the left, it pulls back to the right).

Hooke's law is named after the 17th century British physicist Robert Hooke. He first stated this law in 1676 as a Latin anagram,[1] whose solution he published in 1678 as Ut tensio, sic vis, meaning, "As the extension, so the forceElastic

Objects that quickly regain their original shape after being deformed by a force, with the molecules or atoms of their material returning to the initial state of stable equilibrium, often obey Hooke's law.

We may view a rod of any elastic material as a linear spring. The rod has length L and cross-sectional area A. Its extension (strain) is linearly proportional to its tensile stress Ď&#x192;, by a constant factor, the inverse of its modulus of elasticity, E, hence, or

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Hooke's law only holds for some materials under certain loading conditions. Steel exhibits linear-elastic behavior in most engineering applications; Hooke's law is valid for it throughout its elastic range (i.e., for stresses below the yield strength). For some other materials, such as aluminium, Hooke's law is only valid for a portion of the elastic range. For these materials a proportional limit stress is defined, below which the errors associated with the linear approximation are negligible.

Rubber is generally regarded as a "non-hookean" material because its elasticity is stress dependent and sensitive to temperature and loading rate.

4.11

Shear modulus Expressed in other quantities: G = τ / γ

Shear strain In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of shear stress to the shear strain:[1]

where = shear stress; F is the force which acts A is the area on which the force acts

= shear strain; Δx is the transverse displacement

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I is the initial length

Shear modulus is usually expressed in gigapascals (GPa) or thousands of pounds per square inch (ksi).

Material

Typical values for shear modulus (GPa) (at room temperature)

Diamond[2]

478.

Steel[3]

79.3

Copper[4]

44.7

Titanium[3]

41.4

Glass[3]

26.2

Aluminium[3] Polyethylene

25.5 [3]

Rubber[5]

0.117 0.0006

The shear modulus is one of several quantities for measuring the stiffness of materials. All of them arise in the generalized Hooke's law: 

Young's modulus describes the material's response to linear strain (like pulling on the ends of a wire),



the bulk modulus describes the material's response to uniform pressure, and



the shear modulus describes the material's response to shearing strains.

The shear modulus is concerned with the deformation of a solid when it experiences a force parallel to one of its surfaces while its opposite face experiences an opposing force (such as friction). In the case of an object that's shaped like a rectangular prism, it will deform into a parallelepiped.

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Anisotropic materials such as wood and paper exhibit differing material response to stress or strain when tested in different directions. In this case, when the deformation is small enough so that the deformation is linear, the elastic moduli, including the shear modulus, will then be a tensor, rather than a single scalar value. Waves

Influences of selected glass component additions on the shear modulus of a specific base glass.

In homogeneous and isotropic solids, there are two kinds of waves, pressure waves and shear waves. The velocity of a shear wave, (vs) is controlled by the shear modulus,

where G is the shear modulus Ď is the solid's density. Shear modulus of metals

Shear modulus of copper as a function of temperature. The experimental data[7][8] are shown with colored symbols.

The shear modulus of metals measures the resistance to glide over atomic planes in crystals of the metal. In polycrystalline metals there are also grain boundary factors that have to be considered. In metal alloys, the shear modulus is observed to be higher than in pure metals due to the presence of additional sources of resistance to glide.

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The shear modulus of metals is usually observed to decrease with increasing temperature. At high pressures, the shear modulus also appears to increase with the applied pressure. Correlations between the melting temperature, vacancy formation energy, and the shear modulus have been observed in many metals.[9]

Several models exist that attempt to predict the shear modulus of metals (and possibly that of alloys). Shear modulus models that have been used in plastic flow computations include: 

the MTS shear modulus model developed by[10] and used in conjunction with the Mechanical Threshold Stress (MTS) plastic flow stress model).[11][12]



the Steinberg-Cochran-Guinan (SCG) shear modulus model developed by[13] and used in conjunction with the Steinberg-Cochran-Guinan-Lund (SCGL) flow stress model.



the Nadal and LePoac (NP) shear modulus model[8] that uses Lindemann theory to determine the temperature dependence and the SCG model for pressure dependence of the shear modulus.

MTS shear modulus model The MTS shear modulus model has the form:

where μ0 is the shear modulus at 0K, and D,T0 are material constants. SCG shear modulus model The Steinberg-Cochran-Guinan (SCG) shear modulus model is pressure dependent and has the form

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100

where, μ0 is the shear modulus at the reference state(T = 300 K, p = 0, η = 1), p is the pressure, and T is the temperature. NP shear modulus model The Nadal-Le Poac (NP) shear modulus model is a modified version of the SCG model. The empirical temperature dependence of the shear modulus in the SCG model is replaced with an equation based on Lindemann melting theory. The NP shear modulus model has the form: and μ0 is the shear modulus at 0 K and ambient pressure, ζ is a material parameter, kb is the Boltzmann constant, m is the atomic mass, and f is the Lindemann constant.

