DEPARTMENT OF PHYSICS RV COLLEGE OF ENGINEERING

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1 DEPARTMENT OF PHYSICS RV COLLEGE OF ENGINEERING ENGINEERING PHYSICS NOTES-07-8 COURSE CODE: 6PH/

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3 Semester: I/II ENGINEERING PHYSICS( Theory and practice) Course Code: 6PH/6PH CIE Marks: Hrs/Week: L:T:P:S 4:0::0 SEE Marks: Credits: 05 SEE ( Theory)Duration: Hrs SEE (Practice)Duration: Hrs SYLLABUS: UNIT-I: LASERS AND OPTICAL FIBERS Basic principles of Laser: Absorption and emissions, Einstein s coefficients. Energy density in terms of Einstein coefficients, conditions and requisites of laser. Types of lasers: Helium -Neon Laser, Semiconductor diode Laser. Characteristics of laser beam. Industrial applications of lasers: laser cutting, welding and drilling, measurements of pollutants in atmosphere. 08 h Principle of Optical fibers: propagation mechanism, condition for propagation, acceptance angle and numerical aperture. Modes of propagation, types of optical fibers. Attenuation: Absorption, scattering and radiation loss, attenuation coefficient. Application of optical fiber in point to point communication, advantages of optical fiber communication over electrical mode of communication. UNIT-II: QUANTUM MECHANICS Black body radiation spectrum, Laws of black body radiation spectrum, Plank s quantum theory, Review of Photoelectric effect and Compton effect. Wave particle duality, de-broglie hypothesis. Matter waves: properties of matter waves, wave packet, group velocity, phase velocity and their relations. Application of matter waves: Scanning Electron Microscope (SEM) construction and working. h Uncertainty principle: illustrations-non-confinement of electron inside the nucleus and broadening of spectral lines. Setting up of one dimensional time independent Schrodinger s wave equation-wave function, physical significance of wave function, Eigen function, Eigen values. Application of Schrodinger s wave equation: Free particle, Particle in a one dimensional potential well of infinite depth. Problems.

4 UNIT-III: OSCILLATIONS AND WAVES Simple Harmonic Motion, Characteristics of Simple harmonic motion. Un damped / Free vibrations, differential equations of un damped / free vibrations and solutions. Examples of Simple harmonic oscillators a) Spring and Mass system, b) Torsional Pendulum. Damped vibrations: Differential equations of damped vibrations and solutions. Forced vibrations: Differential equations of forced vibrations and solutions, Resonance. Examples of forced vibrations- LCR circuits. Problems. UNIT-IV: ELECTRICAL CONDUCTIVITY IN METALS AND SEMICONDUCTORS Review of Classical free electron theory, Quantum free electron theory. Fermi energy and Fermi factor in metals, variation of Fermi factor with temperature. Density of states and carrier concentration in metals. Hall effect-determination of number and sign of charge carriers. Band theory of solids, (qualitative approach). Intrinsic semiconductors: carrier concentration, concept of effective mass (qualitative), derivation of electron and hole concentration, intrinsic carrier concentration, Fermi level in intrinsic semiconductors, Expression for the energy gap of intrinsic semiconductors. Extrinsic semiconductors: Types of extrinsic semiconductors, doping methods (qualitative). Variation of carrier concentration in extrinsic semiconductors with temperature, variation of Fermi level in extrinsic semiconductors with temperature and impurity concentration. Hall effect in semiconductors. 09h 0h UNIT-V: DIELECTRICS AND THERMAL CONDUCTIVITY Dielectrics: Electric dipole, Dipole moment, Field due to electric dipole at a point in a plane. Polarization of dielectric materials: Types of polarizations, frequency dependence of polarization mechanisms, dielectric loss. Internal field in solids: for one dimensional infinite array of dipoles (Lorentz field), Clausius - Mossotti equation. Thermal conductivity: conduction of heat in solids, steady state, coefficient of thermal conductivity, thermal conductivity of a good conductor by Searle s method and thermal conductivity of a poor conductor by Lee s and Charlton s method. 09h

5 CONTENTS: UNIT I : LASERS AND OPTICAL FIBERS UNIT II : QUANTUM MECHANICS UNIT III : OSCILLATIONS AND WAVES UNIT IV : ELECTRICAL CONDUCTIVITY IN METALS AND SEMICONDUCTORS UNIT V : DIELECTRICS AND THERMAL CONDUCTIVITY

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7 UNIT- I: LASERS AND OPTICAL FIBERS UNIT- I: LASERS AND OPTICAL FIBERS LASER Introduction: LASER is an acronym for Light Amplification by Stimulated Emission of Radiation. Laser device produces a beam of coherent, monochromatic, intense and directional light. Hence laser light is highly organized when compared with the ordinary light. This is because the waves of a laser beam move in phase with each other, travel in a narrow path in one direction. In the case of an ordinary light it spreads out, travels in different directions and hence it is incoherent. On account of the special properties, lasers are the most versatile and exploited tools in different fields such as Engineering, Medicine, Defense, Entertainment, Communication etc., Other common applications of lasers include reading the bar code, cutting and welding metals, displays in light shows, playing music, printing documents, guiding missile to its target and so on. Basic principles: Interaction of Radiation with Matter Production of laser light is a consequence of interaction of radiation with matter under appropriate conditions. The interaction of radiation with matter leads to transition of the quantum system such as an atom or a molecule of the matter from one quantum energy state to another quantum state. A material medium is composed of identical atoms or molecules each of which is characterized by a set of discrete allowed energy levels. An atom can move from one energy state to another when it receives or releases an amount of energy equal to the energy difference between those two states, which is termed as a quantum jump or transition. Consider a two energy level system with energies E and E of an atom. E is the energy of lower energy state and E is the energy of excited state. The energy levels E and E are identical to all the atoms in the medium. The radiation (either absorbed or emitted) may be viewed as a stream of photons of energy (E -E ) hν, interacting with the material. These interactions lead to any one of the following Induced Absorption of radiation Spontaneous emission of radiation Stimulated emission of radiation

8 UNIT- I: LASERS AND OPTICAL FIBERS Induced Absorption: Excitation of atoms by the absorption of photons is called induced absorption. E E Incident photon hν E - E E E An atom in the lower energy state E absorbs the incident photon of energy (E -E ) and goes to the excited state E. This transition is known as absorption. For each transition made by an atom one photon disappears from the incident beam. for an atom A, A + hν A* (excited state) The number of absorption transitions per second per unit volume occurring in the material at any instant of time will be proportional to (i) The Number of atoms in the ground state N (ii) Energy density of the incident radiation (Uν) Rate of induced absorption B UνN where B is proportionality constant which gives the probability of absorptions and it is called Einstein co-efficient of absorption. Since the number of atoms in the lower energy state is greater, the material absorbs more number of the incident photons. Spontaneous Emission: An atom which is at higher energy state E is unstable spontaneously returns to the lower energy state E on its own during which a single photon of energy (E -E ) hυ is emitted, the process is known as spontaneous emission. This spontaneous transition can be expressed as A * A + hυ E Emitted photon hν(e -E ) E Before After

9 UNIT- I: LASERS AND OPTICAL FIBERS The number of spontaneous transitions per second, per unit volume depends on the number of atoms N in the excited state. Therefore, the rate of spontaneous emission A N Where A is proportionality constant which gives the probability of spontaneous emission and it is called Einstein co-efficient of spontaneous emission of radiation. The process has no control from outside. The instant of transition, directions of emission of photons, phases of the photons and their polarization states are random quantities. There will not be any correlation among the parameters of the innumerable photons emitted spontaneously by the assembly of atoms in the medium. Therefore the light generated by the source will be incoherent (ex: light emitted from conventional sources). Stimulated Emission: Emission of photons by an atomic system with an external influence is called stimulated emission. A mechanism of forced emission was first predicted by Einstein in 96 in which an atom in the excited state need not wait for the spontaneous emission to take place. A photon of energy hυ (E -E ), can induce the excited atom to make downward transition and emit light. Thus, the interaction of a photon with an excited atom triggers it to drop down to the ground state (lower energy) by emitting a photon. The process is known as induced or stimulated emission of radiation. A* + hυ A + hυ E Stimulating photon Stimulated photon E Before After The number of stimulated transitions per sec per unit volume in the material is proportional to (i) The Number of atoms in the excited state N (ii) Energy density of the incident radiation (Uν) Rate of stimulated emission B UνN Where B is proportionality constant which gives the probability of stimulated emissions and it is called Einstein co-efficient of induced (stimulated) emission. The process of stimulated emission has the following properties. (i) The emitted photon is identical to the incident photon in all respects. (It has the same frequency; it will be in phase and will travel in the same direction and will be in the same state of polarization).

10 UNIT- I: LASERS AND OPTICAL FIBERS (ii) The process can be controlled externally. (iii) Stimulated emission is responsible for laser. Some basic definitions. Atomic system It is a system of atoms or molecules having discrete energy levels.. Active medium It is the material medium composed of atoms or ions or molecules supports the basic interaction of radiation with matter in thermal equilibrium condition.. Energy density The energy density U ν refers to the total energy in the radiation field per unit volume per unit frequency due to photons. It is given by the Plank s distribution law U ν 8πhν c e hν kt 4. Population It is the number density (the number of atoms per unit volume) of atoms in a given energy state. 5. Boltzmann factor It is the ratio between the population of atoms in the higher energy state to the lower energy state under thermal equilibrium. If N is the number density of atoms in the energy state E and N is the number density of atoms in the ground state then According to Boltzmann condition N > N N And e hν kt N 6. Population inversion It is the condition such that the number of atoms in the higher energy (N) state is greater than the number of atoms in the ground state (N ). i.e, N > N If N > N, it is non-equilibrium condition and it is called population inversion. Expression for energy density of incident radiation in terms of Einstein coefficients: Consider an atomic system interacting with radiation field of energy density U γ. Let E and E be two energy states of atomic system (E > E ). Let us consider atoms are to be in thermal equilibrium with radiation field, which means that the energy density U γ is constant in spite of the interaction that is taking place between itself and the incident radiation. This is possible only if the number of photons absorbed by the system per second is equal to the number of photons it emits per second by both the stimulated and spontaneous emission processes. 4

11 UNIT- I: LASERS AND OPTICAL FIBERS We know that The rate of induced absorption B UνN, The rate of spontaneous emission A N The rate of stimulated emission B UνN N and N are the number of atoms in the energy state E and E respectively, B, A and B are the Einstein coefficients for induced absorption, spontaneous emission and stimulated emission respectively. At thermal equilibrium, Rate of induced absorption Rate of spontaneous emission + Rate of stimulated emission B N U γ A N + B N U γ or U γ (B N B N ) A N Uγ B AN N B By rearranging the above equation, we get A Uγ () B BN B N N In a state of thermal equilibrium, the populations of energy levels E and E are fixed by the Boltzmann factor. The population ratio is given by, E E ( ) kt N e N E E h kt kt N Ne Ne γ N N Since (E -E hν) Equation () becomes, e hγ kt Uγ A B hγ kt B B e () 5

12 UNIT- I: LASERS AND OPTICAL FIBERS According to Planck's law of black body radiation, the equation for U γ is, U ν 8πhν c hν ekt () Now comparing the equation () and () term by term on the basis of positional identity we have A B 8πhγ and c B B or B B This implies that the probability of induced absorption is equal to the probability of stimulated emission. Due to this identity the subscripts could be dropped, and A and B can be simply represented as A and B and equation () can be rewritten. At thermal equilibrium the equation for energy density is Uγ A B [e hγ kt ] (Think: Even though the probability of induced absorption is equal to the probability of stimulated emission, the rate of induced absorption is not equal to rate of stimulated emission. Why?) Conditions for light amplification:- Conditions for laser emission can be studied by taking the ratios of rate of stimulated emission to spontaneous emission and rate of stimulated emission to absorption. At thermal equilibrium, Stimulated Emission BN U BU ν ν Spon tan eous Emission AN A Since, B /A C /8πhν constant, This suggests that in order to enhance the number of stimulated transitions the radiation density Uν should be made high. Stimulated Emission B Inducedabsorption B N U N U ν ν B B N N (B /B ) The stimulated emission will be larger than the absorption only when N >N. If N >N the stimulated emission dominates the absorption otherwise the medium will absorb the energy. This condition of N >N is known as inverted population state or population inversion. 6

13 UNIT- I: LASERS AND OPTICAL FIBERS Requisites of a laser system: The essential components of a laser are An active medium to support population inversion. Pumping mechanism to excite the atoms to higher energy levels. Population inversion Metastable state An optical cavity or optical resonator. Active medium: It is the material medium composed of atoms or ions or molecules in which the laser action is made to take place, which can be a solid or liquid or even a gas. In this only a few atoms of the medium (of particular species) are responsible for stimulated emission. They are called active centers and the remaining medium simply supports the active centers. Pumping Mechanism: To achieve the population inversion in the active medium, the atoms are to be raised to the excited state. It requires energy to be supplied to the system. The process of supplying energy to the medium with a view to transfer the atoms to higher energy state is called pumping. Important pumping mechanisms are a) Optical pumping: It employs a suitable light source for excitation of desired atoms. This method is adopted in solid state lasers (ex: Ruby laser and Nd:YAG laser). b) Electric discharge: In this process an electric field causes ionization in the medium and raises it to the excited state. This technique is used in gas lasers (ex: Ar + laser). c) Inelastic atom-atom collision : In this method a combination of two types of gases are used, say A and B. During electric discharge A atoms get excited and they now collide with B atoms so that B goes to excited state. This technique is used in gas lasers (ex: He- Ne laser). d) Direct conversion : In this process electrical energy is directly converted into light energy. This technique is used in semiconductor lasers (ex: GaAs laser). Population Inversion and Meta stable state: In order to increase stimulated emission it is essential that N >N i.e., the number of atoms in the excited state must be greater than the number of atoms in the ground state. Even if the population is more in the excited state, there will be a competition between stimulated and spontaneous emission. The possibility of spontaneous emission can be reduce by using 7

14 UNIT- I: LASERS AND OPTICAL FIBERS intermediate state where the life time of atom will be little longer (0-6 to 0 - s ) compared to excited state (0-9 s). This intermediate state is called metastable state and it depends upon the nature of atomic species used in the active medium. Principle of pumping scheme: Consider three energy levels E, E and E of a quantum system of which the level E is metastable state. Let the atoms be excited from E to E state by supply of appropriate energy. Then the atom from the E state undergoes downward transition to either E or E states rapidly. Once the atoms undergo downward transitions to level E they tend to stay, for a long interval of time, because of which the population of E increases rapidly. Transition from E to E being very slow, in a short period of time the number of atoms in the level E is far greater than the level E. Thus Population inversion has been achieved betweene and E. The transition from meta stable state to ground state is the lasing transition. It occurs in between upper lasing E and lower lasing level E. E Non-radiative transition E Meta stable state (Upper lasing level) Lasing transition E (Lower lasing level) Optical resonator: An optical resonator generally consists of two plane mirrors, with the active material placed in between them. One of the mirrors is semitransparent while the other one is 00% reflecting. The mirrors are set normal to the axis of the active medium and parallel to each other. The optical resonant cavity provides the selectivity of photon states by confining the possible direction of photon propagation, as a result lasing action occurs in this direction. The distance between the mirrors is an important parameter as it chooses the wavelength of the photons. Suppose a photon is traveling between two reflectors, it undergoes reflection at the mirror kept at the other end.the reflected wave superposes on the incident wave and forms stationary wave such that the length L of the cavity is given by L n λ Hence λ L n Where, L is the distance between the mirrors. λ is the wavelength of the photon, n is the integral multiple of half wavelength 8

15 UNIT- I: LASERS AND OPTICAL FIBERS The wavelengths satisfying the above condition are only amplified. Hence the cavity is also called resonant cavity. 00% Reflecting mirror Active medium Semi transparent mirror The main Role of the optical resonator is to Provide positive feedback of photons into the active medium to sustain stimulated emission and hence laser acts as a generator of light. Select the direction of stimulated photons which are travelling parallel to the axis of optical resonator and normal to the plane of mirrors are to be amplified. Hence laser light is highly directional. Builds up the photon density (U ν ) to a very high value through repeated reflections of photons by mirrors and confines them within the active medium. Selects and amplifies only certain frequencies of stimulated photons which are to be highly monochromatic and gives out the laser light through the partial reflector after satisfying threshold condition. Helium-Neon (He-Ne) Laser: The Helium Neon laser is a gas laser that produces a continuous laser beam. It is widely used in surgery, communication, printing and scanning. Fig : Schematic diagram of He- Ne Laser Construction: He-Ne laser system consists of a narrow long quarts tube (m long and 0cm diameter) with two electrodes and the tube is filled with He-Ne gas mixture (0: ratio) at a pressure of mm (torr). The tube ends are fitted with Brewster s quartz window, the light travelling parallel to the axis incident at polarizing angle (tanθ B n). Two mirrors one fully silvered (m ) and the other partially silvered (m ) are located outside the ends of the tube such that they are perpendicular to the axis of the tube and perfectly parallel to each other. Mirrors along with 9

16 UNIT- I: LASERS AND OPTICAL FIBERS the quartz tube form the laser cavity. The entire arrangement is placed in an enclosure. Electrodes are connected to a powerful d.c. source. Working: When a d.c. voltage (at 000 V) is applied, the electric field ionizes some atoms of the gas mixture, due to this the electron are released and are accelerated towards the anode and helium and neon ions are accelerated towards cathode. Due to the smaller mass electrons acquire very high velocity. The free electrons while moving towards anode collide with atoms in their way. Collisions are more with the Helium (as Helium and Neon are in 0: ratio) and helium atoms are excited to the levels S and S level which are meta stable state of helium. This is electrical pumping. This collisions are of first kind i.e., He + e He * + e Where He: Helium in ground state, He * : Helium in excited state, e is lesser with energy less than that of e When current is continuously passed through the discharge tube more and more helium atoms are excited to state S and S. The energy levels S (0.66 ev) and S (9.78 ev) of neon atoms are very close to the helium levels S and S. Collisions of nd type takes place helium and neon atoms. In this process Neon atoms are excited to S and S levels and the Helium atoms to ground state. This is called resonance transfer of energy. He * + Ne He + Ne * 0

17 UNIT- I: LASERS AND OPTICAL FIBERS This is the pumping mechanism in the He-Ne laser and Ne atoms are active centers and population inversion sets with lower energy states. It is to be noted that there may be a chance of quick transition from S state to the ground state since it is radiatively connected to the ground state. Such case possible when the probability of decay (S state) to the ground state exceeds that to the p 4 levels which occurs between S and P levels at very low pressures. Since we are applying a high pressure of the order of a mm Hg, the transitions to the ground state undergo complete resonance trapping instead of escaping from the gas i.e, every time a photons is emitted is simply absorbed by another atom in the ground state and ends up in the excited S state through resonance energy transfer. Hence population increases at S levels through continuous process. This leads to increase the life time of S levels to ~00 ns when compared to the life time of P 4 levels of the order of 0 ns. Therefore a favorable lifetime ratio for producing the required population inversion satisfies the lasing action in Ne atoms. There are three types of laser transitions which are as follows, S P transmission with 6. nm (.9μm) which are in IR region S P transition with 6.8 nm (0.6 μm) which are in visible region S P transition with 5. nm (.5 μm) which are in IR region P S transition is spontaneous. The S level is a meta stable state and it should be quickly depopulated. To facilitate this the tube is made narrow and S to ground state transition takes place due to collision with the walls of the quartz tube. The Infra- red (IR) radiations of.9 μm and.5 μm are absorbed by quartz window and 6.8 nm component is amplified in the resonance cavity and comes out through the partially silvered mirror m. Semiconductor diode laser: Semiconductor lasers are very popular in the modern world. The semiconductor lasers are built using highly doped direct band gap semiconductors like Gallium arsenide (GaAs), Gallium arsenide phosphide (GaAsP) materials (Direct band gap semiconductors have high probability of recombination of holes and electrons). They are continuous laser. Fig: Schematic construction of semiconductor laser

18 UNIT- I: LASERS AND OPTICAL FIBERS Construction: A semiconductor laser in its simplest form consists of highly doped GaAs p-n junction. The junction is forward biased with d.c. supply. The p-n junction forms the active region. The resonant cavity is obtained by cleaving (polished) the front and back faces of the semiconducting material The cleaved/polished faces are perfectly parallel and flat. The back face is fully reflecting and front face is partially reflecting. Working: The energy band diagram of p-n junction consists of valence band (V) and conduction band (C ) separated by an energy gap E g. If the semiconductor is assumed to be at absolute zero temperature (T0K), the valence band is completely filled state and the conduction band is completely empty The hatched area represented in figure b (before inversion) shows that the valence band is in a completely filled state of energy. In operation, when the p-n junction is heavily forward biased with a large current, it raises the electrons from the valence band to the conduction band, but this is an unstable state and within a short time (0 - s) the electrons in the conduction band drop to the lowest level in that band itself. At the same time, the electrons near the top of the valence band will drop to the lowest unoccupied levels, leaving behind holes. As a result the lowest level of conduction band is full of electrons (indicated in fig c ) while the top of the valence band is full of holes (indicated by removed hatch in fig c ), which is an indication of population inversion between valence band and conduction band. The narrow depletion region becomes the active region in semiconductor. Now, at some instant, one of the excited electrons from the conduction band falls back into the valence band to recombine with a hole, and the energy associated with this recombination is emitted as a photon of light. This photon as it moves, stimulates the recombination of another excited electron with a hole and release another photon. These two photons are in

19 UNIT- I: LASERS AND OPTICAL FIBERS phase with each other and have the same wavelength, travel together in the same direction and get reflected at the end face. As they travel back they stimulate more electron hole recombination with the release of additional photons and become a part of the monochromatic and coherent laser beam. This beam gets resonated by traveling back and forth and finally leaves through the partially reflecting face. Since the current is being passed continuously (through forward bias), more electrons get excited, rise to the conduction band and new holes generate in the valence band. This situation maintains the population inversion, while the recombination of electron-hole pairs continuous with the generation of laser beam. Note: At very low current, the p n junction acts like an LED Characteristics of Laser beam Directionality: The design of the resonant cavity, especially the orientation of the mirrors to the cavity axis ensures that laser output is limited to only a specific direction. Since laser emits photons in a particular direction, the divergence is less when compared the other ordinary sources. Monochromacity: The laser beam is characterized by a high degree of monochromaticity (single wavelength or frequency) than any other conventional monochromatic sources of light. Ordinary light spreads over a wide range of frequencies, whereas laser contains only one frequency. The spectral bandwidth is comparatively very less when compared to ordinary light. Hence the degree of mono chromaticity is very high in lasers. Coherence: The degree of coherence of a laser beam is very high than the other sources. The light from laser source consists of wave trains that are in identical in phase. Laser radiation has high degree of special (with respect to Space) and temporal (with respective to time) coherence. High Intensity: The laser beam is highly intense. Since wave trains are added in phase and hence amplitudes are added. Laser light emits as a narrow beam and its energy is concentrated in a small region. Since all the energy is concentrated in the particular focus point, it is highly intense and bright. When laser beam is focused on a surface, the energy incident is of the order of millions of joules. Focus ability: Since laser is highly monochromatic, it can be focused very well by a lens. It is so sharp the diameter of the spot will be close to the wavelength of the focused light. It can be focused to a very small area 0.7µm. Since even laser is not ideally monochromatic the spot diameter in actual cases will be 00 to 50 times larger than the wavelength. Applications of Lasers Applications of lasers are wide spread over various scientific disciplines like Physics, Chemistry, Biology, Medicine, Communication, Holography, Material processing, Industry etc., In industry lasers are mainly used in cutting, welding and drilling processes.

20 UNIT- I: LASERS AND OPTICAL FIBERS Laser Light Welding: Welding Spot In the welding process the laser beam is focused on to a spot to be welded. Due to heat generation, the material melts over a tiny area on which beam focused. As a result an impurity such as oxides floats upon the surface. On cooling the material within becomes homogenous solid structure which makes weld strong. In the laser welding process No destruction occurs in the regions around the welded portion. Since it is a contact less process no foreign material comes in contact in to the welded joint. Further since the heat-affected zone is small, laser welding is ideal in areas such as microelectronics. CO lasers are the most popular ones in this particular application. Advantages: High quality and speed Weld dissimilar metals No need of filler Low heat affected Zone Welding can be done in inaccessible areas Cutting: Laser cutting is generally done assisted by gas blowing. In this process a jet of gas is issued through nozzle right at the spot where laser beam is focused. The combustion of the gas burns the metal thus reducing the laser power requirement for cutting. The tiny splinters along with the molten part of metal will be blown away by oxygen jet. The blowing action increases the depth and speed of the cutting process Laser Light Advantages: The quality of cutting is very high. Complicated D cutting profiles is possible. There will be no thermal damage and chemical change, when cutting is done to regions around cut area. High edge quality. No wear and tear of cutting tool. Minimal amount of mechanical distortion. 4

21 UNIT- I: LASERS AND OPTICAL FIBERS Drilling: Drilling of holes is achieved by subjecting materials to laser pulses of duration 0-4 to 0 - second. The intense heat generated over a short duration by the pulses evaporates the material locally, thus leaving a hole. Advantages: In conventional methods tools wear out while drilling whereas the problem doesn't exist with laser setup. Drilling can be achieved at any angle. Very fine holes of diameter 0. to 0.5mm could be drilled with a laser beam Very hard materials or brittle materials could be subjected to laser drilling since there is no mechanical stress with a laser beam. ND-YAG laser is used to drill holes in metals, where as CO laser is used in case of metallic and non -metallic materials. Measurement of pollutant in atmosphere: There are various types of pollutants in the atmosphere, they are oxides of nitrogen, carbon monoxide, sulphur dioxide etc.. In conventional technique the type and the concentration of pollutants in the atmosphere are determined by the chemical analysis. However this is not a real time data. This limitation can be overcome by using laser which yields a real time data. In the measurement of pollutant, laser is made use of the way RADAR system is used. Hence it is often referred to as LIDAR which means Light Detection And Ranging. A LIDAR can be employed to measure distance, altitude & angular coordinates of the object. In LIDAR Ruby laser is used as transmitting source which sends the laser beam through the desired region of the atmosphere. The transmitted light is reflected by the mirrors and reaches a receiver. The receiving part consists of concave mirror collects scattered light. The mirror focuses the light on to a photo-detector which converts the light energy into electrical energy. A narrowband filter is used to cuts off extraneous light and background noise. Then the electrical signal is fed to a computer a data processor, which gives information regarding distance, dimensions of the object etc. 5

22 UNIT- I: LASERS AND OPTICAL FIBERS Two methods can be employed to know the composition of the pollutants (i) Absorption technique (ii) Raman back scattering Absorption technique: When laser beam is made to pass through the atmosphere then molecules can either absorb light of certain frequencies or scatter light of certain frequencies. Depending upon the characteristic absorption pattern, the composition of the pollutant molecules can be determined. Raman back scattering: In this method laser light is passed through the sample, and the spectrum of the transmitted light is obtained. Since laser is highly monochromatic we expect to see only one line in the spectrum but due to Raman scattering, in the spectrum not one but several other lines of weak frequencies can be seen symmetrically. Additional spectral lines are called side bands and they are formed when the oscillating frequencies of the molecules of the gas are added to or subtracted from the incident's light's frequency. Different gases produce different side bands, the shift in frequencies are termed as Raman shifts. Thus by observing Raman spectra of the back-scattered light in the gas sample one can get the information about the composition of the pollutants. 6

23 UNIT- I: LASERS AND OPTICAL FIBERS OPTICAL FIBERS Optical fibers are the light guides used in optical communications as wave-guides. They are thin, cylindrical, transparent flexible dielectric fibers. They are able to guide visible and infrared light over long distances. The working structure of optical fiber consists of three layers. Core- the inner cylindrical layer which is made of glass or plastic. Cladding- which envelops the inner core. It is made of the same material of the core but of lesser refractive index than core. The core and the cladding layers are enclosed in a polyurethane jacket called sheath which safeguards the working structure of fiber against chemical reactions, mechanical abrasion and crushing etc. Propagation mechanism in Optical fiber: In optical fibers light waves can be guided through it, hence are called light guides. The cladding in an optical fiber always has a lower refractive index (RI) than that of the core. The light signal which enters into the core can strike the interface of the core and the cladding at angles greater than critical angle of incidence because of the ray geometry. The light signal undergoes multiple total internal reflections within the fiber core. Since each reflection is a total internal reflection, the signal sustains its strength and also confines itself completely within the core during propagation. Thus, the optical fiber functions as a wave guide. Numerical Aperture and Ray Propagation in the Fiber: Consider an optical fiber consists of core and cladding material placed in air medium. Let n 0, n and n be the refractive indices of surrounding air medium, core and cladding material respectively. The RI of cladding is always lesser than that of core material (n < n ) so that the light rays propagate through the fiber. Let us consider the special case of ray which suffers critical incident at the core cladding interface. The ray travels along AO entering into the core at an angle of θ 0 with respect to the fiber axis. Let it be refracted along OB at an angle θ in the core and further proceed to fall at critical angle of incidence ( 90-θ ) at B on the interface of core and cladding. Since it is critical angle of incidence, the refracted ray grazes along core and cladding interface. 7

24 UNIT- I: LASERS AND OPTICAL FIBERS It is clear from the figure that a ray that enters at an angle of incidence less than θ 0 at O, will have to be incident at an angle greater than the critical angle at the core-cladding interface, and gets total internal reflection in the core material. When OA is rotated around the fiber axis keeping θ 0 same, it describes a conical surface, those rays which are funneled into the fiber within this cone will only be totally internally reflected and propagate through the fiber. The cone is called acceptance cone. The angle θ 0 is called the wave guide acceptance angle or the acceptance cone halfangle which is the maximum angle from the axis of optical fiber at which light ray may enter the fiber so that it will propagate in core by total internal reflection. Sin θ 0 is called the numerical aperture (N.A.) of the fiber. It determines the light gathering ability of the fiber and purely depends on the refractive indices of core, cladding and surrounding medium. Let n 0, n and n be the refractive indices of surrounding medium, core and cladding respectively for the given optical fiber. By applying the Snell s law at O, n 0 sinθ 0 n sinθ sinθ 0 n sinθ n () 0 At the point Bon the core and cladding interface, the angle of incidence 90-θ Applying Snell s law at B n sin(90 θ 0 ) n sin90 or n cosθ n cosθ n () n From equation () sinθ 0 n sinθ n n ( cos θ 0 n ) 0 n n n sinθ 0 n n n 0 n n x n n n 0 n 0 If the medium surrounding the fiber is air, then n 0, Therefore, sinθ 0 n n sinθ 0 Numerical aperture: NA Therefore, NA sinθ 0 n n If θ i is the angle of incidence of an incident ray, then the ray will be able to propagate, If θ i < θ 0 or if sin θi < sin θ0 or sin θ i < NA Fraction Index Change ( ): The fractional index change is the ratio of the difference in the refractive indices between the core and the cladding to the refractive index of core of an optical fiber. It is also known as relative core clad index difference, denoted by. If n and n are the refractive indices of core and cladding, then, 8

25 UNIT- I: LASERS AND OPTICAL FIBERS Relation between NA and ( n n n ) We have RI change Δ n n n Or Δn (n n ) () We know that, N. A. ( n ( n + n + n n n ) n )( n n ) Since n ~ n, (n + n ) n Therefore, N. A. N. A. n n Though an increase in the value of increases NA and thus enhances the light gathering capacity of the fiber, we cannot increase to a very large value, since it leads to a phenomenon called intermodal dispersion which causes signal distortion. Modes of Propagation: The possible number of paths of light in an optical fiber determines the number of modes available in it. It also determines the number of independent paths for light that a fiber can support for its propagation without interference and mixing. We may have a single mode fiber supporting only one signal at a time or multimode fiber supporting many rays at a time. Such number of modes supported for propagation in the fiber is determined by a parameter called V-number. If the surrounding medium is air then the V- number is given by V πd (n λ n ) π d or V ( NA) λ where d is the core diameter, n is the refractive index of core, n is the refractive index of the cladding and λ is the wavelength of the light propagating through the fiber. V π r λ n n (r)π λ Where k is the propagation constant (k π/λ) If fiber is surrounded by a medium of refractive index n 0, then π d V λ n n n 0 ) n n r. k n n. For single mode fiber, the number of modes supported by step index fiber is For graded index fiber, the number of modes are ~ V /4 V 9

26 UNIT- I: LASERS AND OPTICAL FIBERS Types of Optical fibers: The optical fibers are classified under categories. They are a) Step index Single mode fiber (SMF) b) Step index multi mode fiber (MMF) c) Graded index Multi Mode Fiber (GRIN) This classification is done depending on the refractive index profile and the number of modes that the fiber can guide. Refractive Index Profile (RI): Generally in any types of optical fiber, the refractive index of cladding material is always constant and it has uniform value throughout the fiber. But in case of core material, the refractive index may either remain constant or subjected to variation in a particular way. This variation of RI of core and cladding materials with respect to the radial distance from the axis of the fiber is called refractive index profile. This can be represented as follows, RI profile of Step index fiber X axis: Radial distance from the centre of the fiber Intermodal dispersion: In a single mode fiber, a single light ray (called single mode) propagates through the fiber. Since other modes are not interfering with this mode, intermodal dispersion is zero in single mode fiber. In case of multimode fiber, intermodal dispersion is maximum. Different modes are propagating with different distance hence there is a time delay in the propagation Due to this, there is a possibility of dispersion in their path and losses their information. This is called intermodal a) Step index Single mode fiber (SMF): A single mode fiber has a core material of uniform refractive index (RI) value. Similarly cladding also has a material of uniform RI but of lesser value. This results in a sudden increase in the value of RI from cladding to core. Thus its RI profile takes the shape of a step. The diameter value of the core is about 8 to 0 µm and external diameter of cladding is 60 to 70 µm. Because of its narrow core, it can guide just a single mode as shown in Figure. Hence it is called single mode fiber. Single mode fibers are most extensively used ones and they constitute 80% of all the fibers that are manufactured in the world today. They need lasers as the source of light. Though less expensive, it is very difficult to splice them (joining of optical fibers). Since single mode is propagating through the fiber, intermodal dispersion is zero in this fiber. They find particular application in submarine cable system. 8 to 0 µm 60 to 70 µm 0

27 UNIT- I: LASERS AND OPTICAL FIBERS refractive index profile Radial distance Ray propagation Cladding Core Input and output pulse Fig: Step Index Single Mode Fiber light ray b) Step index multimode fiber (MMF): The geometry of a step-index multimode fiber is as shown in below figure. It s construction is similar to that of a single mode fiber but for the difference that, its core has a much larger diameter by the virtue of which it will be able to support propagation of large number of modes as shown in the figure. Its refractive index profile is also similar to that of a single mode fiber but with larger plane regions for the core. The step-index multimode fiber can accept either diode laser or LED (light emitting diode) as source of light. It is the least expensive of all. Since multi modes are propagating through this fiber with different paths, intermodal dispersion is maximum this fiber. Its typical application is in data links which has lower bandwidth requirements. 50 to 00 µm 00 to 50 µm RI profile Radial distance

28 UNIT- I: LASERS AND OPTICAL FIBERS Cladding Core Fig. Step index multimode fiber Input and output pulse c) Graded index multimode fiber (GRIN) Graded index multimode fiber is also denoted as GRIN. The geometry of the GRIN multimode fiber is same as that of step index multimode fiber. Its core material has a special feature that its refractive index value decreases in the radially outward direction from the axis and becomes equal to that of the cladding at the core-cladding interface. But the RI of the cladding remains uniform. Its RI profile is also shown in figure. Either a diode laser or LED can be the source for the GRIN multimode fiber. It is most expensive of all. Its splicing could be done with some difficult. Since multi modes are propagating with different velocity and meets at nodes position, intermodal dispersion is minimized in this fiber. Its typical application is in the telephone trunk between central offices. 50 to 00 µm 00 to 50 µm RI profile Radial distance Input and output pulse Fig. Graded index multimode fiber

29 UNIT- I: LASERS AND OPTICAL FIBERS Differences between single and multimode fibers: Single mode fiber Multi mode fiber Only one mode can be propagated Smaller core diameter Low dispersion of signal Can carry information to longer distances Launching of light and connecting two fibers are difficult Allows large number of modes for light to pass through it Larger core diameter More dispersion of signal Information can be carried to shorter distances only Launching of light and connecting of fibers is easy Differences between step and graded index fibers: Step index fiber Graded index fiber Refractive index of core is uniform Propagation of light is in the form of meridional rays Step index fibers has lower bandwidth Distortion is more (in multimode) No. of modes for propagation N step V / Attenuation in optical fibers: Refractive index of core is not uniform Propagation of light is in the form of skew rays Graded index fibers has higher bandwidth Distortion is less No. of modes for propagation N grad V /4 The total energy loss suffered by the signal due to the transmission of light in the fiber is called attenuation. The important factors contributing to the attenuation in optical fiber are i) Absorption loss ii) Scattering loss iii) Bending loss iv) Intermodal dispersion loss and v) coupling loss. Attenuation is measured in terms of attenuation co-efficient and it is the loss per unit length. It is denoted by symbol α. Mathematically attenuation of the fiber is given by, P 0 log out 0 P α In db/km L Where P out and P in are the power output and power input respectively, and L is the length of the fiber in km. Therefore, Loss in the optical fiber α x L

30 UNIT- I: LASERS AND OPTICAL FIBERS. Absorption loss: There are two types of absorption; one is absorption by impurities and the other intrinsic absorption. In the case of first type, the type of impurities is generally transition metal ions such as iron, chromium, cobalt and copper. During signal propagation when photons interact with these impurities, the electron absorbs the photons and get excited to higher energy level. Later these electrons give up their absorbed energy either as heat energy or light energy. The re-emission of light energy is of no use since it will usually be in a different wavelength or at least in different phase with respect to the signal. The other impurity which would cause significant absorption loss is the OH - (Hydroxyl) ion, which enters into the fiber constitution at the time of fiber fabrication. In the second type i.e., intrinsic absorption it is the absorption by the fiber itself, or it is the absorption that takes place in the material assuming that there are no impurities and the material is free of all inhomogeneities and this sets the lowest limit on absorption for a given material.. Scattering loss: The signal power loss occurs due to the scattering of light energy due to the obstructions caused by imperfections and defects, which are of molecular size, present in the body of the fiber itself. The scattering of light by the obstructions is inversely proportional to the fourth power of the wavelength of the light transmitted through the fiber. Such a scattering is called Rayleigh scattering. The loss due to the scattering can be minimized by using the optical source of large wavelength.. Bending losses (radiation losses): There are two types of bending losses in optical fiber a) macroscopic and b)microscopic bending loss. a) Macroscopic bends: Macroscopic bends occurs due the wrapping of fiber on a spool or turning it around a corner. The loss will be negligible for small bends but increases rapidly until the bending reaches a certain critical radius of curvature. If the fiber is too bend, then there is possibility of escaping the light ray through cladding material without undergoing any total internal reflection at core-cladding interface. b) Microscopic bends: This type of bends occurs due to repetitive small-scale fluctuations in the linearity of the fiber axis. Due to non-uniformities in the manufacturing of the fiber or by non-uniform lateral pressures created during the cabling of the fiber. The microscopic bends cause irregular reflections at core-cladding interface and some of them reflects back or leak through the fiber. 4

31 UNIT- I: LASERS AND OPTICAL FIBERS This loss could be minimized by extruding a compressible sheath over the fiber which can withstand the stresses while keeping the fiber relatively straight. 4. Coupling losses: Coupling losses occur when the ends of the fibers are connected. At the junction of coupling, air film may exist or joint may be inclined or may be mismatched and they can be minimized by following the technique called splicing. Applications of Optical Fibers: Point-to-point Communication The use of optical fibers in the field of communication has revolutionized the modern world. An optical fiber acts as the channel of communication (like electrical wires), but transmits the information in the form of optical waves. A simple p to p communication system using optical fibers is illustrated in the figure. The main components of p to p communication is ) An optical transmitter, i.e., the light source to transmit the signals/pulses ) The communication medium (channel) i.e., optical fiber ) An optical receiver, usually a photo cell or a light detector, to convert light pulses back into electrical signal. 5

32 UNIT- I: LASERS AND OPTICAL FIBERS The information in the form of voice or video to be transmitted will be in an analog electric signal format. This analog signal at first converted into digital electric (binary) signals in the form of electrical pulses using a Coder or converter and fed into the optical transmitter which converts digital electric signals into optic signals. An optical fiber can receive and transmit signals only in the form of optical pulses. The function of the light source is to work as an efficient transducer to convert the input electrical signals into suitable light pulses. An LED or laser is used as the light source for this purpose. Laser is more efficient because of its monochromatic and coherent nature. Hence semiconductor lasers are used for their compact size and higher efficiency. The electrical signal is fed to the semiconductor laser system, and gets modulated to generate an equivalent digital sequence of pulses, which turn the laser on and off. This forms a series of optical pulses representing the input information, which is coupled into the optical fiber cable at an incidence angle less than that of acceptance cone half angle of the fiber. Next the light pulses inside the fiber undergo total internal reflection and reach the other end of the cable. Good quality optical fibers with less attenuation to be chosen to receive good signals at the receiver end. The final step in the communication system is to receive the optical signals at the end of the optical fiber and convert them into equivalent electrical signals. Semiconductor photodiodes are used as optical receivers. A typical optical receiver is made of a reverse biased junction, in which the received light pulses create electron-hole charge carriers. These carriers, in turn, create an electric field and induce a photocurrent in the external circuit in the form of electrical digital pulses. These digital pulses are amplified and re-gain their original form using suitable amplifier and shaper. The electrical digital pulses are further decoded into an analogues electrical signal and converted into the usable form like audio or video etc., As the signal propagates through the fiber it is subjected to two types of degradation. Namely attenuation and delay distortion. Attenuation is the reduction in the strength of the signal because power loss due to absorption and scattering of photons. Delay distortion is the reduction in the quality of the signal because of the spreading of pulses with time. These effects cause continuous degradation of the signal as the light propagates and may reach a limiting stage beyond which it may not be retrieve information from the light signal. At this stage repeater is needed in the transmission path. Reciever Amplifier Transmitter An optical repeater consists of a receiver and a transmitter arranged adjacently. The receiver section converts the optical signal into corresponding electrical signal. Further the electrical signal is amplified and recast in the original form and it is sent into an optical transmitter section where the electrical signal is again converted back to optical signal and then fed into an optical fiber. Finally at the receiving end the optical signal from the fiber is fed into a photo detector where the signal is converted to pulses of electric current which is then fed to decoder which converts the sequence of binary data stream into an analog signal which will be the same information which was there at the transmitting end. 6

33 UNIT- I: LASERS AND OPTICAL FIBERS Advantages over conventional communication: Advantages of optical fibers over conventional cables are ) Large Bandwidth: Optical fibers have a wider bandwidth (when compared to conventional copper cables). This helps in transmitting voice, video and data on a single line and at very fast rates (0 4 bps as compared to about 0 4 bps in ordinary communication line) ) Electromagnetic Interference (EMI): EMI and disturbance in the transmission is a very common phenomenon in ordinary copper cables. However, the optical fiber cables are free from EMI, since Electromagnetic radiation has no effect on the optical wave. Hence, there is no need to provide specially shielded conditions for the optical fiber. ) Low attenuation: Compared to metallic cables, optical fibers have a low attenuation level (as they are relatively independent of frequency). The loss in optical fibers is very low, of the order of 0. to 0.5 db/km of transmission. 4) Electrical Hazards: Since, optical fibers carry only the light signals, there are no problems of short-circuiting and shock hazards. 5) Security: Unlike electrical transmission lines, there is no signal radiation around the optical fiber, hence the transmission is secure. The tapping of the light waves, if done, leads to a loss of signal and can be easily detected. 6) Optical fiber cables are small in size, light weight and have a long life. Disadvantages The disadvantages in the communication systems using optical fibers are Fiber loss is more at the joints if the joints do not match (the joining of the two ends of the separate fibers are called splicing) Attenuation loss is large as the length of the fiber increases. Repeaters are required at regular interval of lengths to amplify the weak signal in long distance communication. Sever bends will increase the loss of the fiber. Hence, the fiber should be laid straight as far as possible and avoid severe bends. Note: Point to Point haul communication system is employed in telephone trunk lines. This system of communication covers the distances 0 km and more. Long-haul communication has been employed in telephone connection in the large cities of New York and Los Angeles. The use of single mode optical fibers has reduced the cost of installation of telephone lines and maintenance, and increased the data rate. Local Area Network (LAN) Communication system uses optical fibers to link the computer-oriented communication within a range of or km. Community Antenna Television (CATV) makes use of optical fibers for distribution of signal to the local users by receiving a multichannel signal from a common antenna. 7