4.12 Bending stiffness The bending stiffness EI of a beam (or a plate) relates the applied bending moment to the resulting deflection of the beam. It is the product of the elastic modulus E of the beam material and the area moment of inertia I of the beam cross-section. According to elementary beam theory, the relationship between the applied bending moment M and the resulting curvature κ of the beam is

where w is the deflection of the beam and x the spatial coordinate. In the literature

A material's property is an intensive, often quantitative property of a material, usually with a unit that may be used as a metric of value to compare the benefits of one material versus another to aid in materials selection.

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101

A material property may be a constant or may be a function of one or more independent variables, such as temperature. Material's properties often vary to some degree according to the direction in the material in which they are measured; a condition referred to as anisotropy. Materials properties that relate two different physical phenomena often behave linearly or approximately so in a given operating range and may then be modeled as a constant for that range. This linearization can significantly simplify the differential constitutive equations that the property describes.

Some material's properties are used in relevant equations to determine the attributes of a system a priori. For example, if a material of a known specific heat gains or loses a known amount of heat, the temperature change of that material can be determined. Materials properties may be determined by standardized test methods. Many such test methods have been documented by their respective user communities and published through ASTM International

There are a variety of other properties to consider in an environmental impact assessment that effect the ecological or human environment that may be difficult to quantify (unlike most of the properties listed on this page)

including

pollution

(extraction,

transportation,

manufacture),

scarcity/abundance, habitat destruction, renewability, recyclability, wars fought over materials, labor exploitation, etc. These can be subjective, dependent on context, or inadequately measured.

4.13 Atomic properties Toughness, in materials science and metallurgy, is the resistance to fracture of a material when stressed. It is defined as the amount of energy per volume that a material can absorb before rupturing. edit] Mathematical definition

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Toughness can be determined by measuring the area (i.e., by taking the integral) underneath the stress-strain curve.It's energy of mechanical deformation per unit volume prior to fracture. The explicit mathematical description is:

Where ε is strain εf is the strain upon failure σ is stress

Another definition is the ability to absorb mechanical (or kinetic) energy up to failure. The Area covered under stress strain curve is called toughness.

If we restrict the upper limit of integration up to the yield point, than the energy absorbed per unit volume is known as modulus of resilience. Thus, mathematically, the modulus of resilience can be expressed by the product of the square of the yield strain, times the Young's modulus, divided by two. Toughness tests Tests can be done by using a pendulum and some basic physics to measure how much energy it will hold when released from a particular height. By having a sample at the bottom of its swing a measure of toughness can be found, as in the Charpy and Izod impact tests. Unit of toughness Toughness is measured in units of joules per cubic metre (J/m3) in the SI system and inch-pound-force per cubic inch (inlbf/in3) in US customary units.

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103

Toughness and strength Strength and toughness are related. A material may be strong and tough if it ruptures under high forces, exhibiting high strains; on the other hand, brittle materials may be strong but with limited strain values, so that they are not tough. Generally speaking, strength indicates how much force the material can support, while toughness indicates how much energy a material can absorb before rupture. 4.13.1 Applications Kitchen knives A cleaver must be tough, to withstand blows against bone. Accordingly, cleavers are made of softer steel, which does not take as sharp an edge as the harder steels used in slicing knives The yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to deform plastically. Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed. Once the yield point is passed some fraction of the deformation will be permanent and non-reversible. In the threedimensional space of the principal stresses (Ď&#x192;1,Ď&#x192;2,Ď&#x192;3), an infinite number of yield points form together a yield surface. Knowledge of the yield point is vital when designing a component since it generally represents an upper limit to the load that can be applied. It is also important for the control of many materials production techniques such as forging, rolling, or pressing. In structural engineering, this is a soft failure mode which does not normally cause catastrophic failure or ultimate failure unless it accelerates buckling

Typical yield behavior for non-ferrous alloys. 1: True elastic limit 2: Proportionality limit

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3: Elastic limit 4: Offset yield strength

It is often difficult to precisely define yielding due to the wide variety of stressstrain curves exhibited by real materials. In addition, there are several possible ways to define yielding:[1]

True elastic limit The lowest stress at which dislocations move. This definition is rarely used, since dislocations move at very low stresses, and detecting such movement is very difficult.

Proportionality limit Up to this amount of stress, stress is proportional to strain (Hooke's law), so the stress-strain graph is a straight line, and the gradient will be equal to the elastic modulus of the material.