34 UNIT- I: LASERS AND OPTICAL FIBERS. Medical application Endoscopes are used in the medical field for image processing and retrieving the image to find out the damaged part of the internal organs of the human body. It consists of bundle of optical fibers of large core diameter whose ends are arranged in the same sequence. Endoscope is inserted to the inaccessible damaged part of the human body. When light is passed through the optical bundle the reflected light received by the optical fibers forms the image of the inaccessible part on the monitor. Hence, the damage caused at that part can be estimated and also it can be treated.. Industry Optical fibers are used in the design of Boroscopes, which are used to inspect the inaccessible machinery parts. The working principle of boroscope is same as that of endoscopes.. Domestic Optical fiber bundles are used to illuminate the interior places where the sunlight has no access to reach. It can also be used to illuminate the interior of the house with the sunlight or the incandescent bulb by properly coupling the fiber bundles and the source of light. They are also used in interior decorating articles. Sl. No SHORT ANSWERE QUESTIONS CO s. Define the terms: a. Spontaneous emission. b. Stimulated Emission. c. Active medium. d. Population inversion.. Explain any one of the industrial applications of laser.. Give any two differences between the laser light and ordinary light. Sl. No LONG ANSWSER QUESTIONS 4. Explain the requisites of a laser system. 5. With energy level diagram of He-Ne gas, explain the working of He-Ne laser. 6. Discuss the conditions required for laser action. 7. Write a note on measurement of pollutants in atmosphere using laser. 8. Explain the three processes which take place when radiation interacts with matter. 9. Explain the terms stimulated emission and population inversion. Obtain & an expression for energy density of photons in terms of Einstein s co-efficient. 0. Explain the characteristics of a laser beam.. Explain the working principle of a semiconductor laser using band diagram and discuss its advantages. 8

35 UNIT- I: LASERS AND OPTICAL FIBERS S. No Questions CO Give reason Optical fibers are immune to electromagnetic interference. Intermodal dispersion is minimum in GRIN compared to MMSI fiber. Repeaters are used in the path of optical fibers in point to point communication system. Explain the terms a. Acceptance angle b. Cone of acceptance c. Numerical aperture d. Modes of propagation e. Attenuation. Explain attenuation losses in optical fibers. 4 Write any two advantages of optical fiber communication over normal communication. 5 Explain propagation mechanism in optical fibers. 6 Distinguish between step and graded index fibers. LONG ANSWER QUESTIONS S. No Questions CO s With the help of ray diagram, explain the working principle of optical fibers. & Derive an expression for acceptance angle of an optical fiber in terms of & refractive indices. What is numerical aperture? Obtain an expression for numerical aperture in terms of refractive indices of core and cladding and arrive at the condition for propagation. 4 Explain the terms ) modes of propagation and ) types of optical fibers 5 What is attenuation? Explain the different losses in optical fibers. 6 With the help of a block diagram explain point to point communication. 7 Discuss the advantages and disadvantages of an optical communication system. 9

36 UNIT- I: LASERS AND OPTICAL FIBERS PROBLEMS:. The ratio of population of two energy levels out of which one corresponds to metastable state is.059x0-0. Find the wavelength of light emitted at 0K. N 0.059x0, T0K, λ? N Constants h6.6x0-4 Js, K.8x0 - J/K, C x0 8 m/s Using the relation for Boltzmann s factor N hν hc e e N kt λkt λ 6.8nm.. Calculate the ratio of i) Einstein Coefficients, ii) Stimulated to spontaneous emissions, for a system at 00K in which radiations of wavelength.9µm are emitted. A 8πhν 8πh 5 6.x0 B c λ A 5 Since B B we can write 6.x0 B We have B N Uν B Rate of stimulated emission/rate of spontaneous emission Uν AN A 8πhν But A U ν Uν hν hν c B kt kt e e B Therefore rate of stimulated emission/rate of spontaneous emission A A x B e hν kt hν 0-5 kt e. Calculate on the basis of Einstein s theory, the number of photons emitted per second by a He-Ne laser source emitting light of wavelength 68A o with an optical power of 0mW. E n E nhc P t t λt n P λ Hence t hc.8x0 6 0

37 UNIT- I: LASERS AND OPTICAL FIBERS 4. Calculate the numerical aperture, relative RI difference, V- number and number of modes in an optical fibre of core diameter 50µm. Core and cladding Refractive indices.4 and.40 at λ 80nm. (NA) (n - n ) n n n NA Hence no of modesv / 5. π d ( n n ) V λ 5. An optical fibre has clad of RI.50 and NA 0.9. Find the RI of core and the acceptance angle. (NA) n - n θ o sin (0.9) n - (.50).96 o n The NA of an OF is 0. when surrounded by air. Determine the RI of its core. Given The RI of cladding as.59. Also find the acceptance angle when it is in a medium of RI.. (NA) (n n ) θ o 8.64 o n A glass clad fibre is made with core glass of RI.5 and cladding is doped to give a fractional index difference of Determine a) The cladding index. b) The critical internal reflection angle. c) The external critical acceptance angle d) The numerical aperture 8. The attenuation of light in an optical fibre is estimated at.db/km. What fractional initial intensity remains after km & 6km? L : P out /P in 6.% L6 : P out /P in 4.79% 9. Find the attenuation in an optical fibre of length 500m, when a light signal of power 00mW emerges out of the fibre with a power 90mW. pout 0log0( ) Pin α L α 0.95dB/km 0. A semiconductor laser emits green light of 55 nm. Find out the value of its band gap. E g hc/λ.5 ev

38 UNIT- I: LASERS AND OPTICAL FIBERS. The probability of spontaneous transition is given as in a laser action which results with the radiation of 6.8 nm wavelength. Calculate the probability of stimulated emission. (Ans..x0 ). Calculate the critical angle if the refractive indices of optical fiber are.5 &.48. (Ans. θ c 80.6 o ). The optical fiber power after propagating through a fiber of.5 km length is reduced to 5% of its original value. Compute the fiber loss in db/km. (Ans. 4 db/km)

39 UNIT- II: QUANTUM MECHANICS UNIT II: QUANTUM MECHANICS Introduction: At the beginning of the 0 th century, Newton s laws of motion were able to successfully describe the motion of the particles in classical mechanics (the world of large, heavy and slow bodies) and Maxwell s equations explained phenomena in classical electromagnetism. However the classical theory does not hold in the region of atomic dimensions. It could not explain the stability of atoms, energy distribution in the black body radiation spectrum, origin of discrete spectra of atoms, etc. It also fails to explain the large number of observed phenomena like photoelectric effect, Compton Effect, Raman Effect, Quantum Hall effect, superconductivity etc. The insufficiency of classical mechanics led to the development of quantum mechanics. Quantum mechanics gives the description of motion and interaction of particles in the small scale atomic system where the discrete nature of the physical world becomes important. With the application of quantum mechanics, most of the outstanding problems have been solved. Black Body Radiation Black-body radiation is electromagnetic radiation emitted by a black body held at constant, uniform temperature. Black Body Spectrum: It is a graph showing the variation of the energy of the black body radiations as a function of their wavelengths or frequencies. The energy distribution in the black body spectrum is explained by Wien s distribution law in the lower wavelength region and Rayleigh Jeans law explains the energy distribution in the larger wavelength region. Wien s law: E λ Rayleigh Jeans law λ dλ 5 Aλ e 4 E dλ 8πk Tλ dλ B λt dλ Fig : Blackbody Radiation spectrum

40 UNIT- II: QUANTUM MECHANICS Neither, Wien s law nor Rayleigh- jean s law could explain the energy distribution in the entire blackbody spectrum. The energy distribution in the entire blackbody spectrum was successfully explained by Max. Planck by quantum Theory. Planck s quantum theory The energy distribution in the black body radiation spectrum was successfully explained by Max Planck in the year 900. According to Planck s quantum theory thermal energy is not emitted or absorbed continuously, but it is emitted or absorbed in discrete quantities called quanta. Each quantum has an energy hν where h is the Planck s constant and υ is the frequency of the radiation. Applying the Planck s quantum theory an expression for the energy distribution in the black body spectrum was obtained and it is called Planck s formula. The Planck s formula is as follows 8πhc Eλdλ dλ 5 ( hc ) λ λkt e Where k is the Boltzmann s constant; h- Planck s constant and c is the velocity of light, λ is the wavelength of the black-body radiation and ω is the angular frequency of the radiation. Photoelectric effect: When the light of a suitable wavelength shines on certain materials, then electrons are spontaneously emitted from the surface of material. It can be observed in any material but most readily and steadily in metals and good conductors. This phenomenon is known as the photoelectric effect. The materials that exhibit photoelectric effect are called photosensitive materials and the emitted electrons are called photoelectrons. Heinrich Hertz first observed this phenomenon in 887. The electrons are emitted only when the photons reach or exceed a threshold frequency (energy) and below that threshold, no electrons are emitted from the metal regardless of the light intensity or the length of time of exposure to the light. To explain this phenomenon, Albert Einstein proposed that light be seen as a collection of discrete bundles of energy (photons), each with energy hυ, where υ is the frequency of the light that is being quantized and h is known as the Planck constant. Einstein s photoelectric equation: Einstein, in 905, proposed that the light energy is localized in small packets similar to the Planck s idea of quanta, and named such packets as photons. According to Einstein, in photoelectric effect one photon is completely absorbed by one electron, which thereby gains the quantum of energy and may be emitted from the metal. Thus the photon energy is used in the following two parts: i). A part of its energy is used to free the electron from the atoms of the metal surface. This energy is known as a photoelectric work function of metal (W o ) ii) The other part is used in giving kinetic energy (½ mv ) to the electron. Thus hν W + mv o where v is the velocity of the emitted electron. 4

41 UNIT- II: QUANTUM MECHANICS This equation is known as Einstein s photoelectric equation. When the photon s energy is of such a value that it can just liberate the electron from metal, then the kinetic energy of the electron will be zero. Then the above equation reduces to h ν o W o, where ν o is called the threshold frequency. Threshold frequency is defined as the minimum frequency which can cause photoelectric emission. Below this frequency no emission of electron takes place. Compton Effect: When a monochromatic beam of high frequency radiation (X rays, γ rays, etc.) is scattered by a substance, then the scattered radiation contains two components - one having a lower frequency or greater wavelength called as modified radiation and the other having the same frequency or wavelength called as unmodified radiation. This phenomenon is known as Compton effect and was discovered by Prof. A.H. Compton in 9. The process of recoiling of electron and scattering of photon is as shown in the following figure: Scattered photon Incident photon Φ----- is the recoil angle θ is the scattering angle. Electron at rest θ Φ Recoil electron Schematic diagram of Compton Effect According to the quantum concept of radiation, the radiation is constituted by energy packets called photons. The energy of photon is hν, where h is Planck s constant and ν is the frequency of radiation. The photons move with velocity of light c, possess momentum hν/c and obey all the laws of conservation of energy and momentum. According to Compton, the phenomenon of scattering is due to an elastic collision between two particles, the photon of incident radiation and the electron of the scatterer. When the photon of energy hν collides with the electron of the scatterer at rest, it transfers some energy to the electron, i.e., it loses the energy. The scattered photon will therefore have a smaller energy and consequently a lower frequency (ν ) or greater wavelength (λ ) than that of the incident photon. The observed change in wavelength of the scattered radiation is known as Compton Shift. In the scattering process, the electron gains kinetic energy and thus recoils with a velocity v. The change in wavelength of the photon scattered through an angle θ is given by ' h λ ( λ λ) [ cos ϑ] mc 0 5

42 UNIT- II: QUANTUM MECHANICS Wave and particle duality of radiation: To understand the wave and particle duality, it is necessary to know what a particle is and what a wave is. A particle is a localized mass and it is specified by its mass, velocity, momentum, energy, etc. In contrast a wave is a spread out disturbance. A wave is characterised by its wavelength, frequency, velocity, amplitude, intensity, etc. It is hard to think mass being associated with a wave. Considering the above facts, it appears difficult to accept the conflicting ideas that radiation has a wave particle duality. However this acceptance is essential because the radiation exhibits phenomena like interference, diffraction, polarization, etc., and shows the wave nature and it also exhibits the particle nature in the black-body radiation effect, photoelectric effect, Compton Effect etc. Radiation, thus, sometimes behave as a wave and at some other time as a particle, this is the wave particle duality of radiation. De Broglie s concept of matter waves: Louis de-broglie in 94 extended the wave particle dualism of radiation to fundamental entities of physics, such as electrons, protons, neutrons, atoms, molecules, etc. de-broglie put a bold suggestion that like radiation, matter also has dual characteristic, at a time when there was absolutely no experimental evidence for wavelike properties of matter waves. de Broglie Hypothesis of matter waves is as follows. In nature energy manifests itself in two forms, namely matter and radiation. Nature loves symmetry. As radiation can act like both wave and a particle, material particles (like electrons, protons, etc.) in motion should exhibit the property of waves. These waves due to moving matter are called matter waves or de-broglie waves or pilot waves. Wavelength of matter waves: The concept of matter waves is well understood by combining Planck s quantum theory and Einstein s theory. Consider a photon of energy E, frequency γ and wavelength λ. hc By Planck s theory E hγ λ By Einstein s mass-energy relation E mc By equating and rearranging the above equations, we get hc mc λ h h λ mc p Where, p is the momentum of the photon and h is a Planck s constant. Now consider a particle of mass m moving with a velocity v and momentum p. According to the do-broglie hypothesis matter also has a dual nature. Hence the wavelength λ of matter waves is given by 6

43 UNIT- II: QUANTUM MECHANICS λ h mv This is the equation for the de-broglie wavelength. h p de Broglie wavelength of an electron Consider an electron of mass m accelerated from rest by an electric potential V. The electrical work done (ev) is equal to the kinetic energy E gained by the electron. E ev Also E mv m v me mv me Therefore wavelength of electron wave h h λ p mv h λ me But E ev h h λ me mev V Substituting the values of e, m and h, we get h o.8a me.8 λ A o V Note: Instead of a an electron, if a particle of charge q is accelerated through a potential difference V, then h λ mqv Properties of matter waves and how they are different from electromagnetic waves:. Lighter the particle, greater would be the wavelength of matter waves associated with it.. Smaller the velocity of the particle, greater would be the wavelength.. For p 0, λ is infinity ie., the wave becomes indeterminate. This means that matter waves are associated with moving particles only. 4. Matter waves are produced by charged or uncharged particles in motion. Whereas electromagnetic waves are produced only by a moving charged particle. Hence matter waves are non electromagnetic waves. 7

44 UNIT- II: QUANTUM MECHANICS 5. In an isotropic medium the wavelength of an electromagnetic wave is a constant, whereas wavelength of a matter wave changes with the velocity of the particle. Hence matter waves are non- electromagnetic waves. 6. A particle is a localized mass and a wave is a spread out disturbance. So, the wave nature of matter introduces a certain uncertainty in the position of the particle. 7. Matter waves are probability waves because waves represent the probability of finding a particle in space. Wave packet: A wave packet refers to the case where two (or more) waves exist simultaneously. A wave packet is often referred to as a wave group. This situation is permitted by the principle of superposition. In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wave numbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere. Each component wave function, and hence the wave packet, are solutions of a wave equation. Depending on the wave equation, the wave packet's profile may remain constant (no dispersion) or it may change (dispersion) while propagating. Quantum mechanics ascribes a special significance to the wave packet. It is interpreted as probability amplitude. The modulus square of the probability amplitude describes the probability density that a particle in a particular state will be measured to have a given position or momentum. The equation that describes the evolution of this wave packet is the Schrödinger equation. It can be Time independent or Time dependent. A monochromatic wave has velocity called the phase velocity given by ω v p υλ where ω k π is the angular frequency, k is the wave number, υ is the frequency. However, if we λ have a compound wave(wave packet) that is composed of individual waves with a range of frequencies, each individual wave has its phase velocity, but the amplitudes of the waves add up to produce a wave packet which has a velocity all of its own. This velocity is called the group velocity and is usually different from the individual phase velocities of the waves that ω ω make up the packet. It is given by the expression v g k k Schematic diagram of Wave Packet formed by superposition of waves 8

45 UNIT- II: QUANTUM MECHANICS Relation between group velocity and phase velocity: ω v p > ω v pk k ω ( v pk) v p π v p λ vg v p + k v p + ( ) v k k k λ λ k π v p π v p vg v p + ( ) ( λ v p λ λ λ 4π ) λ Relation between group velocity and particle velocity: p π v p π + ( ) ( ) λ λ k We know that E hυ hω π...( ) while h p h hk π...( ) (Where E, p is the Energy, momentum of the λ π k particle). Thus we can show that the group velocity is given by: ω E ( p / m) p mv particle v g v particle k p p m m Thus we have shown that for a free particle, the group velocity equals the particle velocity, while the phase velocity is half the group velocity. Relation between group velocity, phase velocity and light velocity: v p ω k E p mc mv particle c v particle c v g Thus we get the relation v. v c g p Application of matter waves: Scanning Electron Microscope The wave nature of electrons has been practically applied in diverse fields, of these the most important is in the development of scanning electron microscope, having magnification and resolving power much higher than the optical microscope. Principle: The smallest distance resolved by a microscope is called Limit of resolution (Δx) and the resolving power of a microscope is the reciprocal of the limit of resolution. λ x µ sinθ Resolving Power /Δx μsinθ/λ where µ sinθ is called the numerical aperture of the objective. Thus shorter the wavelength(λ) of the radiation, greater is the resolving power. 9

46 UNIT- II: QUANTUM MECHANICS It is this resolving power that sets a limit to the magnification of the microscope, because a mere increase in the magnifying power without a corresponding increase in the resolving power would only produce large blurred image. Optical microscope uses a light beam of wavelength 600 nm so that the smallest distance resolvable 00 nm(assuming numerical aperture ). Hence, if instead of light waves we could use electron beam the resolving power can be increased by more than 0 6 fold. Electrons are known to exhibit wave characteristics under suitable conditions, the wavelength of electron wave is governed by de Broglie relation. h λ mev The magnification can be increased by making use of de Broglie waves associated with electrons, accelerated to approximately 75 kv which has a wavelength of 4.48x0 - m which is roughly 0 5 times shorter than visible radiation. In optical microscopes, light waves produce the image and magnification is obtained by optical lenses. An electron microscope generally uses electron waves and the electron beam is focused by suitable magnetic lenses. Magnetic lens: The magnetic fields which are axially symmetric have a focusing effect on the electron beam passing through them. The axially symmetric magnetic fields are produced by short solenoids. Such solenoids are called magnetic lens. The focal length of such magnetic lens depends on the quality of coil and current. By adjusting the current through the coils the intensity of the magnetic field and the focal length can be adjusted. It is widely used in electron microscopes. Scanning Electron Microscopy (SEM) The SEM uses electrons instead of light to form an image. SEM is a powerful and commonly used instrument, in both academia and industry for imaging the surfaces of almost any material with a resolution down to about nm. Most of the scanning electron microscopes have magnification ranges from X0 to X SEM image gives a characteristic three-dimensional appearance and useful for adjudging the surface structure of the sample. Working Principle of the SEM: A beam of electrons are produced by heating a thermionic filament. And it is operated with an acceleration voltage of KV to 40 KV for the electrons. The electron beam follows a vertical path and it makes its way through a set of magnetic lenses which focuses and directs the beam towards the sample. These electrons interact with the sample to a depth of ~μm and generate a number of different types of signals, which are emitted from the area of the specimen where the electron beam is impinging. The induced signals are detected and the intensity of one of the signals (at a time) is amplified and used as the intensity of a pixel on the image on the computer screen. The electron beam, then moves to the next position on the sample and the detected intensity gives the intensity in the second pixel and so on. This 40

47 UNIT- II: QUANTUM MECHANICS produces an image with depth-of-field which is usually times better than that of an optical microscope, and also enables a three-dimensional image to be obtained. Construction: A schematic of SEM is shown in the following figure. Fig. Basic construction of SEM. Electrons are thermionically emitted from a tungsten filament (cathode) are drawn to an anode and focused by two successive magnetic lenses into a beam with a very fine spot size that is typically 0Å in diameter. Pairs of scanning coils located at the objective (magnetic) lens deflect the beam either linearly or in raster fashion over a rectangular area of the specimen surface. Upon impinging on the specimen, the primary electrons decelerate and several processes such as elastic scattering viz., forward scattering and backscattering of the incoming electrons and inelastic scattering viz., generation of secondary electrons, Auger electrons, Bremsstrahlung, characteristic x-rays, electron-hole pairs (in insulators and semiconductors), long-wavelength electromagnetic radiation etc. A number of different types of signals, which are emitted from the area of the specimen where the electron beam is impinging, are as shown in fig. Fig. Different types of signals produced when high-energy electrons impinges on a material. 4

48 UNIT- II: QUANTUM MECHANICS Various SEM techniques are differentiated on the basis of what is subsequently detected and imaged, and the principle images produced in the SEM are of three types: Secondary electron images, Backscattered electron images and Elemental X-ray maps. The secondary electrons are the electrons which are generated from the surface of the specimen when it is bombarded by high energy primary electrons. These electrons are collected by detector which creates a pattern of light and dark areas on a computer screen corresponding to the emission of secondary electrons from the specimen. As the number of electrons produced at any given point can be related directly to the topography of the specimen with respect to the detector, the patterns created on the viewing screen represent the surface topography of the specimen. Backscattered electrons are the high energy electrons that are elastically scattered and essentially possess the same energy as the incident or primary electrons. The probability of backscattering increases with the atomic number of the sample material. Therefore the primary electrons arriving at a given detector position can be used to yield images containing information on both topology and atomic composition. An additional electron interaction in the SEM is that the primary electron collides with and ejects a core electron from an atom in the sample. The excited atom will decay to its ground state by emitting either a characteristic X-ray photon or an Auger electron. By analyzing energies of characteristic of the x-ray photon the atoms can be identified. Further the concentration of atoms in the specimen can be determined by counting the number of X- rays emitted. Applications: SEM is used in a wide range of fields as a tool for observing surfaces at the nanometer level. It is an indispensable instrument for not only basic research but also for highly integrated semiconductor devices and new advanced materials. The SEM is used in the investigation of surface unevenness of the material (topography), the shape and size of the particle making up the object (Morphology), heterogeneity of the materials to visualize various mineral components in their distinct growth forms and their relation in terms of overall micro fabric and texture (composition) and to visualize the arrangement of atom in an object (crystallographic information). Besides surface topographic studies the SEM can also be used for determining the chemical composition of a material, its fluorescent properties, and the formation of magnetic domains and so on. Heisenberg s uncertainty principle Heisenberg s uncertainty principle is a direct consequence of the dual nature of matter. In classical mechanics, a moving particle at any instant has a fixed position in space and a definite momentum which can be determined if the initial values are known (we can know the future if we know the present) In wave mechanics a moving particle is described in terms of a wave group or wave packet. According to Max Born s probability interpretation the particle may be present 4

49 UNIT- II: QUANTUM MECHANICS anywhere inside the wave packet. When the wave packet is large, the momentum can be fixed, but there is a large uncertainty in its position. On the other hand, if the wave packet is small the position of the particle may be fixed, but the particle will spread rapidly and hence the momentum (or velocity) becomes indeterminate. In this way certainty in momentum involves uncertainty in position and the certainty in position involves uncertainty in momentum. Hence it is impossible to know within the wave packet where the particle is and what is its exact momentum. (We cannot know the future because we cannot know the present). Thus we have Heisenberg s uncertainty principle. According to the Heisenberg s uncertainty principle It is impossible to specify precisely and simultaneously certain pairs of physical quantities like position and momentum that describe the behavior of an atomic system. Qualitatively, this principle states that in any simultaneous measurement the product of the magnitudes of the uncertainties of the pairs of physical quantities is equal to or greater than h/4π (or it is of the order of h) Considering the pair of physical quantities such as position and momentum, we have ΔpΔx h/4π.... Where Δp and Δx are the uncertainties in determining the momentum and the position of the particle. Similarly, we have other canonical forms as ΔEΔt h/4π ΔJΔθ h/4π Where ΔE and Δt are uncertainties in determining energy and time while ΔJ and Δθ are uncertainties in determining angular momentum and angular position. Illustration of Heisenberg s uncertainty principle:. Nonexistence of electrons in the atomic nucleus: If an electron is to exist inside the nucleus, then the maximum uncertainty in its position x must not exceed the size of the nucleus. The diameter of the nucleus is of the order of 0-4 m. i.e,. x 0-4 m. Heisenberg s Uncertainty principle states that, x p x h...( ) 4π h P x 4π x Therefore Uncertainty in momentum is, x 0 P x P kgms x 0 This is the minimum uncertainty in the momentum of the electron. The momentum of the electron is greater than the uncertainty in the momentum. Thus p x kgm / s... ( ) 4

50 UNIT- II: QUANTUM MECHANICS If the momentum is as per the equation () then the velocity of the electron is much greater than the velocity of light. Hence we have to consider the relativistic equation for the energy of the electron. We now show the derivation of this equation as done by Einstein. According to the theory m0c E mc....() of relativity the energy v equivalence E, of mass m c is given by, where m o and m is the rest mass and relativistic mass of the particle and v its velocity. Squaring the above equation we get, E m0 c v c 4. c m 0 c 6 v...() The momentum of the particle is given by p mv. m 0 v v c...(4) p m c 0 v c v...(5) Multiplying by c, we get m p c c Subtracting equation (6) from equation (), we have E p c m 0 c 4 ( c ( c v v ) 0 v c v )... )...(6) Thus we get the equation E p c + m c...( 9) 4 0 ( 7) E p c + m 0 c 4...( 8) Where E is the relativistic energy of the particle. Now from equation (8), 4 E p c + m c c p m c o + o Since, the rest mass of the electron m o 9. x 0 - kg. Now by using the condition (9) in the above equation, we can say that, in order that the electron to exist within the nucleus, its energy E must be such that, the energy E of the electron in the nucleus is approximately is E.58454x0 J E 9.89MeV 44

51 UNIT- II: QUANTUM MECHANICS In order that an electron may exist inside the nucleus, its energy must be greater than or equal to 9.89 MeV. But, the experimental investigations on beta decay shows that the kinetic energy of the beta particles (electrons) is of the order of to 4 MeV. This clearly indicates that electrons cannot exist within the nucleus.. Broadening of spectral lines When an atom absorbs a photon, it rises to the excited state and it will stay in the excited state for certain time called the lifetime. Lifetime of atoms in the excited levels is of the order of 0-8 s. When the atom comes to the ground state it emits a photon of energy exactly equal to the energy difference between the two levels as shown in the figure. Δλ 0 E hν E hν Intensity λ λ ΔE Δλ FWHM E hν E hν Intensity Fig. Line width for emitted photons λ The energy of the emitted photon is given by hc E hν...() λ. where h is a Planck s constant, ν is the frequency, c is the velocity of light and λ is the wavelength. Thus for the first set of diagrams, where there is only one sharply defined excited state, with ΔE 0, there is only a single wavelength of the photons emitted and there would be no broadening of the spectral lines. However, in nature, there is always a finite amount of uncertainty in the energy of the lines (ΔE) that would lead to a finite broadening in the wavelength of the photons emitted as shown in the second set of diagrams. Differentiating equation () with respect to wavelength (λ), we get hc λ Ε λ 45

52 UNIT- II: QUANTUM MECHANICS hc λ E...() λ According to Heisenberg s uncertainty principle, the finite lifetime Δt of the excited state means there will be an uncertainty in the energy of the emitted photon is given by h Ε 4π t Substituting for ΔE from () and applying the condition of minimum uncertainty, we get hc λ h λ 4π t λ or λ 4π c t This shows that for a finite lifetime of the excited state, the measured value of the emitted photon wavelength will have a spread of wavelengths around the mean value λ. This uncertainty in the measured value of wavelength demands for very narrow spread, the lifetime of the excited state must be very high (of the order of 0 - s). Such excited levels are called Metastable states. This concept is adopted in the production of highly monochromatic laser light. Schrödinger s Wave Equation: In96 Schrödinger starting with de-broglie equation (λ h/mv) developed it into an important mathematical theory called wave mechanics which found a remarkable success in explaining the behavior of the atomic system and their interaction with electromagnetic radiation and other particles. In water waves, the quantity that varies periodically is the height of water surface. In sound waves it is pressure. In light waves, electric and magnetic fields vary. The quantity whose variation gives matter waves is called wave function (ψ). The value of wave function associated with a moving body at a particular point x in space at a time t is related to the likelihood of finding the body there at a time. A wave function ψ(x,t) that describes a particle with certain uncertainty in position, moving in a positive x-direction with precisely known momentum and kinetic energy may assume any one of the following forms: Sin( ωt - kx), cos( ωt - kx), e i(ωt kx), e -i(ωt kx) or some linear combinations of them. Schrödinger wave equation is the wave equation of which the wave functions are the solutions. It cannot be derived from any basic principles, but can be arrived at, by using the de-broglie hypothesis in conjunction with the classical wave equation. Time Independent one dimensional Schrödinger wave equation (TISE): In many situations the potential energy of the particle does not depend on time explicitly; the force that acts on it, and hence the potential energy vary with the position of the particle only. The Schrödinger wave equation for such a particle is time independent wave equation. Let ψ(x,t) be the wave function of the matter wave associate with a particle of mass m moving with a velocity v. The differential equation of the wave motion is as follows. 46

53 UNIT- II: QUANTUM MECHANICS 47...() t v x ψ ψ The solution of the Eq.() as a periodic displacement of time t is ψ(x,t) ψ 0 (x) e -iωt..() Where ψ 0 (x) is the amplitude of the matter wave. Differentiating Eq. partially twice w.r.t. to t, we get ω ψ i t ψ 0 (x) e -iωt ω ψ i t ψ 0 (x) e -iωt ψ ω t ψ 0 (x) e -iωt t ψ - ω ψ...() Substituting Eq. in Eq. x v ψ ω ψ (4) We have 4 λ π λ π ω k v Substituting this in Eq4, we get...(5) 4 ψ λ π ψ x 0...(6) 4 + ψ λ π ψ x Substituting the wavelength of the matter waves λh/mv in Eq.6 we get 0...(7) 4 + ψ π ψ h v m x If E and U are the total and potential energies of the particle respectively, then the kinetic energy E k of the particle U E mv E k ) ( U E m v m Substituting this in Eq.7, we get 0...(8) ) ( 8 + ψ π ψ U E h m x Hence ψ is a function of x alone and is independent of time. This equation is called the Schrödinger time- independent one dimensional wave equation.

54 UNIT- II: QUANTUM MECHANICS Physical significance of the wave function: The wave function ψ(x, t) is the solution of Schrödinger wave equation. It gives a quantummechanically complete description of the behavior of a moving particle. The wave function ψ cannot be measured directly by any physical experiment. However, for a given ψ, knowledge of usual dynamic variables, such as position, momentum, kinetic energy, etc., of the particle is obtained by performing suitable mathematical operations on it. The most important property of ψ is that it gives a measure of the probability of finding a particle at a particular position. ψ is also called the probability amplitude. In general ψ is a complex quantity, whereas the probability must be real and positive. Therefore a term called probability density is defined. The probability density P (x,t) is a product of the wave function ψ and its complex conjugate ψ *. * Pxt (,) ψψ ψ ( x, t) Normalization of wave function If ψ is a wave function associated with a particle, then ψ dτ is the probability of finding the particle in a small volume dτ. If it is certain that the particle is present in a volume τ then the total probability in the volume τ is unity i.e., ψ d τ. This is the normalization condition. In one dimension the normalization condition is ψ dx Note: When the particle is bound to a limited region the probability of finding the particle at * infinity is zero i.e., ψψ at x is zero. x τ Properties of wave function: The wave function ψ should satisfy the following properties to describe the characteristics of matter waves.. ψ must be a solution of Schrödinger wave equation.. The wave function ψ should be continuous and single valued everywhere. Because it is related to the probability of finding a particle at a given position at a given time, which will have only one value.. The first derivative of ψ with respect to its variables should be continuous and single valued everywhere, because ψ is finite. 4. Ψ must be normalized so that ψ must go to 0 as x ±, so that ψ d τ over all the space be a finite constant. Eigen functions and Eigen values: The Schrödinger wave equation is a second order partial differential equation; it will have many mathematically possible solutions (ψ). All mathematically possible solutions are not physically acceptable solutions. The physically acceptable solutions are called Eigen functions (ψ). 48

55 UNIT- II: QUANTUM MECHANICS The acceptable wave functions ψ has to satisfy the following conditions:. ψ is single valued.. ψ and its first derivative with respect to its variable are continuous everywhere.. ψ is finite everywhere. Once the Eigen functions are known, they can be used in Schrödinger wave equation to evaluate the physically measurable quantities like energy, momentum etc., these values are called Eigen values. In an operator equation O ψ λψ where is an operator for the physical quantity and ψ is an Eigen function and λ is the Eigen value. For example: H ψ Eψ Where H is the total energy (Hamiltonian) operator, ψ is the Eigen function and E is the total energy in the system. Similarly other eigen value equations are : P ψ pψ Where P stands for the momentum operator and p stands for momentum eigen value. Similarly we have x ψ xψ Where x stands for position operator and position eigen value. Applications of Schrodinger s wave equation:. For a Particle in an one-dimensional potential well of infinite depth (Particle in a box) Consider a potential energy landscape extending from x 0 to x a, where it is throughout constant and we therefore set it to zero for our convenience. Outside this region, the potential energy U is infinity. This type of landscape can be represented by the figure below. Consider a particle of mass m moving freely in x- direction in the region from x0 to xa. Outside this region potential energy U is infinity and within this region U0. As the particle cannot exist outside this region, due to infinitely high potential energy, we take ψ 0 outside this region/box. Inside the box U 0, hence the Schrodinger s equation is given by, Ψ 8π m + EΨ 0 x h Ψ + k Ψ 0...() x 8mπ E where, k...() V0 h U Discussion of the solution The solution of the above equation is given by Ψ A cos kx + Bsin kx...() where A & B are constants which depending on the boundary conditions of the well. Now apply boundary conditions for this, ^ ^ O x0 x xa 49

56 UNIT- II: QUANTUM MECHANICS Condition: I at x 0, U, thus the particle is not present. Hence ψ 0. Substituting the condition I in the equation 4, we get A 0 and B 0. (If B is also zero for all values of x, ψ is zero. This means that the particle is not present in the well.) Now the equation can be written as Ψ Bsin kx...(4) Condition: II at x a, U, thus the particle is not present. Hence ψ 0 Substituting the condition II in equation 5 we get 0 B sin(ka) Since B 0, sin ka 0 ka nπ where, n,,. nπ k a n π k a Substitute the value of k in equation (). 8mπ E n π h a n h E...(5) 8ma The equation (5) gives energy values or Eigen value of the particle in the well. When n0, from eqn (4) for all values of x between x 0 to x a, ψ n 0. This would imply that the particle is not present inside the box, which is not true. Hence the lowest physically allowed value of n for this problem is. The lowest energy corresponds to n is called the zero-point energy or Ground state h energy. Ezero po int 8ma All the states of n > are called excited states. To evaluate B in equation (), one has to perform normalization of wave function. Normalization of wave function: nπ Consider the equation, ψ Bsin kx Bsin x a The total probability of finding the particle inside the well is unity because there is only one particle within the box. But sin θ ( cos θ ) a Ψ dx 0 B a 0 nπ sin xdx a 50

57 UNIT- II: QUANTUM MECHANICS 5 a B a B n n a a B a x n n a x B dx a x n dx B dx a x n B a a a a 0 sin sin cos ` cos π π π π π π Thus the normalized wave function of a particle in a one-dimensional box is given by, x a n a n Ψ π sin where, n,, This equation gives the Eigen functions of the particle in the box. The Eigen functions for n,,.. are as follows. x a a x a a x a a Ψ Ψ Ψ π π π sin sin sin Since the particle in a box is a quantum mechanical problem we need to evaluate the most probable location of the particle in a box and its energies at different permitted state. Let us consider first three cases Case (): n This is the ground state and the particle is normally found in this state. For n, the Eigen function is x a a Ψ π sin At x 0, a ψ 0 and ψ 0 At x a/, ψ a and ψ /a

58 UNIT- II: QUANTUM MECHANICS U U Ψ Ψ (/a) /a x0 xa/ xa x0 xa/ xa The plot of Ψ, the wave function/amplitude and Ψ the probability density versus x is as shown above. From the figure, it indicates the probability of finding the particle at different locations inside the box. This means that in the ground state the particle cannot be found at the walls of the box and the probability of finding the particle is maximum at the central region. The Energy in the ground state is given by h E 8ma. Case : n This is the first excited state. The Eigen function for this state is given by π Ψ sin x a a At x 0, a/, a ψ 0 and ψ 0 At x a/4 ψ At x a/4 ψ - a a and ψ /a and ψ /a These facts are seen in the following plot. U U Ψ Ψ /a a/4 a/4 (/a) a/4 a/4 This means that in the first excited state the particle cannot be observed either at the walls or at the center. The energy is E 4E. Thus the energy in the first excited state is 4 times the zero point energy. 5

59 UNIT- II: QUANTUM MECHANICS Case : n This is the second excited state and the Eigen function for this state is given by π Ψ sin x a a At x 0, a/, a/,a ψ 0 and ψ 0 At x a/6, 5a/6 ψ a and ψ /a At x a/ ψ - and ψ /a a U U Ψ Ψ (/a ) a/6 5a/6 a/ /a a/6 a/ 5a/6 a/ a/ a/ a/ The energy corresponds to second excited state is given by E 9E. Note: The particle in an infinite well/box, closely resembles the real life problem of electrons trapped in the attractive force field of the positively charged nucleus. The energy levels in which the electrons can reside are quantized as we know! Further there is not one potential well but an array of such wells which are stacked periodically, with the periodicity given by the details of the material that we are considering. The height of the barrier is not infinity in real life as otherwise it would be impossible to get free/itinerant electrons as seen in metals.. Free Particle: Free particle means, it is not under the influence of any kind of field or force. Thus, it has zero potential energy, i.e., U 0 over the entire region. Hence Schrodinger s equation becomes, Ψ 8π m + EΨ 0 x h h Eψ π m Ψ x 8 h is nothing but the Kinetic energy operator which acts on ψ to give the total 8π m x energy E times ψ, as the potential energy is zero throughout. Further there is no restriction on the value of momentum or energy as there are no boundary conditions setting ψ 0 anywhere. Hence all energy values are allowed, and there is no quantization of energy levels. 5

60 UNIT- II: QUANTUM MECHANICS Q.No Sample Questions CO. What is Wien s law and Rayleigh Jeans law and discuss the limitations. How does Planck s law extrapolate between the two?. State De Broglie hypothesis.. What is wave function? Give its physical significance and properties. 4. With a neat labeled diagram, explain the construction and working of Scanning & Electron Microscope. 5. State Heisenberg s uncertainty principle. By applying Heisenberg s uncertainty principle, illustrate the broadening of spectral lines. & 6. State Heisenberg s uncertainty principle. Using Heisenberg s uncertainty & principle, prove the non-existence of electron within the nucleus of an atom. 7. Using Heisenberg s uncertainty principle explain the broadening of spectral lines. 8. What are Eigen functions? Mention their properties. 9. Setup time independent one-dimensional Schrodinger s wave equation for a matter wave. 0. Apply the time independent Schrodinger s wave equation to find the solutions & for a particle in an infinite potential well of width a. Hence obtain normalized wave function. Solve the Schrodinger s wave equation for a free particle. & P No. Problems CO. Calculate the de Broglie wavelength associated with a proton moving with a velocity equal to (/0) th of the velocity of light. To be found: de Broglie wavelength, λ Solution: 4 h λ.64 0 m 7 8 mv.67 0 (/ 0) 0. An electron and a proton are accelerated through the same potential difference. Find the ratio of their de Broglie wavelengths. To be found: Ratio of de Broglie wavelength, λ Solution: De Broglie Wavelength, λ λ h, λα me m For electron, λα For proton, λα e m e p m p λe Ratio of De Broglie Wavelengths, λ p m m p e 54

61 UNIT- II: QUANTUM MECHANICS. Compare the energy of a photon with that of a neutron when both are associated with wavelength of A o. Given that the mass of neutron is kg. To be found: Comparison of energy of photon with that of neutron Solution: Energy of neutron, 4 P h ( ) E. 0 J 0.08eV n 0 7 m λ m ( 0 ) hc Energy of photon, E J 4.9eV p 0 λ 0 Ep 5 Ratio of energies, E n 4. An electron has a speed of 4.8 x 0 5 m/s accurate to 0.0 %. With what accuracy with which its position can be located. To be found: Uncertainty in position, Δx Solution: h Uncertainty principle is given by, x p 4π Uncertainty in speed, Δv 4.8 x 0 5 x m/s 4 h Uncertainty in position, x 0 m 4π m v The inherent uncertainty in the measurement of time spent by Iridium-9 nuclei in the excited state is found to be.4x0-0 s. Estimate the uncertainty that results in its energy in the excited state. To be found: Uncertainty in energy, ΔE h Solution: Uncertainty principle is given by, E t 4π 4 h E.77 0 J 0 4π t The position and momentum of kev electron are simultaneously determined and if its position is located within Å. What is the percentage of uncertainty in its momentum? To be found: Percentage of uncertainty in momentum of electron, Δp h Solution: Uncertainty principle is given by, x p 4π Uncertainty in momentum, 4 h p kg. m / s 0 4π x Momentum, p me kgm / s p Percentage of uncertainty in momentum of electron, 00. p 55

62 UNIT- II: QUANTUM MECHANICS 7. Show that the energy Eigen value of a particle in second excited state is equal to 9 times the zero point energy. To be found: Energy Eigen value for second excited state is equal to 9 times the zero point energy. nh Solution: Energy Eigen value equation is given by, E 8ma h n, zero-point energy state, E 8ma 9h n, second excited state, E 9E 8ma 8. An electron is bound in a one-dimensional potential well of width Å, but of infinite height. Find the energy value for the electron in the ground state. To be found: Energy Eigen value Solution: nh Energy Eigen value equation is given by, E 8ma For n, ground state energy, 4 ( ) 0 ( ) E 7.65eV An electron is bound in one dimensional potential well of infinite potential of width 0. nm. Find the energy values in the ground state and also the first two exited state. To be found: Energy Eigen value Solution: nh Energy Eigen value equation is given by, E 8ma For n, ground state energy, 4 ( ) 9 ( ) E 6.6eV For n,first excited state, For n,second excited state, E 4E 04.6eV E 9E 5.44eV 0. An electron is trapped in a potential well of a width 0.5nm. If a transition takes place from the first excited state to the ground state find the wavelength of the photon emitted. To be found: Wavelength of the photon emitted, λ Solution: For n, ground state energy, 4 ( ) 9 ( ) E.4 0 J.507eV 56

63 UNIT- II: QUANTUM MECHANICS For n,first excited state E E J ev Energy difference, E E E J ev hc Wavelength of the photon emitted, λ 74.8nm 9 E 7. 0 J Exercise:. Taking Planck s law as the starting point try to derive Wien s law and Rayleigh Jeans law in the two limits in which they are valid.. How does the concept of superposition of waves leading to a wave packet naturally reconcile the dual wave and particle nature of matter?. Is there any connection between the position momentum and energy-time uncertainty relationship? Discuss. 4. Is the derivation for particle in a box done above valid for highly energetic relativistic particles? Discuss. 57