Elastic limit (yield strength) Beyond the elastic limit, permanent deformation will occur. The lowest stress at which permanent deformation can be measured. This requires a manual load-unload procedure, and the accuracy is critically dependent on equipment and operator skill. For elastomers, such as rubber, the elastic limit is much larger than the proportionality limit. Also, precise strain measurements have shown that plastic strain begins at low stresses.[2][3]

Offset yield point (proof stress) This is the most widely used strength measure of metals, and is found from the stress-strain curve as shown in the figure to the right. A plastic strain of 0.2% is usually used to define the offset yield stress, although other values may be used depending on the material and the application.

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The offset value is given as a subscript, e.g., Rp0.2=310 MPa. In some materials there is essentially no linear region and so a certain value of strain is defined instead. Although somewhat arbitrary, this method does allow for a consistent comparison of materials.

Upper yield point and lower yield point Some metals, such as mild steel, reach an upper yield point before dropping rapidly to a lower yield point. The material response is linear up until the upper yield point, but the lower yield point is used in structural engineering as a conservative value. If a metal is only stressed to the upper yield point, and beyond, luders bands can develop.[4]

4.14 Yield criterion A yield criterion, often expressed as yield surface, or yield locus, is an hypothesis concerning the limit of elasticity under any combination of stresses. There are two interpretations of yield criterion: one is purely mathematical in taking a statistical approach while other models attempt to provide a justification based on established physical principles. Since stress and strain are tensor qualities they can be described on the basis of three principal directions, in the case of stress these are denoted by , and

The following represent the most common yield criterion as applied to an isotropic material (uniform properties in all directions). Other equations have been proposed or are used in specialist situations.

4.15 Isotropic yield criteria Maximum Principal Stress Theory - Yield occurs when the largest principal stress exceeds the uniaxial tensile yield strength. Although this criterion

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106

allows for a quick and easy comparison with experimental data it is rarely suitable for design purposes.

Maximum Principal Strain Theory - Yield occurs when the maximum principal strain reaches the strain corresponding to the yield point during a simple tensile test. In terms of the principal stresses this is determined by the equation:

Maximum Shear Stress Theory - Also known as the Tresca yield criterion, after the French scientist Henri Tresca. This assumes that yield occurs when the shear stress exceeds the shear yield strength

Total Strain Energy Theory - This theory assumes that the stored energy associated with elastic deformation at the point of yield is independent of the specific stress tensor. Thus yield occurs when the strain energy per unit volume is greater than the strain energy at the elastic limit in simple tension. For a 3-dimensional stress state this is given by:

Distortion Energy Theory - This theory proposes that the total strain energy can be separated into two components: the volumetric (hydrostatic) strain energy and the shape (distortion or shear) strain energy. It is proposed that yield occurs when the distortion component exceeds that at the yield point for a simple tensile test. This is generally referred to as the Von Mises yield criterion and is expressed as:

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Based on a different theoretical underpinning this expression is also referred to as octahedral shear stress theory.

Other commonly used isotropic yield criteria are the 

Mohr-Coulomb yield criterion



Drucker-Prager yield criterion



Bresler-Pister yield criterion



Willam-Warnke yield criterion

The yield surfaces corresponding to these criteria have a range of forms. However, most isotropic yield criteria correspond to convex yield surfaces.

4.16 Anisotropic yield criteria When a metal is subjected to large plastic deformations the grain sizes and orientations change in the direction of deformation. As a result the plastic yield behavior of the material shows directional dependency. Under such circumstances, the isotropic yield criteria such as the von Mises yield criterion are unable to predict the yield behavior accurately. Several anisotropic yield criteria have been developed to deal with such situations. Some of the more popular anisotropic yield criteria are: 

Hill's quadratic yield criterion.



Generalized Hill yield criterion.



Hosford yield criterion.

4.17 Factors influencing yield stress The stress at which yield occurs is dependent on both the rate of deformation (strain rate) and, more significantly, the temperature at which the deformation occurs. Early work by Alder and Philips in 1954 found that

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the relationship between yield stress and strain rate (at constant temperature) was best described by a power law relationship of the form

where C is a constant and m is the strain rate sensitivity. The latter generally increases with temperature, and materials where m reaches a value greater than ~0.5 tend to exhibit super plastic behaviour.

Later, more complex equations were proposed that simultaneously dealt with both temperature and strain rate: where Îą and A are constants and Z is the temperature-compensated strain-rate - often described by the Zener-Hollomon parameter:

where QHW is the activation energy for hot deformation and T is the absolute temperature.