64 UNIT- II: QUANTUM MECHANICS 58

65 UNIT III: OSCILLATIONS AND WAVES UNIT III: OSCILLATIONS AND WAVES Introduction: Motion of bodies can be broadly classified into three categories: [] Translational motion [] Rotational motion [] Vibrational / Oscillatory motion Translational motion: When the position of a body varies linearly with time, such a motion is termed as translational motion. Example: A car moving on a straight road, a ball moving on the ground. Rotational motion: When a body as a whole does not change its position linearly with time but rotates about its axis, this motion is said to be rotational motion. Example: rotation of earth about its axis, rotation of a fly wheel on ball bearings. Vibrational /Oscillatory motion: When a body executes back and forth motion which repeats over and again about a mean position, then the body is said to have Vibrational/oscillatory motion. If such motion repeats in a regular interval of time then it is called Periodic motion or Harmonic motion and the body executing such motion is called harmonic oscillator. In harmonic motion there is a linear relation between force acting on the body and displacement produced. Example: bob of a pendulum clock, motion of prongs of tuning fork, motion of balance wheel of a watch, the up and down motion of a mass attached to a spring. Note:. If there is no linear relation between the force and displacement then the motion is called un harmonic motion. Many systems are un harmonic in nature.. In some oscillatory systems the bodies may be at rest but the physical properties of the system may undergo changes in oscillatory manner Examples: Variation of pressure in sound waves, variation of electric and magnetic fields in electromagnetic waves. Parameters of an oscillatory system. Mean position: The position of the oscillating body at rest.. Amplitude: The amplitude of an SHM is the maximum displacement of the body from its mean position.. Time Period: The time interval during which the oscillation repeats itself is called the time period. It is denoted by T and its unit in seconds. Period T π Displacement Acceleration 4. Frequency: The number of oscillations that a body completes in one second is called the frequency of periodic motion. It is the reciprocal of the time period T and it is denoted by n. Frequency n T 59

66 UNIT III: OSCILLATIONS AND WAVES 5. Phase: It is the physical quantity that expresses the instantaneous position and direction of motion of an oscillating system. If the SHM is represented by ya sin (ωt+φ ), together with ω as the angular frequency. The quantity (ωt+φ ) of the sine function is called the total phase of the motion at time t and φ is the initial phase or epoch. All periodic motions are not vibratory or oscillatory. In this chapter we shall study the simplest vibratory motion along one dimension called Simple Harmonic Motion (SHM) which usually occurs in mechanical systems. SIMPLE HARMONIC MOTION (SHM) A body is said to be undergoing Simple Harmonic Motion (SHM) when the acceleration of the body is always proportional to its displacement and is directed towards its equilibrium or mean position. A particle or a system which executes simple harmonic motion is called Simple Harmonic Oscillator. Examples of simple harmonic motion are motion of the bob of a simple pendulum, motion of a point mass fastened to spring, motion of prongs of tuning fork etc., TYPES OF SHM Simple harmonic motion can be broadly classified in to two types, namely linear simple harmonic motion and angular simple harmonic motion. Linear Simple Harmonic Motion: If the body executing SHM has a linear acceleration then the motion of the body is linear simple harmonic motion. Examples: motion of simple pendulum, the motion of a point mass tied with a spring etc., Angular Simple Harmonic Motion: If the body executing SHM has an angular acceleration then the motion of the body is angular simple harmonic motion. Examples: Oscillations of a torsional pendulum () UN DAMPED OR FREE VIBRATIONS If a body oscillates without the influence of any external force then the oscillations are called Free oscillations or un damped oscillations. In free oscillations the body oscillates with its natural frequency and the amplitude remains constant (Fig.) Figure : Free Vibrations In practice it is not possible to eliminate friction completely.actually the amplitude of the vibrating body decreases to zero as a result of friction. Hence, practical examples for free oscillation/vibrations are those in which the friction in the system is negligibly small. 60

67 UNIT III: OSCILLATIONS AND WAVES DIFFERENTIAL EQUATION OF FREE OSCILLATIONS Consider a block of mass m suspended from a rigid support through a mass less spring. The restoring force produced in the spring obeys Hooke s Law. Hooke s Law: The restoring force produced in a system is proportional to the displacement. When the mass is displaced through a distance y then the restoring force (F) produced is F k y () The constant k is called force constant or spring constant. It is a measure of the stiffness of the spring. The negative sign in equation () indicates that the restoring force is in the direction of its equilibrium position. Figure : A one-dimensional harmonic oscillator. In equilibrium condition the linear restoring force (F) in magnitude is equal to the weight mg of the hanging mass, shown in figure. i.e, F mg mg ky mg and k () y Now, if the load is displaced down through a distance y from its equilibrium position and released then it executes oscillatory motion. dy d y Velocity of the body v and Acceleration of the body a dt dt From Newton s second law of motion for the block we can write dy dy F ma m () dt from equation () substitue F -ky -ky m dt dy k or + y 0 dt m k m Substitue ω, i.e., d y dt + ω y (4) Whereωis the angular natural frequency. 6

68 UNIT III: OSCILLATIONS AND WAVES Equation (4) is the general differential equation for the free oscillator. The mathematical solution of the equation (4), y(t) represents the position as a function of time t. Let y(t) A sin (ωt +φ ) be the solution. y(t) Asin(ωt +φ ) (5) Where A is the amplitude and (ωt +φ ) is called the phase, φ is the initial phase (i.e., phase at t0). Period of the oscillator (T) We have F ma, also F ky At equilibrium, magnitude of restoring force ( F)is equal to the weight mg of the hanging mass. ma ky ky Acceleration a m Period T π Displacement Acceleration Tπ y m π π π k y k ω ω m π Period of the oscillator T π ω m k Although the position and velocity of the oscillator are continuously changing, the total energy E remains constant and is given by E mv ky + The two terms in (6) are the kinetic and potential energies, respectively. (6) VELOCITY AND ACCELERATION We can find velocity from the expression for displacement, y A sin (ωt+φ ) Velocity in SHM The rate of change of displacement is the velocity of the vibrating particle. By differentiating the above expression with time, we get 6

69 UNIT III: OSCILLATIONS AND WAVES dy v dt Aωcos ωt ( + φ ) φ v Aω -sin (ωt+ ) y As sin(ωt+ φ) A y v Aω - A vω A -y This is the expression for velocity of the particle at any displacement y. The maximum velocity is obtained by substituting y 0, v max ω A Since y 0 corresponds to its mean position, the particle has maximum velocity when it is at its mean position. At the maximum displacement, i.e., at the extreme position y A of the particle, the velocity is zero. Acceleration in SHM The rate of change of velocity is the acceleration of the vibrating particle. Differentiating velocity expression with respect to time t, we get acceleration dy a - dt d y a -ω y dt Aω sin ω ( t+ φ ) The above equation gives acceleration of the oscillating particle at any displacement. This equation is the standard equation of SHM. Since y0 corresponds to its mean position, the particle has minimum acceleration. At the maximum displacement, i.e., at the extreme position ya, maximum acceleration is ω A Graphical representation of displacement, velocity and acceleration Let the displacement of a particle executing SHM be y Asin ωt+ ( φ ) dy velocity v Aωcos ωt+ dt ( φ ) dy and Accleration a -Aω sin ωt+ dt ( φ ) 6

70 UNIT III: OSCILLATIONS AND WAVES Table : Displacement, Velocity and Acceleration for various values of time t Time ωt Displacement Asin ωt + φ ( ) Velocity Aωcos ωt + φ ( ) Acceleration Aω sin ωt + φ t Aω 0 T t 4 π +A 0 ω A T t π 0 - Aω 0 T t 4 π -A 0 + ω A t T π 0 + Aω 0 ( ) The variation of displacement, velocity and acceleration with time t is shown graphically: Energy in simple harmonic motion The total energy (E) of an oscillating particle is equal to the sum of its kinetic energy and potential energy under ideal conditions (no friction). The velocity of a particle executing SHM at a position where its displacement is y from its mean position is v ω A -y Kinetic energy: Kinetic energy of the particle of mass m is K m ω A -y K mω A -y -- () 64

71 UNIT III: OSCILLATIONS AND WAVES Potential energy: Let F be the restoring force and dy be displacement against the restoring force. Let dw be the work done by the force during the small displacement dy. dw -F.dy -(-ky) dy ky dy Total work done for the displacement y is given by W dw kydy y 0 y 0 ( ) W mω ydy kmω w mω y () This work done is stored in the body as potential energy U mω y Total energy E K + U mω A - y + mω y m ω A Thus we find that the total energy of a particle executing simple harmonic motion is mωa Special cases: (i) When the particle is at the mean position y 0, from eqn () the kinetic energy is maximum and from eqn. () the potential energy is zero. Hence the total energy is wholly kinetic E K max m ωa (ii) When the particle is at the extreme position y +A, from eqn. () the kinetic energy is zero and from eqn. () the Potential energy is maximum. Hence the total energy is wholly potential. E U max m ωa The variation of energy of an oscillating particle with the displacement can be represented in a graph as shown in the Figure. Figure. Variation of KE and PE with respect to displacement 65

72 UNIT III: OSCILLATIONS AND WAVES DIFFERENTIAL EQUATION OF ANGULAR SHM AND ITS SOLUTION EXAMPLE: TORSIONAL PENDULUM Simple harmonic motion can also be angular. In this case, the restoring torque required for producing SHM is directly proportional to the angular displacement and is directed towards its mean position. A heavy metal disc suspended by means of a wire from a rigid support, constitutes a Torsional Pendulum (Fig. 4).When the wire is twisted by an angle θ, a restoring torque/couple acts on it, tending it to return to its mean position. Restoring torque or couple (τ) C θ.. () Figure 4: Torsional Pendulum Where, C is the torsional constant. It is equal to the moment of the couple required to produce unit angular displacement. Its unit is Nmrad. The negative sign indicates that the torque is acting in the opposite direction to the angular displacement. d θ The angular acceleration produced by the restoring couple in the wire, α and I be the dt moment of inertia of the metal disc about the axis of the wire. According to Newton s second law, the equation of torque or couple acting on it is given by d θ τ I α I () dt d θ Therefore, on equating () and () we have, I -Cθ dt d θ C We get angular acceleration, θ, dt I C d θ Substituting ω, we have ωθ I dt The above relation shows that the angular acceleration is proportional to angular displacement and is always directed towards its mean position. Therefore, the motion of the disc is always simple harmonic motion (SHM). d θ + ωθ 0 () dt This is the differential equation of angular simple harmonic motion of the Torsional Pendulum. Therefore, the time period and frequency of oscillator is given by relation, π I C Period T π Where ω and Frequency ω C I n C T π I Note: For a given wire of length l, radius r and rigidity modulus η 4 πηr Torsionalconstant of the wire C l 66

73 UNIT III: OSCILLATIONS AND WAVES () DAMPED VIBRATIONS Free vibrations are oscillations in which the friction/resistance considered is zero or negligible. Therefore, the body will keep on vibrating indefinitely with respect to time. In real sense if a body set into vibrations, its amplitude will be continuously decreasing due to friction/resistance and so the vibrations will die after some time. Such vibrations are called damped vibrations (Fig. 5). Figure 5: Damped vibrations DIFFERENTIAL EQUATION OF DAMPED VIBRATIONS AND SOLUTION A block of mass m connected to the free end of a spring is partially immersed in a liquid (Fig. 6) and is subjected to small vibrations. The damping force encountered is more when the block moves in the liquid and hence the amplitude of vibration decreases with time and ultimately stops vibrating; this is due to energy loss from viscous forces. Let y to be the displacement of the body from the equilibrium state at any instant of time t, then dy/dt is the instantaneous velocity. Figure 6: Spring and mass system The two forces acting on the body at this instant are: i. A restoring force which is proportional to displacement and acts in the opposite direction, it may be written as F restoring ky ii. A frictional or damping force is directly proportional to the velocity and is oppositely directed to the motion, it may be written as dy F r damping dt 67

74 UNIT III: OSCILLATIONS AND WAVES The net force on an oscillator subjected to a linear damping force that is linear in velocity is simply the sum is thus given by F F restoring + F damping d y r dy k dy -ky -r dt dy But by Newton'slaw of motion, F m dt dy wheremis themassof thebody and is the accleration of the body. dt d y dy Then, m -ky-r dt dt or + + y () dt m dt m Let r m b and k m ω, then the above equation takes the form, d y dy + b + ω y0 dt () dt This is the differential equation of second order. In order to solve this equation, we assume its solution as αt y Ae () Where A and α are constants. Differentiating equation () with respect to time t, we have dy αt d y αt Aαe and Aα e dt dt By substituting these values in equation (), we have αt αt αt Aα e + baαe + ω A e 0 or αt Ae ( α + bα + ω ) 0 For the above equation to be satisfied, either y0, or ( α bα ω ) Since y0, corresponds to a trivial solution, one has to consider the solution α + bα + ω 0 ( ) The standard solution of the above quadratic equation has two roots, it is given by, 4b ± 4b 4ω α α b± b ω Therefore, the general solution of equation () is given by ( b + b ω ) t ( b b ω ) t y Ce + De (4) Where C and D are constants, the actual solution depends upon whetherb > ω, b b < ω. ω and 68

75 UNIT III: OSCILLATIONS AND WAVES Case : Heavy damping ( b > ω ) In this case, b ω is real and less than b, therefore in equation (4) both the exponents are negative. It means that the displacement y of the particle decreases continuously with time. That is, the particle when once displaced returns to its equilibrium position quite slowly without performing any oscillations (Fig.7). Such a motion is called overdamped or aperiodic motion. This type of motion is shown by a pendulum moving in thick oil or by a dead beat moving coil galvanometer. Case : Critical damping ( b Figure 7: Heavy damping or overdamped motion ω ) By substituting b ω in equation (4) the solution does not satisfy equation (). Hence we consider the case when b ω is not zero but is a very small quantity β. The equation (4) then can be written as ( b + β) t ( b β) t y Ce + De bt βt y e ( Ce βt + De ) sinceβ issmall, we can approximate, βt βt e + βt and e βt,on the basis of exponentialseries expansion bt y e C( + βt) + D( βt) [ ] [ β ] bt y e ( C+ D) + tc ( D) which is the product of terms bt As t is present in e and also in the term βtc ( D), both of them contribute to the variation of y with respect to time. But by the virtue of having t in the exponent, the term e bt predominantly contributes to the equation. Only for small values of t, the term βtc ( D) bt contributes in a magnitude comparable to that of e. Therefore, though y decreases throughout with increase of t, the decrement is slow in the beginning, and then decreases rapidly to approach the value zero. i.e., the body attains equilibrium position (Fig.8). Such damping motion of a body is called critical damping. This type of motion is exhibited by many pointer instruments such as voltmeter, ammeter...etc in which the pointer moves to the correct position and comes to rest without any oscillation. Figure 8: Critical damping motion. 69

76 UNIT III: OSCILLATIONS AND WAVES Case : Low damping ( b < ω ) This is the actual case of damped harmonic oscillator. In this case b ω is imaginary. Let us write b-ω i ω -b iβ where β ω -b and i -. Then,equation (4) now becomes y Ce +De (-b + iβ )t (-b - iβ )t -bt iβt -iβt y e (Ce + De ) -bt y e (C(cosβ t+isinβ t)+d(cosβ t-isinβ t)) [ ] -bt y e (C+D)cosβ t+i(c-d)sinβ t) RewritingC+DAsinφ and i(c-d)acos φ, Where A andφ are constants. y e -bt [ Asinφcosβ t+acosφsinβ t] -bt y Ae sin(β t+ φ) bt This equation y Ae sin( β t + φ) represents the damped harmonic oscillations. The bt oscillations are not simple harmonic because the amplitude ( Ae ) is not constant and decreases with time (t). However, the decay of amplitude depends upon the damping factor b. This motion is known as under damped motion (Fig.9). The motion of pendulum in air and the motion of ballistic coil galvanometer are few of the examples of this case. Figure 9: Under damped motion The time period of damped harmonic oscillator is given by π π π T β ω -b k r - m 4m The frequency of damped harmonic oscillator is given by k r n T π m 4m 70

77 UNIT III: OSCILLATIONS AND WAVES () FORCED VIBRATIONS In the case of damped vibrations, the amplitude of vibrations decrease with the time exponentially due to dissipation of energy and the body eventually comes to a rest. When a body experiences vibrations due to the influence of an external driving force the body can continue its vibration without coming to a rest. Such vibrations are called forced vibrations. For example: When a tuning fork is struck on a rubber pad and its stem is placed on a table, the table is set in vibrations with the frequency of the fork. These oscillations of the table are the forced oscillations/vibrations. So forced vibrations can also be defined as the vibrations in which the body vibrates with frequency other than natural frequency of the body, and they are due to applied external periodic force. DIFFERENTIAL EQUATIONS OF FORCED VIBRATIONS AND SOLUTIONS Suppose a particle of mass m is connected to a spring. When it is displaced and released it starts oscillating about a mean position. The particle is driven by external periodic force ( F sin ω t). The oscillations experiences different kinds of forces viz, o d ) A restoring force proportional to the displacement but oppositely directed, is given by F restoring -ky, where k is known as force constant. ) A frictional or damping force proportional to velocity but oppositely directed, is given by dy F damping - r, where r is the frictional force/unit velocity. dt ) The applied external periodic force is represented by ( Fosin ωdt), where F o is the maximum value of the force and ω d is the angular frequency of the driving frequency. The total force acting on the particle is given by, dy F net F external + Frestore +Fdamp Fo sinω t - r - ky d dt dy r dy k Fo sinω t + + y dt m dt m d () m dy By Newton's second law of motion F m. dt d y dy Hence,m F osinω t - r - ky dt d dt The differential equation of forced vibrations is given by r k F m m m o Substitute b, ω and f, then equation () becomes d y dt dy dt +b +ω yf sinωdt- Where b is the damping coefficient, o () F is the amplitude of the external driving force and ω is the angular frequencyof theexternalforce. d 7

78 UNIT III: OSCILLATIONS AND WAVES The solution of the above equation () is given by y Asin(ω t φ) () d Where, A is the amplitude of the forced vibrations. By differentiating equation () twice with respect to time t, we get dy d y ωdacos( ωdt φ) and ω sin( ) da ωdt φ dt dt By substituting the values of y, dy/dt and d y/dt in equation (), we get d d d d d d d { } ω Asin ( ω t φ) + bω Acos ( ω t φ) + ω Asin ( ω t φ) f sin ( ω t φ) + φ or A( ω ω )sin ( ω t φ) + bω Acos ( ω t φ) f sin ( ω t φ)cosφ + f cos ( ω t φ)sin φ (4) d d d d d If equation (4) holds good for all values of t, then the coefficients of sin (ω d t- φ) and cos (ω d t- φ) must be equal on both sides, then A( ω ωd ) f cos φ (5) and bωd A f sin φ (6) By squaring and adding equations (5) and (6), we have A ( ω ωd) + 4b ωda f f A (7) ( ω ωd) + 4b ωd By dividing equation (6) by (5), we get bωd bωd tan φ, Phase φ tan (8) ω ωd ω ωd Substituting the value of A in equation () we get f y sin( ω ) d t φ (9) ( ω ω ) + 4b ω d d This is the solution of the differential equation of the Forced Harmonic Oscillator. From equations (7) and (8), it is clear that the amplitude and phase of the forced oscillations depend upon ( ω ω d ), i.e., these depend upon the driving frequency (ω d ) and the natural frequency (ω) of the oscillator. We shall study the behaviour of amplitude and phase in three different stages of frequencies i.e, low frequency, resonant frequency and high frequency. 7

79 UNIT III: OSCILLATIONS AND WAVES Case I: When driving frequency is low i.e., ( ωd << ω). In this case, amplitude of the vibrations is given by f A ( ω ω ) + 4b ω d d d ( d) ( ) As ω ω, ω -ω ω ω and 4b ω 0 ω 0 o o o A f and ω d f F m F F k ω km k m m bω d and φ tan tan (0) 0 ω ωd d This shows that the amplitude of the vibration is independent of frequency of the driving force and is dependent on the magnitude of the driving force and the force-constant (k). In such a case the force and displacement are always in phase. Case II: When ωd ω i.e., frequency of the driving force is equal to the natural frequency of the body. This frequency is called resonant frequency. In this case, amplitude of vibrations is given by f A ( ω ωd) + 4b ωd f Fo m Fo A bω ( rm) ω rω d d d bω d bωd π and φ tan tan tan ( ) ω ω d 0 Under this situation, the amplitude of the vibrations become maximum and is inversely proportional to the damping coefficient. For small damping, the amplitude is large and for large damping, the amplitude is small. The displacement lags behind the force by a phase π/. Case III: When ( ωd >> ω) i.e., the frequency of force is greater than the natural frequency of the body. In this case, amplitude of the vibrations is given by f A (ω -ω ) +4b ω d d 4 d d d ω ω (ω -ω ) ω f F m F since ω is very large,ω 4b ω, A 4 o o d d >>> d ωd ωd mωd - bω d - b - and φ tan tan - tan (-0)π ω -ωd ωd This shows that amplitude depends on the mass and continuously decreases as the driving frequency ω d is increased and phase difference towards π. 7

80 UNIT III: OSCILLATIONS AND WAVES RESONANCE: If we bring a vibrating tuning fork near another stationary tuning fork of the same natural frequency as that of vibrating tuning fork, we find that stationary tuning fork also starts vibrating. This phenomenon is known as Resonance. Resonance is a phenomenon in which a body vibrates with its natural frequency with maximum amplitude under the influence of an external vibration with the same frequency. Theory of resonant vibrations: (a) Condition of amplitude resonance. In case of forced vibrations, the expression for amplitude A and phase φ is given by, f A bω d and φ tan ( ω ωd) + 4b ωd ω ωd The amplitude expression shows the variation with the frequency of the driving force ω d. For a particular value of ω d, the amplitude becomes maximum. The phenomenon of amplitude becoming a maximum is known as amplitude resonance. The amplitude is maximum when ( ω ωd) + 4b ωd is minimum. If the damping is small i.e., b is small, the condition of f maximum amplitude reduced to A max. bω d (b) Sharpness of the resonance. We have seen that the amplitude of the forced oscillations is maximum when the frequency of the applied force is at resonant frequency. If the frequency changes from this value, the amplitude falls. When the fall in amplitude for a small change from the resonance condition is very large, the resonance is said to be sharp and if the fall in amplitude is small, the resonance is termed as flat. Thus the term sharpness of resonance can be defined as the rate of fall in amplitude, with respect to the change in forcing frequency on either side of the resonant frequency. Figure 0: Effect of damping on sharpness of resonance 74

81 UNIT III: OSCILLATIONS AND WAVES Figure 0, shows the variation of amplitude with forcing frequency for different amounts of damping. Curve () shows the variation of amplitude when there is no damping i.e., b0. In this case the amplitude is infinite atωd ω. This case is never realized in practice due to friction/dissipation forces, as a slight damping factor is always present. Curves () and () show the variation of amplitude with respect to low and high damping. It can be seen that the resonant peak moves towards the left as the damping factor is increased. It is also observed that the value of amplitude, which is different for different values of b (damping), diminishes as the value of b increases. This indicates that the smaller is the damping, sharper is the resonance or large is the damping, flatter is the resonance. EXAMPLE FOR FORCED VIBRATIONS: SERIES LCR CIRCUIT An L-C-R circuit fed by an alternating emf is a classic example for a forced harmonic oscillator. Consider an electric circuit containing an inductance L, capacitance C and resistance R in series as shown in the figure. An alternating emf has been applied to a circuit is represented by E sinω t. o d Figure : Series LCR circuit. Let q be the charge on the capacitor at any instant and I be the current in the circuit at any instant. The potential difference across the capacitor is q, the back emf due to self inductance C in the inductor is di L dt and, the potential drop across the resistor is IR. The sum of voltages across the three LCR elements illustrated must be equal the voltage supplied by the source element. Hence the voltage equation at any instant is given by, V + V +V V (t) L R C S di q L + IR+ Eosinωdt dt C Differentiating,weget d I di dq L +R + ω deocos dt dt C dt ω d t dq d I di I but I, L +R + ω deocosωdt dt dt dt C d o + + I cosω dt d I R di ωe dt L dt LC L d I R di ωe d o π + + I sin ωdt+ () dt L dt LC L 75

82 UNIT III: OSCILLATIONS AND WAVES This is the differential equation of the forced oscillations in the electrical circuit. It is similar to equation of motion of a mechanical oscillator driven by an external force. d y dy F o +b +ω y sinω t m d dt dt () The explicit and precise connection with the mechanical oscillation equation is given below: Displacement : y I Velocity : dy dt di dt Damping constant : b R/L Natural frequency : ω / LC External force : F(t) E o And F o /m ω d E o /m Forced oscillations (refer previous section) Starting with equation from forced vibrations, we have d y dy F +b +ω y o sinω t m d dt dt Amplitude of mechanical oscillations/ vibrations is given by LCR Circuit application The equivalent equation for LCR circuit is given by d o + + I sin ω dt+ d I R di ωe π dt L dt LC L Amplitude of current I o is given by A F m o d + b d ( ω ω ) 4 ω I0 ω d Eo L ( ω ωd) + 4b ωd R Substitute for ω and 4 b, LC L we get Io LC ω d Eo L R ωd + ω d L 76

83 UNIT III: OSCILLATIONS AND WAVES In the denominator,multiply and divde the term by ωd and ωd by L, we get LC Io Io ω d Eo L ω d L R ωd ω d LC + ω L d L ω d Eo L d R ( Lωd) + C L ωd L ω ω d ω d Eo Io L ω d Lωd + R L Cω d substitute XC (Capacitive reactance) Cωd and Lωd X L(Inductive reactance) Io Eo E o ( XC XL) + R ( XL XC) + R The solution of the equation () for the current at any instant in the circuit is of the form Eo I sin( ω ) sin( ) () dt φ Io ωdt φ R + ωd L ωdc Electrical Impedance: The ratio of amplitudes of alternating emf and current in ac circuit is Eo Eo called the electrical impedance of the circuit. ie.., Z Io I Z Eo Eo The amplitude of the current is I o Z R + ωdl- ωc d o Impedance of the circuit Z R + ωdl- (4). ωc d 77

84 UNIT III: OSCILLATIONS AND WAVES The quantity ωdl- is the net reactance of the circuit which is the difference between ωc d the inductive reactance (X L) ωl d and the capacitive reactance (X C). ω C. Equation () shows that the current I lags in phase with the applied emf ωd L ω angle φ and is given by dc XL XC φ tan tan R R d E sinω t by an The following three cases arise:. When X L >X C, φ is positive, that is, the current lags behind the emf. Circuit is inductive.. When X L <X C, φ is negative, that is, the current leads behind the emf. Circuit is capacitive.. When X L X C, φ is zero, that is, the current is in phase with the emf. Circuit is resistive. o d Electrical Resonance: According to equation (4), the current have its maximum amplitude when X L X C 0 or ω dl 0 or dl or d ω C ω ω C ω LC d Where ω d is the angular frequency of the applied emf, while ω LC is the (angular) natural frequency of the circuit. Hence, the maximum amplitude of the current oscillations occurs when the frequency of the applied emf is exactly equal to the natural (un-damped) frequency of the electrical circuit. This is the condition of electrical resonance. Sharpness of resonance and Bandwidth: When an alternating emf is applied to an LCR circuit, electrical oscillations occur in the circuit with the frequency ω d is equal to the applied emf. The amplitude of these oscillations (current amplitude) in the circuit is given by Eo Eo Io, Where Z is theimpedance of the circuit. Z R + ωd L ωdc d 78

85 UNIT III: OSCILLATIONS AND WAVES At resonance, when the frequency ω d of the applied emf is equal to the natural frequency ω of the circuit, the current amplitude I o is maximum and is equal to E o /R. Thus at LC resonance the impedance Z of the circuit is R. At other values of ω d, the current amplitude I o is smaller and the impedance Z is larger than R. E o / ω ω ω Figure : Graphical plot of current vs frequency. The variation of the current amplitude I o with respect to applied emf frequency ω d is shown in the figure. I o attains maximum value (E o /R) when ω d has resonant value ω and decreases as ω d changes from ω. The rapidity with which the current falls from its resonant value (E o /R) with change in applied frequency is known as the sharpness of resonance. It is measured by the ratio of the resonant frequency ω to the difference of two frequencies ω and ω taken at of the resonant (ω) value. ω i.e., Sharpness of resonance (Q), ω ω ω and ω are known as the half power frequencies. The difference of half power frequencies, ω - ω is known as band-width. The smaller is the bandwidth, the sharper is the resonance. 79

86 UNIT III: OSCILLATIONS AND WAVES SI Sample Questions (one mark each) CO No. What is simple harmonic motion?. Write the general equation representing SHM.. List any two characteristics of SHM. 4. A particle executes a S.H.M. of period 0 seconds and amplitude of meter. Calculate its maximum velocity. 5. Hydrogen atom has a mass of.68x0-7 kg, when attached to a certain massive molecule it oscillates as a classical oscillator with a frequency0 4 cycles per second and with amplitude of 0-0 m. Calculate the acceleration of the oscillator. 6. A body executes S.H.M such that its velocity at the mean position is 4cm/s and its amplitude is cm. Calculate its angular velocity. 7. What is free vibration? 8. What is damped vibration? 9. Give any two examples for damped vibration. 0. What is forced vibration?. Given two vibrating bodies what is the condition for obtaining resonance?. Explain why a loaded bus is more comfortable than an empty bus?. What is a simple harmonic oscillator? 4. What is torsional oscillation? 5. Displacement of a particle of mass 0g executes SHM given by x 5sin πt + φ and its displacement at t0 is cm where the amplitude T isn5cm. Calculate the initial phase of the particle. 6. Name the two forces acting on a system executing damped vibration. 7. How critical damping is beneficiary in automobiles? 8. What is restoring force? Descriptive Questions. Every SHM is periodic motion but every periodic motion need not be SHM. Why? Support your answer with an example.. What is the phase difference between (i) velocity and acceleration (ii) acceleration and displacement of a particle executing SHM?. Show graphically the variation of displacement, velocity and acceleration of a particle executing SHM. 4. Setup the differential equation for SHM. 5. Define the terms (i) time period (ii) frequency and (iii) angular frequency of oscillations. 6. What is phase of SHM? Explain the term phase difference. 7. Derive an expression for the time period of a body when it executes angular SHM. 8. Explain the oscillations of a mass attached to a horizontal spring. Hence deduce an expression for its time period. 9. Distinguish between linear and angular harmonic oscillator? 0. What are damped vibrations? Establish the differential equation of motion 80

87 UNIT III: OSCILLATIONS AND WAVES for a damped harmonic oscillator and obtain an expression for displacement. Discuss the case of heavy damping, critical damping and low damping.. What is damping? On what factors the damping depends?. What is forced vibration? Give an example.. What do you mean by forced harmonic vibrations? Discuss the vibrations of a system executing simple harmonic motion when subjected to an external force. 4. What is driven harmonic oscillator? How does it differ from simple and damped harmonic oscillator? 5. What is resonance? Explain the sharpness of resonance. 6. Illustrate an example to show that resonance is disastrous sometimes. SI Problems No. A particle executes SHM of period.4 second and amplitude 5cm. Calculate its maximum velocity and maximum acceleration. Solution: The maximum velocity at y 0 in v ω A -y, v max ωa π π ω 0.radian T.4 vmax cm / sec 0.0 m / s At the maximum displacement, i.e., at the extreme position xa, maximum acceleration is -ω A a m/ s. A circular plate of mass 4kg and diameter 0.0 metre is suspended by a wire which passes through its centre. Find the period of angular oscillations for small displacement if the torque required per unit twist of the wire is 4 x 0 - N- m/radian. Solution: MR 4(0.05) Moment of Inertia I 0.005kgm CO I Time period is given by T π.4 7.0s. C 4x0 8

88 UNIT III: OSCILLATIONS AND WAVES. A mass of 6kg stretches a spring 0.m from its equilibrium position. The mass is removed and another body of mass kg is hanged from the same spring. What would be the period of motion if the spring is now stretched and released? Solution: F mg m Fky, k 96N/m and Tπ s y y 0. k The particle performing SHM has a mass.5g and frequency of vibration 0Hz. It is oscillating with amplitude of cm.calculate the total energy of the particle. Given :Frequency ωπn0πs -, Amplitude Acm0.0m, mass m.5 x 0 - kg Solution: - - Total energy, E mω A.5 0 ( 0π ) ( 0.0 ).97 0 J 5. A vibrating system of natural frequency 500 cycles/sec, is forced to vibrate with a periodic force/unit mass of amplitude 00 x 0-5 N/kg in the presence of a damping/ unit mass of 0.0 x 0- rad/s. Calculate the maximum amplitude of vibration of the system. Given :Natural frequency 500 cycles/sec, Amplitude of the force / unit mass, Fo/m00 x 0-5 N/kg, Damping coefficient, r/m 0.0 x 0 - rad/s Solution: Maximum amplitude of vibration f Fo m A bω ( rm) ω A 0.08 m π 500 Themaximumamplitudeof vibration of the system is 0.08meter 6. A circuit has an inductance of π henry and resistance 00Ω. An A. C. supply of 50 cycles is applied to it. Calculate the reactance and impedance offered by the circuit. Solution: The inductive reactance is X L ω d L πnl Here ω d πnπ x 50 00π rad/sec and L π Henry. X L ω d LπnL π x 50x π 00Ω. Z R + X The impedance is L + Ω 7. A series LCR circuit has LmH, C0.µF and R0Ω.Calculate the resonant frequency of the circuit. Solution: The resonant angular frequency of the circuit is given by ω LC Here LmH and C0.µF 5 ω 0 rad / s 7 0 x0 8

89 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS UNIT- IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS Introduction: Conducting materials play a vital role in Engineering. It is very essential to know the electrical properties of materials for specific application of the materials. The properties of metals such as electrical conduction, thermal conduction, specific heat etc., are due to the free electrons or conduction electrons in metals. The first theory to explain the electrical conductivity of metals is Classical free electron theory and it was proposed by Drude in the year900 and later developed and refined by Lorentz. Hence classical free electron theory is known as Drude-Lorentz theory. Assumptions of Classical Free Electron Theory:. A metal is imagined as a three dimensional ordered network of positive ions with the outermost electrons of the metallic atoms freely moving about the solid. The electric current in a metal, due to an applied field, is a consequence of drift velocity of the free electrons in a direction opposite to the direction of the field.. The free electrons are treated as equivalent to gas molecules and thus assumed to obey the laws of kinetic theory of gases. As per kinetic theory of gases, in the absence of the field the energy associated with each electron at a temperature T is kt, where k is Boltzmann constant. It is related to the kinetic energy through the relation kt mv th where v th is the thermal velocity of the electrons.. The electric potential due to the ionic core (lattice) is taken to be essentially constant throughout the metal. 4. The attraction between the free electrons and the lattice ions and the repulsion between the electrons are considered insignificant. Drift Velocity Initially the electrons in the metal which are in thermal equilibrium will move in random directions and often collide with ions with no net displacement. When electric field is applied, the equilibrium condition is disturbed and there will be net displacement in randomly moving free electron s positions, with time in a direction opposite to the direction of the field. This displacement per unit time is called drift velocity which will be constant for the free electrons in the steady state. This accounts for the current in the direction of the field. 8

90 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS If E is the electric field applied to the metal, τ is mean collision time, then drift velocity for conduction electron in a metal is given by ee v d m τ where e and m are charge and mass of electron respectively. Current density (J): It is the current per unit area of cross section of an imaginary plane held normal to the direction of current in a current carrying conductor. i.e. J I/A where A is the area of cross section. Electric Field (E): Electric field across homogeneous conductor is defined as the potential drop per unit length of the conductor. If L is the length of a conductor of uniform cross section and uniform material composition and V is the potential difference between its two ends, then electric field E is given by E V/L Mean Free Path ( λ ): It is the average distance travelled by the conduction electrons between successive collisions with lattice ions. Mean Collision Time (τ ): It is the average time that elapses between two consecutive collisions of an electron with the lattice ions. Relation between v, τ and λ : If v is the total velocity of the electrons, then the mean collision time τ is given by λ τ v Resistivity ( ρ ): For a material of uniform cross section, the resistance R is directly proportional to length L and inversely proportional to area of cross section A 84

91 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS L R α A i.e. L R ρ A where ρ is called resistivity. It is the property of the material and gives the measure of opposition offered by the material during the current flow in it. RA ρ L Conductivity (σ ): It is reciprocal of resistivity. It is a physical property that characterizes conducting ability of a material. L σ ρ RA Relation between J, σ and E: From ohms law VIR I. ρl A I V. A l ρ I J and σ A ρ J σe Expression for electric current in a conductor: (I) I nevd A n - Number of free electrons in unit volume of the conductor vd - Drift velocity of electrons A - Area of cross section of the conductor e Charge of an electron Conductivity: Eeτ I neav d V d m Eeτ ne τ ne τ V I ( nea )( ) ( ) AE A. m m m d m l V.. I ne τ A ρl V. I A V RI 85

92 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS Mobility of electrons: m ρ ne τ ne τ σ ρ m Mobility of electrons ( µ ) is defined as the magnitude of drift velocity acquired by the electron in unit field. i.e. Mobility in terms of Electrical conductivity: We have equation from ohm s law, J σe vd eeτ ee µ E E m m J σ E Hence σ neµ I AE nev A AE d neµ Thermal conductivity of Metals: It is defined as the ratio of flow of heat across a unit cross section of a conductor with a unit temperature gradient between the opposite faces perpendicular to the direction of the heat flow. Q It is given by K t dθ A dx where negative sign shows that the heat energy flows from the hot end to the cold end. Wiedemann-Franz Law: This law states that the ratio of the thermal conductivity to the electrical conductivity is K K directly proportional to the absolute temperature. α T or cons tant, This constant L is σ σt known as Lorentz number. Failures of classical free electron theory: Although electrical and thermal conductivity in metals can be explained successfully through classical free electron theory, it failed to account for many other experimental facts such as specific heat, temperature dependence of σ & dependence of electrical conductivity on electron concentration.. The molar specific heat of a gas at constant volume is Cv R, where R is a universal constant. But the experimental value of electronic specific heat is C v 0-4 RT which the classical theory could not explain. Also the experimental value shows that the electronic 86

93 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS specific heat is temperature dependent, whereas the classical free electron theory says that it is temperature independent.. The electrical conductivity of a metal is inversely proportional to temperature. According to classical free electron theory, electrical conductivity is inversely proportional to the square root of temperature, i.e. σ. T ne τ. Electrical conductivity is given as σ m According to classical electron theory electrical conductivity is directly proportional to the electron concentration. But monovalent metals like copper found to have high electrical conductivity than the divalent & trivalent metals like Zinc and Aluminium. Hence CFET fails to explain the observation. 4. Though metals are expected to exhibit negative Hall co-efficient since the charge carriers in them are electrons, some metals like zinc have positive Hall co-efficient. The free electron theory could not explain the positive Hall co-efficient of metals. Assumptions of quantum free electron theory: The main assumptions of quantum free electron theory are. The energy values of free electrons are quantized. The allowed energy values are realized in terms of a set of energy levels.. The distribution of electrons in the various allowed energy levels follows Pauli s exclusion principle.. Distribution of electrons in energy states obey Fermi-Dirac statistics. 4. The free electrons travel in a constant potential inside the metal but stay confined within its boundaries. 5. The attraction between the free electrons and lattice ions, the repulsion between the electrons themselves are ignored. Fermi level and Fermi energy: If we assume the number of electrons per unit volume as n e then these electrons should be accommodated in the various energy levels. At absolute zero temperature, the electrons occupy the lowest available energy levels. The highest occupied level in metals at zero Kelvin is called as the Fermi level and the corresponding energy value of that level is called as the Fermi energy, it is denoted by E F. Thus at 0K all levels up to the Fermi level are occupied while the levels above it are vacant. E F E 0 Energy band The dotted level is the Fermi level. Levels from E o up to E F are occupied while levels above E F are empty. 87

94 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS Fermi-Dirac statistics: Under thermal equilibrium the free electrons are distributed in various available energy states. The distribution of electrons among the energy levels follows statistical rule known as Fermi- Dirac statistics. Fermi-Dirac statistics is applicable to fermions. Fermions are indistinguishable particles with half integral spin. Since electron has half spin they obey Fermi-Dirac statistics and they are called Fermions. Fermi factor represents the probability that a quantum state with energy E is occupied by an electron, is given by Fermi-Dirac distribution function, f ( E) E EF + exp kt Where k is the Boltzmann s constant, T is the temperature in Kelvin, E is the energy and E F is the Fermi energy. Dependence of Fermi factor on temperature: The dependence of Fermi factor on temperature at T0K is given in the figure. At T0K E F E 0 f (E) Case : the probability of occupation for E < E F at T 0K Substituting the value of T 0K in the Fermi function we get f ( E) ( ) E E F kt + e + 0+ e fe) implies that all the levels below E F are occupied by electrons. Case : the probability of occupation for E>E F at T 0K. Substituting the value of T 0K in the Fermi function, we get f ( E) 0 E E ( F kt ) + e. + e This shows that all energy levels above E F are vacant. 88

95 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS Case : probability of occupation at temperature > 0K. At ordinary temperatures, though the value of probability remain for E<< E F, it starts decreasing from as the values of E become closer to E F.. The value of f(e) become ½ at EE F. This is because at E E F f ( E) 0 ( ) E E F + kt e + + e. For values just beyond E F, f(e)>0 4. Further above E > E F, the probability value falls off to zero rapidly. It implies that the probability of occupancy of Fermi level at any temperature other than 0K is 0.5 Hence Fermi level is defined as the energy level at which the probability of electron occupancy is half. Also, Fermi energy, E F is the average energy possessed by the electrons which participate in conduction process in conductors at temperatures above 0K. Density of states g (E): The permitted energy levels for electrons in a solid will be in terms of bands. Each energy band spread over an energy range of few ev. The number of energy levels in each band will be extremely large and hence the energy values appear to be virtually continuous over the band spread. Each energy level consists of two states and each state accommodates only one electron. Therefore, energy level can be occupied by two electrons only, having opposite directions of spin. The exact dependence of density of energy states on the energy is realized through a function denoted as g (E) and is known as density of states function. It is defined as, the number of available states per unit volume per unit energy interval. The number of states lying in the range of energies between E and E+dE is given by 4π g( E) de (m) / E / de. h Where E is the kinetic energy of the electron in the energy level E. 89

96 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS Carrier concentration in metals and Fermi energy at 0K Number of free electrons /unit volume which possess energy in the range E and E+dE is given by N (E) de g(e) de f(e) The number of free electrons/unit volume of the material, i.e., n is equal to the total number of electrons that are distributed in various energy levels upto E F. Thus we have g(e) de is given by, n n E F E 0 E F E 0 N( E) de g( E) f ( E) de But, f (E), at T 0K n E F E 0 g( E) dex 4π g( E) de (m) h 4π n (m h ) / / E / E F E E 0 4π / / n (m) ( E ) F h 8 / / n πm x ( E ) F h 8π / / n (m) ( E ) F h This is the equation of concentration of electrons in a metal at 0K. h n / Expression for the Fermi energy at 0K is given by E F ( ) 8m π / E F Bn h / Where B ( )( ) is a constant5.85x0-8 J. 8m π Success of Quantum Free Electron theory. The theory could successfully explain the specific heat capacity of metals.. It could also explain temperature dependence of electrical conductivity.. It explained the dependence of electrical conductivity on electron concentration. 4. It also explained photoelectric effect, Compton effect, Black body radiation, Zeeman effect etc., de / de 90

97 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS HALL EFFECT: When a transverse magnetic field B is applied perpendicular to current carrying conductor, a potential difference is developed across the specimen in a direction perpendicular to both current and the magnetic field. This phenomenon is called the Hall effect. The Voltage so developed is called Hall voltage. Hall effect helps to i) determine the sign of charge carrier in the material ) determine the charge carrier concentration and iii) determine the mobility of charge carrier, if conductivity of material is known. Hall effect measurement showed that the negative charge carrier electrons are responsible for conduction in metal and it also shows that there exist two types of charge carriers in semiconductor. BAND THEORY OF SOLIDS: The energy band structure of a solid determines whether it is a conductor, an insulator, or a semiconductor The electron of an isolated atom has certain definite energies such as s,s, p, s, etc. Between two consecutive allowed values of energy there is forbidden gap. As we bring together large number of identical atoms to form a solid, significant changes take place in the energy levels. The energy levels of each atom will interact with the other identical atoms. The wave functions of each atom will overlap and as a result the energy levels of each atom are distributed slightly and split into a number of levels corresponding to the number of atoms. The split energy levels are very close to each other and they form a narrow band known as energy band. The range of energies possessed by electrons in a solid is known as energy band. The energy band formed by the energy levels of the valence electrons is called valence band. The energy band immediately above the valence band where the conduction electrons are present is called conduction band. The separation between the upper level of valence band and the bottom level of conduction band is known as forbidden energy gap, E g. The forbidden energy gap is a measure of the bondage of valence electrons to the atom. The greater the energy gap more tightly valence electrons are bound. When energy is supplied, electrons from the valence band jump to the conduction band and thereby the material starts conducting. SEMICONDUCTORS Pure semiconductors are the materials having electrical conductivity greater than that of insulators but significantly lower than that of a conductor at room temperature. They have conductivity in the range of 0-4 to 0 4 S/m. The interesting feature about semiconductors is that they are bipolar and current is transported by two types of charge carriers of opposite sign namely electrons and holes. The number of carriers can be drastically enhanced by doping the 9