4.18 Strengthening mechanisms There are several ways in which crystalline and amorphous materials can be engineered to increase their yield strength. By altering dislocation density, impurity levels, grain size (in crystalline materials), the yield strength of the material can be fine tuned. This occurs typically by introducing defects such as impurities dislocations in the material. To move this defect (plastically deforming or yielding the material), a larger stress must be applied. This thus causes a higher yield stress in the material. While many material properties depend only on the composition of the bulk material, yield strength is extremely sensitive to the materials processing as well for this reason. These mechanisms for crystalline materials include ď&#x201A;ˇ

Work Hardening

ď&#x201A;ˇ

Solid Solution Strengthening

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

Particle/Precipitate Strengthening



Grain boundary strengthening

109

Work Hardening Where deforming the material will introduce dislocations, which increases their density in the material. This increases the yield strength of the material, since now more stress must be applied to move these dislocations through a crystal lattice. Dislocations can also interact with each other, becoming entangled.

The governing formula for this mechanism is: where σy is the yield stress, G is the shear elastic modulus, b is the magnitude of the Burgers vector, and ρ is the dislocation density.

Solid Solution Strengthening By alloying the material, impurity atoms in low concentrations will occupy a lattice position directly below a dislocation, such as directly below an extra half plane defect. This relieves a tensile strain directly below the dislocation by filling that empty lattice space with the impurity atom.

The relationship of this mechanism goes as:

where τ is the shear stress, related to the yield stress, G and b are the same as in the above example, C_s is the concentration of solute and ε is the strain induced in the lattice due to adding the impurity.

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while moving through the matrix, will be forced against a small particle or precipitate of the material. Dislocations can move through this particle either by shearing the particle, or by a process known as bowing or ringing, in which a new ring of dislocations is created around the particle.

In these formulas,

is the particle radius,

is the

surface tension between the matrix and the particle,

is the

distance between the particles.

Grain boundary strengthening Where a buildup of dislocations at a grain boundary causes a repulsive force between dislocations. As grain size decreases, the surface area to volume ratio of the grain increases, allowing more buildup of dislocations at the grain edge. Since it requires a lot of energy to move dislocations to another grain, these dislocations build up along the boundary, and increase the yield stress of the material. Also known as Hall-Petch strengthening,

this

type

strengthening

governed

the

formula:

where Ď&#x192;0 is the stress required to move dislocations, k is a material constant, and d is the grain size.

Testing Yield strength testing involves taking a small sample with a fixed crosssection area, and then pulling it with a controlled, gradually increasing force until the sample changes shape or breaks. Longitudinal and/or transverse strain is recorded using mechanical or optical extensometers.

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Indentation hardness correlates linearly with tensile strength for most steels. [5] Hardness testing can therefore be an economical substitute for tensile testing, as well as providing local variations in yield strength due to e.g. welding or forming operations.

4.19 Implications for structural engineering Yielded structures have a lower stiffness, leading to increased deflections and decreased buckling strength. The structure will be permanently deformed when the load is removed, and may have residual stresses. Engineering metals display strain hardening, which implies that the yield stress is increased after unloading from a yield state. Highly optimized structures, such as airplane beams and components, rely on yielding as a fail-safe failure mode. No safety factor is therefore needed when comparing limit loads (the highest loads expected during normal operation) to yield criteria.

4.23 Summary The bending stiffness EI of a beam (or a plate) relates the applied bending moment to the resulting deflection of the beam. It is the product of the elastic modulus E of the beam material and the area moment of inertia I of the beam cross-section. A material property may be a constant or may be a function of one or more independent variables, such as temperature. Material's properties often vary to some degree according to the direction in the material in which they are measured; a condition referred to as anisotropy.

4.24 Keywords Stiffness: The stiffness, k, of a body is a measure of the resistance offered by an elastic body to deformation.

Young's modulus: In solid mechanics, Young's modulus, also known as the tensile modulus, is a measure of the stiffness of an isotropic elastic

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material. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds.

4.25 Self Assessment Questions 1. Explain Transitions to inelasticity. 2. Describe Stiffness. 3. Discuss Relationship to elasticity. 4. Define Young's modulus 5. Explain Directional materials. 6. Discuss Force exerted by stretched or compressed material. 7. Define Hooke's law.

4.26 References Hibbeler, R.C., Engineering Mechanics: Dynamics, 7th edition. PrenticeHall, Englewood Cliffs, N.J.,1995. Reif, F., Fundamentals of Statistical and Thermal Physics. McGraw-Hill Inc., 1965. Kittel, C. and Kroemar, H., Thermal Physics, 2nd edition. W.H. Freeman, 1980. Demarest, Engineering Electromagnetics. Prentice-Hall. Staelin, D.H., Morgenthaler, A.W. and Kong, J.A., Electromagnetic Waves. Prentice Hall, 1994. Kroemer, H., Quantum Mechanics for Engineering, Materials Science & Applied Physics. Prentice Hall, NJ, 1994.

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