98 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS semiconductor with suitable impurities. The doped semiconductor which exhibits higher conductivity is called an extrinsic semiconductor. The conductivity of an extrinsic semiconductor depends on the doping level which is amenable to control. The current transportation in extrinsic semiconductor occurs through two different processes namely drift and diffusion. Pure semiconductors are of relatively less importance whereas extrinsic semiconductors are widely used in fabricating devices. These devices are more generally known as solid-state electronic devices. INTRINSIC SEMICONDUCTORS A semiconductor in an extremely pure form is known as an intrinsic semiconductor. Intrinsic carriers in pure semiconductors At room temperature in pure semiconductors, a single event of breaking of bonds leads to two carriers; namely electron and hole. The electron and hole are created as a pair & the phenomenon is called electron-hole pair generation. At any temperature T the number of electrons generated will be equal to the number of holes generated. If n denotes number density of electrons in the conduction band & p is the number of holes in the valence band then n p n i where, n i is called intrinsic concentration or the intrinsic density After the generation, the carriers move independently; the electrons move in the conduction band & the holes move in the valence band. The motion of these two carriers is random in their respective band as long as no external field is applied. Concept of Effective Mass of the Electron and Holes: Consider an isolated electron of mass m and charge e in an electric field of strength E. The electric force acting on it is ee. The electron gets accelerated, then -ee ma. However, an electron within a crystal is in a periodic potential due to positive ion cores. The neighboring ions and electrons in the crystal do exert some force on the electron in a crystal. Then ma - ee + force due to neighboring ions and electrons. Since the latter force is not known quantitatively, we can write the above equation as m e *a -ee or m e * -ee/a where m e * is called the effective mass of the electron within the crystal. Thus it is inferred that the effective mass of an electron depends on its location in the energy band. Electrons near the bottom of the conduction band have an effective mass which is almost equal to the effective mass of a free electron. Electrons near the bottom of the valence band have negative effective mass. The removal of an electron with a negative effective mass is identical to creating a particle of positive mass. Thus hole is given the status of particle with positive effective mass m h *. Carrier concentration in intrinsic semiconductor The actual number of electron in the conduction band is given by top of the band n f ( E) g ( E) de () Ec c 9

99 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS Since F-D function describes the probability of occupancy of energy state. Under thermal equilibrium condition, the electron concentration obtained from eqn. () is the equilibrium concentration. As f(e) rapidly approaches zero for higher energies, the integral in eqn. () can be re-written as n f ( E) gc ( E) de Ec 4π * g c ( E) (me ) ( E) de Where E is the kinetic energy of the electron. h E E c Conduction band In the above fig. the bottom edge of conduction band E C corresponds to the potential energy of an electron at rest in conduction band. Therefore the quantity (E E C ) represents the kinetic energy of conduction level electron at high energy level. 4 * g c ( E) π (me ) ( E Ec ) de () h n 4 π * (me ) h E c E EF ( E E c) ( E E + exp kt E EF E EF kt kt kt As E > E F : e >> : + e e ( E Therefore e E EF kt + e Using this eqn in eqn. () we get F EF ) kt de ) () ( E EF ) * kt e ) ( E Ec ) e de Ec ( EF Ec ) ( E Ec ) * kt kt (me ) e ( E Ec ) e Ec 4 n π (m h 4 n π h Let E-E c x Lower limit when EE c x E c E c 0 Upper limit when E x - E c Therefore 4π * n (m e ) h e then dx de ( EF EC ) kt 0 x e ax dx (4) de 9

100 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS The integral is similar to standard integral. The solution of eqn.(4) is given by / ax x e dx 0 π, where a/kt a a 4π * ( EF Ec) kt π / n ( me ) e ( KT ) h Rearranging the term we get * / me kt ( EC EF ) kt π e n h Let * π me kt NC h ( EC EF ) kt NC / n e (6) N c is temperature-dependent material constant known as effective density of states in the conduction band. Expression for hole concentration in valence band If f (E) is the probability for occupancy of an energy state at E by an electron, then probability that energy state is vacant is given by [- f(e)]. Since hole represents the unoccupied state in valence band, the probability for occupancy of state at E by a hole is equal to probability of absence of electron at that level. The hole concentration in valence band is therefore given by E v [ ] p f ( E) gv ( E) de (7) bottomband (5) E v E Valence band - f(e ) rapidly approaches to zero for lower energy levels, the above equation rewritten as E v [ ] p f ( E) g ( E) de p v E v 4π * f ( E) m h Ev E Now f ( E) [ ] ( ) ( ) de h E EF kt e (8) E EF E EF + e + e kt kt 94

101 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS E EF For E<E F (E-E v ) is negative. Therefore e 0 kt ( E F E ) E EF kt Therefore + e and equation 8 reduces to - f(e) e kt E * ( m ) 4π p h ( Ev E) h 4π * p (m h ) h Let E v -Ex then -de dx or de -dx LowerLt UpperLt x E x E v x ( v ) + x E x 0 E E v v e EF Ev E v kt E v 0 E E 0 e ( E E ) F kt de Ev E kt ( E E) e de F v v 4π * kt p (m ) e de h v h 4π * p (m h ) h Above equation is of the standard form 4π p h e EF Ev kt 0 v E E kt ( E E) e ( ) Ev E kt ( E E) e de v ax π x e dx where E v -E x and a a a kt 0 EF Ev * kt π (m ) e ( kt ) h πm p h / * h kt e ( EF Ev ) kt * π mh kt Let Nv h where N v is temperature-dependent material constant known as effective density of states in the valence band. p N v e ( EF Ev ) kt Fermi level in intrinsic semiconductor In an intrinsic semiconductor electron and hole concentrations are equal. Therefore n p N c e ( EC EF ) ( EF EV ) kt NV e kt Taking logarithm on both side and rearranging the term, we get 95

102 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS ( E EF ) NV ( EF E > ln kt N C kt Multiplying by kt throughout C V ) NV > E C + EF kt ln EF + EV N C NV > E F kt ln EC + EV N + C EC + EV NV > EF + kt ln N C Substituting the values of N V and N C and after simplification we get As kt is small and the effective mass E eqn. () may be ignored. F * EC + Ev m h + kt ln * 4 me * me and () * mh do not differ much, the second term in the If * m e * m h, then we get EC + Ev EF we can write eqn. () as EC + Ev + Ev Ev EC Ev EF + Ev but EC Ev Eg Eg EF + Ev Eg If top of the valence band E v is taken as zero level, then E F () Thus Fermi level in the intrinsic semiconductor lies at the centre of the energy gap as shown below: Ec E Eg Ev k 96

103 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS INTRINSIC DENSITY, n i In an intrinsic semiconductor at T0K, the electron concentration in the conduction band is identical to hole concentration in the valence band. npn i From this, we get npn i n i N C ( Ec EF ) kt e ( N C N V ) e N v ( EF EV ) kt e ( Ec EV ) kt But E c -E v E g n n i i ( N ( N C C N N V V ) e / ) Eg / kt e Eg / kt Substituting the values of N c and N v we get, πk / * * / 4 / Eg / kt ni [ ] ( me mh ) T e h The following important points may be inferred from the above relation. The intrinsic density is independent of Fermi level position.. The intrinsic density is a function of band-gap E g, which represents the energy needed to break a bond.. The intrinsic density strongly depends on the temperature. The contribution of temperature increase to n i is mostly due to the exponential term and only to a marginal extent due to the term T /. Expression for the band gap of a Semiconductor: The band gap is the energy separation between the conduction band and the valence band of a semiconducting material. The conductivity of an intrinsic semiconductor is given by σ n eµ + n eµ σ n ie( µ e + µ h ) Substituting the value of n i, we get * * πktm me m h Eg σ x exp e( µ ) e + µ h h m kt Eg The above equation can be written as σ Aexp kt * * πktm me m h Where A x e( µ ) e + µ h h m Eg As σ : ρ Bexp ρ kt 4 4 i e i h 97

104 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS RA l Eg We know that ρ then R B exp l A kt Taking log on both sides Eg R C exp where kt Eg ln R lnc + kt Therefore (ln R lnc) kt The band gap is given by kt (ln R lnc) E g E g Bl C A Eg ln R lnc + is of the form y mx + c : By taking ln R in the y-axis and in the x-axis, kt T if a graph is plotted, a straight line is obtained as shown in below figure. L ln R N Eg M Slope m kt Therefore E g ( m) kt ln C /T By finding the slope of the straight line, the band gap of the semiconductor is determined using the relation, E g k x slope of the straight line drawn between ln R and /T. Extrinsic semiconductor The intrinsic semiconductor has low conductivity which is not amenable to control. However a judicious introduction of impurity atoms in a intrinsic semiconductor produces useful modification of its electrical conductivity. The method of introduction of controlled quantity of impurity into an intrinsic semiconductor is called doping. The impurity added is called dopant. The semiconductor doped with impurity atoms is called extrinsic semiconductor. There are two types of extrinsic semiconductor namely p-type & n-type which are produced depending on the group of impurity atoms. n-type semiconductors are produced when pure semiconductors are doped with pentavalent impurity atoms such Phosphorous, Arsenic etc. p-type semiconductors are produced when pure semiconductors are doped with trivalent impurity atoms such as Aluminum, Boron etc. 98

105 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS Temperature variation of carrier concentration in extrinsic semiconductor The dependence of electron concentration on temperature for n-type semiconductor is as shown in the figure below. n Ionization region Depletion region Intrinsic region I II III T d T T i At 0K the donor levels are filled which means that all the donor electrons are bound to the donor atoms. At low temperature, corresponding to region- I, there is no enough energy to ionize all the donors and not at all enough to break covalent bond. As temperature increases, the donor atoms get ionized and donor atoms go into the conduction band. The region-i is known as ionization region. Occasionally a covalent band maybe broken out, but number of such events will be insignificantly small. At about 00K all donor atoms are ionized, once all electrons from donor level are excited into conduction band, any further temperature increase does not create additional electrons and the curve levels off. The region (region-ii) is called depletion region. In the depletion region the electron concentration in the conduction band is nearly identical to the concentration of dopant atom. If N D is donor concentration then n n N D (depletion region) where n n electron concentration in n-type As temperature grows further, electron transitions from valence band to conduction band increases. At high temperature (region-iii) the number of electron transition becomes so large that the intrinsic electron concentration exceeds the electron concentration due to donor. This region is therefore called intrinsic region. In intrinsic region, n n n i Similarly in p-type semiconductor, the acceptor levels are vacant at 0K & valence band is full. As temperature increases in the ionization region, the electrons from the valence band jump into acceptor level. However the electrons do not acquire enough energy to jump into conduction band levels. At the temperature T s, the acceptor levels are saturated with electrons. The region- II lying between T s (saturation temperature) and T i is called the saturation region. In case of p-type materials within this temperature interval the hole concentration remains constant as thermal energy is not yet sufficient to cause electron 99

106 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS transition from valence band to conduction band. In the saturation region, the hole concentration is equal to the acceptor impurity concentration. Thus p p N A With increase of temperature beyond T, electron transition due to intrinsic process commence & hole concentration due to intrinsic process far exceeds that due to impurity atom. In region-iii, p p n i Fermi level in extrinsic semiconductor The carrier concentration in extrinsic semiconductors varies with temperature as discussed earlier. It follows that the probability of occupancy of respective bands & position of Fermi level varies with temperature. In n-type semiconductor, in low temperature region the electron in the conductor band is only due to the transition of electrons from donor levels. Therefore Fermi level lies between the donor level E D & the bottom edge of conduction band. As temperature increases the donor level gradually gets depleted & the Fermi level shifts downward. At the temperature of depletion T d, the Fermi level coincides with the donor level E D i.e. E Fn E D. As temperature increases further above T d, the Fermi level shifts downward approximately in linear fashion, though electron concentration in the conduction band Ec + ED kt N c remains constant. This is in accordance with the relation EF n ln. N D At temperature T i, where intrinsic process contributes to electron concentration significantly, the Fermi level approaches the intrinsic value E Fi E g /. With further increase in temperature the behavior of extrinsic semiconductor transitions into that of an intrinsic type & Fermi level stays at E Fi.Thus E Fn E Fi E g /. N-type semiconductor 00

107 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS P-type semiconductor In case of p-type semiconductor the Fermi level E Fp rises with increasing temperature from below the acceptor level to intrinsic level E Fi as shown in fig. EA + Ev EFp (ionization region) As temperature increases further above T s, the Fermi level shifts downward approximately in linear fashion, though hole concentration in the valence band remains constant. This is in Ev + E A kt N v accordance with the relation EF p + ln. N A E Fp E A (at TT s ) and E Fp E g /. Fig. Effect of variation of impurity concentration: The addition of donor impurity to an intrinsic semiconductor leads to the formation of discrete donor level below the bottom edge of conduction band. At low impurity concentrations the impurity atom are spaced far apart & do not interact with each other. With an increase in the impurity concentration the impurity atom separation in the crystal decreases & they tend to interact. Consequently the donor level also undergoes splitting & form energy band below the conduction band. The larger the doping concentration, the broader is the impurity band & at one stage it overlaps with the conduction band.. The broadness of donor levels into a band is accompanied by a decrease in the width of forbidden gap & also the upward displacement of Fermi level. The Fermi level moves closer & closer to the conduction band with increasing impurity concentration & finally moves into the conduction band as donor band overlaps the conduction band. In similar way, in p-type semiconductor, the acceptor level broadens and forms into a band with increasing impurity concentration which ultimately overlaps the valence band. The Fermi level moves down closer to the valence band and finally at high impurity concentration it will shift in to valence band. 0

108 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS n-type semiconductor HALL EFFECT IN SEMICONDUCTORS: Let us consider a rectangular plate of p-type semiconductor. When potential difference is applied across its ends, a current I flows through it along x direction. If holes are majority charge carriers in p-type semiconductors then the current is given by I paev d () Where p- concentration of holes A- Area of cross section of end face e- charge on the hole v d - drift velocity of holes therefore current density J I/A pev d () Any plane perpendicular to current flow direction is an equipotential surface. Therefore potential difference between front and rare faces is zero. If magnetic field is applied normal to crystal surface and also to the current flow, a transverse potential difference is produced between the faces F & F /. It is called Hall voltage V H. 0

109 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS Before the application of magnetic field B, the holes move in an orderly eay parallel to faces F & F /. On the application of magnetic field B, the holes experience a sideway deflection due to the Lorentz force F C. The magnitude of this force is given by F L e B v d Because of this force, holes are deflected towards the front dace F and pile up there. Initially the material is electrically neutral everywhere. However as holes pile up on the front side, a corresponding equivalent negative charge is left on the rare dace F /. As a result an electric field is produced across F & F /.The direction of electric field will be from front face to rare face. It is such that it opposes the further pile up of holes on the front face F. A condition of equilibrium is reached when F E due to transverse electric field E H balances the Lorentz force. The transverse electric field E H is known as Hall field. In equilibrium condition F E F H F E e E H e(v H /w) () Where w- width of semiconductor plate From eqn () v d J x /Pe BJx Therefore F L (4) p Equating () and (4) we get evh BJx w p wbjx wbi VH pe pea If t is the thickness of the semiconductor plate, Awt. Then above equation reduces to BI VH pet Hall field per unit current density per unit magnetic field is called Hall co-efficient R H Thus EH VH w wbjx R H BJx BJx wpebjx R H (6) pe Substitute (6) in (5) we get BI VH R H (7) t Vt H R H (8) BI The Hall voltage is a real voltage & can be measured with a voltmeter with the direction of magnetic field & current depicted in this fig, the sign of Hall voltage is +ve. For n-type semiconductor Hall voltage will be ve, when the direction of current is same as in the fig. Therefore by knowing the sign of Hall voltage the type of semiconductor & the sign of the majority charge carriers will be known. 0

110 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS The carrier concentration is given by BI p R e V te In case of n-type semiconductor BI n R e V te H H H H SOLVED NUMERICALS:. What is the probability of a level lying 0.0 ev below the Fermi level not being occupied by electrons at T 00K? Solution: Probability not being occupied by electrons -f(e) - (e (E-E F )/k B T + ) - - (e 0.0/ ) - /(.47 + ) Find the temperature at which there is % occupancy probability of a state 0.5 ev above Fermi energy. Solution: f(e) 0.0 /[e (E-E F )/k B T + ] for E-E F 0.5 ev Solving we get 0.0 /[e 5797/T + ]. Thus e 5797/T / Taking log we get, T 5797/ K. The effective mass of holes in a material is 4 times that of electrons. At what temperature would the Fermi energy be shifted by 0% from the middle of the forbidden energy gap? Given band gap ev. Solution: E F ( E C + E V )/ + (kt/4) log(m h /m e ) Fermi level is shifted by 0% 0. ev. Originally Fermi energy was 0.5 ev above E V. Now it is ev above E F. (E V + 0.6) ev (E C +E V )/ + (kt/4)log(4) ----() And (E V + 0.5) ev (E C +E V )/ () Subtracting from we get 0. ev (kt/4)log(4) 0.60x0-9 J (x.8x0 - xt/4)0.60. T 6K 4. For an intrinsic semiconductor with gap width E g 0.7eV. Calculate the concentration of intrinsic charge carriers at 00K assuming that m e *m o (rest mass of electron). / πmkt ni e h n i.49x0 8 /m Eg / kt 04

111 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS 5. Calculate the free electron concentration, mobility and drift velocity of electrons in an Aluminium wire of length 5m and resistance 60m-ohm, if it carries a current of 5A assuming that each Aluminium atom contributes free electron for conduction. Solution: nooffreeelectronsperatomxn Ax0 xdensity n atomicweight 6 x6.05x0 x.7x0 n n.8x0 m. µ ρen 8 9.7x0 x0 x.8x0 9 µ.7x0 m v s. V µ d E µ IR.79x0 x5x60x0 4 V d µ E.x0 m / s L 5 6. The Fermi level in silver is 5.5eV at 0K. Calculate the number of free electrons per unit volume and the probability occupation for electrons with energy 5.6eV in silver at the same temperature. Solution: / h / E F n 0 8m π 8 n 5.84x0 m. 7. Calculate the probability of an electron occupying an energy level 0.oeV above the Fermi level at 00K and 400K in a material. f ( E) E EF + e kt f ( E) x.6x0.8x0 x00 + e f(e) 0.6 at 400K. 8. A semiconducting material mm long, 5 mm wide and mm thick has a magnetic flux density of 0.5 Wb/m applied perpendicular to the largest faces. A current of 0 ma flows through the length of the sample, and the corresponding voltage measured across its width is 7µV. Find the Hall coefficient of the semiconductor. Solution: Hall coefficient R H E y /J x B z -/ne. Since E y V y /w, R H V y /wj x B z Thus R H (7x0-6 x0 - )/(0x0 - x0.5).7x0-6 m C - 05

112 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS Electrical Properties of matter S. No Questions. An electron is accelerated by an electric field of 4V/cm, is found to have mobility 8x 0 - m / Vs. What is its drift velocity?. How many valence electrons will a donor impurity has in a n-type semiconductor?. With increase in temperature, how does resistance of a pure semiconductor vary? 4. What is a hole in context of semiconductors? 5. In Hall effect experiment what is the polarity of Hall voltage for a n-type semiconductor? 6. What will be the Fermi velocity of an electron in copper if Fermi energy (E F ) 6 ev? 7. At 00K, if probability for occupancy of an energy state E by an electron is 0.75, calculate probability for occupancy of the same state by a hole? 8. Write any two assumptions of Drude-Lorentz theory? 9. Sketch the graph of Fermi factor f(e) verses E for the case EE F at at T> 0K in metals. 0. State density of states in metals.. Write an expression for density of states in metals.. Sketch the variation of fermi level with temperature for n type semiconductor.. What are Fermions? 4. Outline the phenomenon of Hall effect in materials. 5. For silicon semiconductor with band gap.ev, determine the position of the Fermi level at 00K if m * e 0.m o and m * h 0.8m o. 6. Distinguish between intrinsic and extrinsic semiconductors. 7. Find the probability that energy level at 0. ev below Fermi level being occupied at temperature 000K? 8. What is the value of Fermi function when EE f at T>0K? 9. What is the effect of increase of impurity concentration on band gap in extrinsic semiconductors? 0. Mention any two demerits of classical free electron theory.. Find the probability of a level lying 0.0 ev below the Fermi level being occupied by electrons at T 0K?. What is the magnitude of Lorentz force in Hall effect experiment?. With neat sketch, show the Fermi level position in p-type semiconductor. 4. Give the expression for Ohm s law in terms of J,σ and E. 5. What is Fermi factor in Fermi Dirac distribution? 6. Find the relaxation time of conduction electrons in a metal if its resistivity is.5x0-8 Ωm and it has 5x0 8 conduction electrons/m 7. Sketch the position of Fermi level at 0K in a band diagram of a n-type semiconductor, at low doping 8. Find the Fermi velocity of conduction electron if the Fermi energy of silver is 8eV 9. Determine the probability of occupancy of an energy level situated 0.05eV above the Fermi energy at temperature of 0K 06

113 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS 0. Write an expression for carrier concentration of p-type semiconductor.. Write any two postulates of classical free electron theory of solids.. Give the relation between electrical conductivity and mobility of charge carriers in a conductor.. If a system is composed of indistinguishable, half integral spin particles and obeys Pauli exclusion principle, then what statistics is obeyed by the system? 4. What is the value of the Fermi factor for metals at room temperature? 5. Plot variation of Fermi factor with temperature in a metal. 6. Distinguish between free electron theory and band theory of solids in terms of influence of lattice on the electron moving in a metal. 7. Give expression for Fermi level at 0 k in an intrinsic semiconductor. 8. Find the temperature at which there is % probability that a state with energy 0.5 ev above Fermi energy is occupied? 9. What is Hall Effect? 40. Sketch the variation in the energy of the Fermi level in a n type semi-conductor as a function of temperature? 4. Describe in words Wiedemenn-Franz Law 4. What is the formula for intrinsic carrier density (n i )? 4. A wire of diameter 0. meter contains 0 8 free electrons per cubic meter. For an electric current of 0A, calculate the drift velocity for free electrons in the wire? 44. The fermi level in an intrinsic semi-conductor is at.5 ev. What is the width of the band gap? 45. Electrical conductivity of Cu is Ω - m -. If the free electron density of Cu is m -. Find the mobility of electrons? 46. The fermi energy for an intrinsic semiconductor is at 5 ev. At 0K, calculate the probability of occupation of electrons at E 5.5eV? 47. A sample of silicon is doped with 0 7 phosphorous atoms/cm. Find the Hall voltage, if the sample is 00µm thick, I x ma and B z 0-5 Wb/m? 48. Write any one drawback of classical free electron theory? 49. Write the relation for specific heat of a metal as per quantum free electron theory 50. Write the condition at which the value of f(e) at 0 K. 5. Mention any two assumptions of quantum free electron theory. 5. Find the relaxation time of conduction electrons in a metal of resistivity Ωm. If the metal has conduction electrons per m. 5. Why is that only the electrons near the Fermi level contribute to electrical conductivity? 54. Find the probability with which an energy level 0.0 ev below Fermi level will be occupied at room temperature of 00K. 55. A copper strip of.0 cm wide and.0 mm thick is placed in a magnetic field of 5000 gauss. If a current of 00 A is setup in the strip with the Hall voltage appears across the strip is found to be 0.8 V. calculate the Hall coefficient. 56. Which statistical rule is obeyed by electrons in quantum free electron theory? 57. Where does the Fermi level lie in case of n type semiconductor with high impurity 07

114 UNIT IV: ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS concentration? 58. Electron concentration in a semiconductor is 0 0 m. Calculate Hall coefficient? 59. What is doping in semiconductors? 60. Evaluate the probability of occupation of an energy level 0.4 ev below the Fermi energy level in metal at zero Kelvin. 6. Copper has electrical conductivity of 9x07Ω - m - and thermal conductivity of 00 Wm - K - at 05K. Find the Lorentz s number on the basis of classical free electron theory? 6. If the probability of absence of electron in an energy level of valance band of semiconductor is 0.65 what is the probability of occupation in the same level by a hole? 6. In the band diagram of a p-type semiconductor show the position of the Fermi level when the doping concentration is low? 64. Graphically show the variation of ln(n p ) with increasing temperature in Kelvin where n e is the electron concentration in an intrinsic semiconductor. 65. A wire of 4 mm radius carries a current of 8A. Find the current density? Sample Questions ) Give the postulates of classical free electron theory and explain the failures of classical free electron theory. ) Give the success of Quantum free electron theory. ) Discuss the variation of Fermi factor in metals with temperature. 4) Explain density of states in metals. 5) Explain Fermi Dirac distribution function. Show that at temperatures above 0K probability of occupancy of Fermi level in metals is 50%. 6) Define Fermi energy at 0K and at above 0K in metals. 7) Derive an expression for the electron concentration in metals at 0K. 8) Derive an expression for the electron concentration in intrinsic semiconductor. 9) Derive an expression for the hole concentration in intrinsic semiconductor. 0) Show that Fermi level of an intrinsic semiconductor lies in the middle of the band gap. ) With a neat sketch explain how Fermi level changes in n-type semiconductor with the increase in temperature. ) With a neat sketch explain how Fermi level changes in p-type semiconductor with the increase in temperature. ) Give an account of effect of carrier concentration on Fermi level. 4) What is Hall Effect? Obtain an expression for Hall coefficient. 5) Derive expressions for Hall voltage and Hall coefficient in n-type semiconductors. 08

115 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY UNIT V: DIELECTRICS AND THERMAL CONDUCTIVITY Introduction: Materials such as glass, ceramics, polymers and paper are non-conducting materials. They prevent the flow of current through them, therefore they can be used for insulation purposes. When the main function of non-conducting materials is to provide electrical insulation they are called Insulators. distance Figure : Band diagram of an insulator When non-conducting materials are placed in an electric field, they undergo appreciable changes as a result of which they act as stores of electric charges. When charge storage is the main function, the insulating materials are called Dielectrics. For a material to be a good dielectric, it must be an insulator. Hence any insulator is a dielectric. The forbidden energy gap (E g ) between the valence band and conduction band is very large (fig.) in dielectrics and excitation of electrons from valence band to conduction band is not possible under ordinary conditions. Therefore conduction cannot occur in a dielectric. Even if the dielectric contains impurities, extrinsic conduction cannot occur as observed in case of extrinsic semiconductors. The resistivity of an ideal dielectric is infinity, in practise dielectrics conduct electric current to a negligible extent and their resistivity s range from 0 0 to 0 0 Ωm. Electric dipole and dipole moment: Figure : Electric dipole 09

116 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY A pair of equal and opposite charges separated by a small distance is called an electric dipole and the product of the magnitude of one of the charges and the distance of their separation is called dipole moment (µ). Consider two charges q and +q with a distance of separation is a where a is the distance from the centre of dipole to one of the charge as shown in figure. The dipole moment for this arrangement is given by µ (a)q () Field due to an electric dipole at any point in a plane ) Field due to electric dipole at axial point: Figure A configuration of two equal and opposite charges separated by a distance is a dipole. Let P be a point at a distance r from the centre O on the axial line of the dipole, at which the intensity of the field is to be determined (figure ). The field E at P due to the charge -q is towards q and is given by q E () 4πε 0 ( r a) Similarly the field E at P due to +q is away from +q and is given by q E () 4πε 0 ( r + a) The resultant field intensity at P will be E E - E Substituting for E and E from the above equations & q q E 4πε 0 ( r a) 4πε 0 ( r + a) q[( r + a) E 4πε ( r a) ( r + a) q[ r 0 + a ( r a) + ra r ] a E 4πε 0 ( r a ) 4qra E 4πε 0 ( r a ) If r is very much greater that a, then (r +a ) r 4 4qra ( q.a) µ E 4 4πε 0r 4πε 0r πε 0r where µq.a and is the electric dipole moment. + ra] 0

117 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY ) Field due to an Electric dipole at Equatorial point Figure 4 Let P be a point on the perpendicular bisector (equatorial line) of the dipole at a distance r from the centre of O of the dipole. E and E are the fields due to charges +q and q respectively (Figure 4). The resultant field at P is E E + E The field E along AP extended is q E.() 4πε 0 AP where AP OP +OA r +a substituting for AP in equation () q E 4πε 0 ( r + a ) The field E along PB is q E.. () 4πε 0 BP where BP OP +OA r +a substituting for BP in equation () q E...() 4πε ( r + ) 0 a q E E E..(4) 4πε 0 ( r + a ) when E and E are resolved parallel to axial line and along the equatorial line, the component along the equatorial line gets cancelled and along the line parallel to axial line gets added up. The vector sum of these two fields has the magnitude E E cos θ+ E cos θ Ecosθ (5)

118 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY From the right angled triangle AOP OA cos θ AB Substituting equation 4 & 6 in equation 5 we get E ( r qa a + a 4πε 0 ( r + a ) ( r + a q(a) 4πε ) )...(6) qa / 4πε 0 ( r + a ) µ / / 0 ( r + a ) 4πε 0( r + a ) If r is very much greater a then, r +a r and (r +a ) / r µ E 4πε r 0 where µ is electric dipole moment, ε 0 is the permittivity of free space and r is the perpendicular distance from the centre of the dipole to point P. Polar and Non-polar dielectrics: A dielectric material doesn t possess any free electrons. All the electrons are bound very strongly to the respective nuclei of the atoms of the parent molecules. Each molecule consists of equal number of positive and negative charges. All the positive charges are concentrated in the nuclei, which are surrounded by electron clouds in which all the negative charges are distributed. If in the molecules of some dielectric materials, the effective centre of the negative charge distribution coincides with the effective centre of the positive charge distribution such materials are called non-polar dielectrics.eg Hydrogen, carbondioxide etc. In some dielectric materials, the effective centres of the negative and positive charges in the molecules do not coincide with each other in the absence of an external electric field. Each molecule behaves like a permanent dipole and the materials comprising of such dipoles are called polar dielectrics. Ex: HCl, H O etc. Polarization If a dielectric is placed in an electric field of strength E, the electron cloud will be displaced in the direction opposite to E by a distance d with respect to the nucleus. The centres of gravity of positive and negative charges in the atom no more coincide. The atom is equivalent to the system of two charges, qze of equal magnitude but opposite in sign separated by a distance d and the atoms behaves like a dipole and it is called induced dipole. The atom is said to be polarized. E0 E>0

119 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY Figure 5 The induced dipole sets up its own electric field which is opposite in direction to the external field. The dipole moment µ is a vector, directed along the axis of the dipole from the negative charge to the positive charge. When the molecule is polarized, restoring forces due to coulomb attraction come into play which tends to pull the displaced charges together. The charges separate until the restoring force balances the force due to the electric field. The induced dipole moment is proportional to the field strength. The larger the field, greater the displacement of charges and hence larger the induced dipole moment. The induced dipole moment is given by µαe α is the polarizability of the molecule. It characterizes the capacity of electric charges in a molecule to suffer displacement in an electric field. The unit of polarizability is Fm. The induced dipole moment vanishes as soon as the electric field is switched off. Dipole in an electric field: Figure 6: An electric dipole in a uniform electric field When a polar molecule is placed in a uniform electric field E (figure 6), the field exerts a force +qe on the charge +q and qe on q. The net force on the dipole is zero since the two forces acting on it are equal and opposite to each other. Therefore, there is no translational force on the dipole in the uniform field. The two forces are anti parallel and separated by perpendicular distance, hence constitutes a couple, which tends to rotate the dipole. The torque on the dipole is given by τ qe asinθ µ sinθ E (qaµ) Displacement of positive and negative charges in the molecules of a dielectric under the action of applied electric field leading to the development of dipole moment is known as dielectric Polarization. Figure 7

120 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY Consider an electrically neutral slab inserted between the plates of a parallel plate capacitor as shown in figure 7. Dielectric is imagined to be divided into large number of identical cells of volume dv. Under the action of external electric field, charges are induced in each cell and each cell acquires a dipole moment dµ. Then intensity of polarization P is defined as the total dipole moment per unit volume of the material. dµ µ v P dv Dielectric constant: For isotropic materials the electric flux density E and the electric induction (or electric displacement) D are related by the equation D ε 0 ε r E where ε x0 - F/m, is the dielectric constant of vacuum and ε r is the dielectric constant or relative permittivity for the material. It has no units. Dielectric susceptibility: The magnitude of polarization is directly proportional to the intensity of the electric field. Thus, Pχ ε 0 E (for linear dielectrics) χ (chi) is the proportionality constant and is called the dielectric susceptibility of the material. It characterizes the ease with which the dielectric material can be influenced by an external field. P is a measure of the polarization produced in the material per unit electric field. Relation between ε r and χ: In order to describe the combined effects of the applied electric field E and electric polarization P, an auxiliary vector D called Electric displacement vector is introduced. D ε 0 E+P Substituting for Pχ ε 0 E in the above equation D ε 0 E+ χ ε 0 E D (+ χ) ε 0 E D ε 0 ε r E where ε r +χ ε 0 is the absolute permittivity of the free space and ε r is the relative permittivity or the dielectric constant of the material. Note : Dielectric breakdown or Dielectric strength: When a dielectric is subjected to very high electric fields, a considerable number of co-valent bonds are torn away and electrons get excited to energy levels in the conduction band. These electrons acquire a large kinetic energy and cause localized melting, burning and vapourization of material leading to irreversible degradation and failure of the material. It results in high electrical conductivity and total loss of charge storage property of the dielectric. The formation of such conducting paths in a dielectric under the action of an electric field is termed as dielectric breakdown. Dielectric strength is the limiting electric field intensity above which a breakdown occurs and charge storage property of the dielectric disappears. 4

121 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY Types of Polarization ) Electronic or Atomic Polarization: Figure 8 This is the polarization that results from the displacement of electron clouds of atoms or molecules with respect to the heavy fixed nuclei to a distance that is less than the dimensions of atoms or molecules (figure 8). This polarization sets in over a very short period of time, of the order of s. It is independent of temperature. The polarization is given by P e Nα e E..() where N is the number of atoms/unit volume, α e is electronic polarizability. P We have Pχ ε 0 E or χ () ε 0 E Dielectric constant ε r +χ.() Substituting eqn. () in eqn. () P ε r + e..(4) ε 0E Substituting for P e from eqn () in eqn (4) Nα ε r + e E E ε 0 Nα ε r + e ε 0 ( ε r ) (5) or α e ε 0 N ε r is the dielectric constant of a non-polar gaseous dielectric. The above equation indicates that the dielectric constant depends on the polarizability of a molecule and the number of molecules in a unit volume of the dielectric. ) Ionic Polarization: Ionic polarization occurs in ionic crystals. It is brought about by the elastic displacement of positive and negative ions from their equilibrium position. Eg. Sodium chloride crystal. A NaCl molecule consists of Na + ion bound to Cl - ion through ionic bond. If the interatomic distance is d, the molecule exhibits an intrinsic dipole moment equal to qd where q is the charge of the electron and d is the distance of separation. Figure 9 5

122 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY When ionic solids are subjected to an external electric field, the adjacent ions of opposite sign undergo displacement (figure 9). The displacement causes an increase or decrease in the distance of separation between the atoms depending upon the location of the ion pair in the lattice. This polarization takes s to build up and is independent of temperature. Ionic polarization is given by P i Nα i E For most materials, the ionic polarisability is less than electronic polarization. Typically α i 0 αe ) Orientation or dipole Polarisation: This polarization is a characteristic of polar dielectrics which consists of molecules having permanent dipole moment. In the absence of external electric field, the orientation of dipoles is random resulting in a complete cancellation of each others effect (figure 0). When the electric field is impressed the molecular dipoles rotate about their axis of symmetry and tend to align with the applied field and the dielectric acquires a net dipole moment and it is orientation polarization. Figure 0 The dipole alignment is counteracted by thermal agitation. Higher the temperature, the greater is the thermal agitation. Hence, orientation polarization is strongly temperature dependent. In case of solids, the rotation of polar molecules may be highly restricted by the lattice forces, leading to a great reduction in their contribution to orientation polarization. Because of this reason, while the dielectric constant of water is about 80, that for solid ice is only 0. As the process of orientation polarization involves rotation of molecules, it takes relatively longer time than other two polarisations. The build up time is of the order of 0-0 s or more. The orientation polarizability µ α 0 kt Nµ E and orientation polarization P 0 kt Orientation polarization is inversely proportional to temperature and proportional to the square of the permanent dipole moment. 4) Space charge or Interface polarization: This polarisation occurs in multiphase dielectric materials in which there is a change of resistivity between different phases, when such materials are subjected to an electric field, especially at high temperatures, the charges get accumulated at the interface, because of sudden change in conductivity across the boundary (figure ). Since the accumulation of 6

123 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY charges with opposite faces occurs at opposite parts in the low resistivity phase, in effect it leads to the development of dipole moment within the low resistivity phase domain. Eg. Non homogenous materials such as composites. Figure Relaxation time: It is the time required for the dipole to reach the equilibrium orientation from the disturbed position in alternating field conditions. The reciprocal of relaxation time is the relaxation frequency. (or) The average time taken by the dipoles to reorient in the field direction is known as relaxation time (τ). Dielectric loss: When a dielectric material is subjected to the influence of an electric field, its molecules come into a state of electrostatic stress due to polarisation. The molecules behave as electric dipoles. If the polarity of the applied voltage is reversed, the stress is also reversed because of which the dipoles switch their orientation. In order to do so, the dipoles are required to overcome a sort of internal friction which involves a loss of energy. This occurs every time the switching or realignment of the dipoles takes place. The energy loss due to friction is always dissipated as heat by the dielectric material to the surroundings. This is termed as dielectric loss and defined as, loss of energy in the form of heat in a dielectric medium due to internal friction that is developed as a consequence of switching action of electric dipoles under ac conditions. This part of energy is lost or wasted as heat and no useful work is done by it. Power loss in dielectrics, when subjected to dc voltages will be very small due to high resistance of dielectric materials. The power loss in an ac field will be quite large. In an ideal dielectric material, loss is zero. Frequency dependence of Dielectric constant: Dielectric constant (ε r) remains unchanged when the material is subjected to a dc voltage. But the dielectric constant (ε r)` changes when the material is subjected to the influence of an ac voltage, the changes depend on the frequency of the applied voltage. 7

124 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY Figure : Variation of dielectric constant with frequency All the four polarization mechanisms that occur in a dielectric material will be effective in static field conditions. But, each of them respond differently at different frequencies under alternating field conditions, since the relaxation frequency of different polarization processes are different as shown in figure. If τ e, τ i and τ o are the relaxation times for electronic, ionic and orientation polarisations, then in general τ e < τ i < τ o When the frequency of applied field matches the relaxation frequency of a given polarisation mechanism, the absorption of energy from the field becomes maximum. When the frequency of applied field becomes greater than the relaxation frequency for a particular polarisation mechanism, the switching action of the dipoles cannot keep in step with that of changing field and the corresponding polarization mechanism is halted. Thus as the frequency of applied ac is increased, different polarization mechanisms disappear in the order- interface, orientation, ionic and electronic. f 0 <f i <f e In addition, ε r becomes a complex quantity and is expressed as ε r * ε r -j ε r where ε r and ε r are the real and imaginary parts of the dielectric constant. ε r represents part of the dielectric constant that is responsible for the increase of capacitance of a capacitor when the dielectric is placed between the plates of a capacitor. ε r represents the loss. The loss that occurs in a dielectric material is essentially due to the phase lag of voltage behind current in the capacitor between the plates of which dielectric material lies. Such a loss in a capacitor is expressed by a factor called tan δ. A large value of tan δ signifies higher dielectric loss. It is also referred to as tangent loss " ε r tan δ ε ' r 8

125 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY Internal field in a solid for one dimensional infinite array of dipoles: When a dielectric material, either solid or liquid is subjected to an external electric field, each of the atoms develope a dipole moment and acts as an electric dipole. Hence the resultant field at any given atom will be the sum of applied electric field and the electric field due to the surrounding dipoles. The resultant local field is called the internal field (E i )and is defined as the electric field that acts at a site of any given atom of a solid or liquid dielectric subjected to an external electric field and is the resultant of the applied field (E) and the field due to all the surrounding dipoles (E ' ). E i E+ E ' Figure : Linear array of atoms in an electric field Consider a dielectric material kept in an external uniform electric field of strength E. In the material let us consider an array of equidistant atomic dipoles arranged parallel to the direction of the field as shown in figure. In the linear array let us consider an atom X. The distance from X to the dipole A is d, to the dipole B is d and so on ( d is the distance between two consecutive dipoles). The dipole moment of each of the atomic dipole is µ. The total field at X due to all the other dipoles in the array is evaluated as follows. µ µ Field at X due to A () 4πε 0 d πε 0d Consider the dipole A, since it is situated symmetrically on the other side of X, µ µ its field at X will also be 4πε 0 d πε 0d Therefore the field E at X due to both the dipoles A and A gets added up µ µ E [ ].() πε 0d πε 0 d Similarly if we consider the two dipoles B and B each of which is located at a distance d, then the field E at X due to both of them is µ µ E [ ] () πε 0 (d) πε 0 (d) Similarly the field at X due to the dipoles C and C is 9

126 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY µ E πε ( d 0 ).() Therefore the total field E ' induced at X due to all the dipoles in the linear array is µ πε 0 d By summation of infinite series E ' E + E + E + E n µ µ µ µ πε 0 d πε 0 ( d) πε0 ( d ) πε 0 (nd) µ [ ] πε 0 d n n. n n where n,,,. E '.µ πε 0d Therefore the total field at X which is the internal field E i, is the sum of the applied field (E) and the field due to all the dipoles (E ' ) E i E+ E '.µ E+ πε 0d Thus, the combined effect of induced dipoles of neighbouring atoms is to produce a net field at the location of a given atom, which is larger than the applied field. Clausius Mosotti Equation: Let us consider a solid dielectric, which exhibits electronic polarizability. If α e is the electronic polarizability per atom, it is related to the bulk polarization P through the relation Pα e NE i P Therefore α e. () NE i where N is the number of atoms per m and E i is the local field. P From Lorentz field equation E i E+. () ε 0 Substituting equation () in equation () we get α e We have E P P N[ E + ] ε 0 P ε ( ε ) 0 r Substituting for E in the above equation, we obtain 0

127 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY α P e P P N[ + ] ε ( ε ) Nαe ε [ ε 0 r ε ( ε r ) Nα e ( ε + ) ε r r 0 ( ε r ) Nαe ( ε + ) ε r 0 ] ε r + [ ] ( ε ) The above equation is known as Clausius Mosotti equation. r Appendix : (Additional information) Loss Angle and Loss Tangent Let us consider a parallel plate capacitor C, constituted by plates of area A and separated by a distance d. Let a dielectric having permittivity ε r fill the space between the plates. Let a sinusoidal voltage V of angular frequency ω be applied to the capacitor. The current through the capacitor is given by V I jω CV + R IjI r +I a The above relations indicate that two kinds of current flow through the dielectric - the conduction current I a V/R and the displacement current I r given by I r ωcv The resultant current I I + lags behind the displacement current by an angle δ. r I a I r I C δ φ I a In case of an ideal dielectric, R and I a 0, and it would not absorb electric energy. In such a case, the resultant current I would be ahead of the voltage V precisely by a phase angle ϕ90 0 and the current would have been purely reactive current I r.

128 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY wε r AV I I 0 ε r d However, for a lossy dielectric, the total current is II a +j I r The phase angle ϕ between V and I is now slightly less than The angle δ(90-ϕ) is called the loss angle. It is given by I a tanδ I r V / R or tan δ ωcv ωcr tan δ is known as the loss tangent. It represents the electrical power lost which is in the form of heat. Hence it is also called dissipation factor. The real power loss in the dielectric is given by P L VI a VI r tan δ ωcv tan δ. ωε 0ε r AV tanδ d Complex relative permittivity: In most of the materials, the dielectric behaviour is more complex indicating the presence of other sources of dielectric loss. To include losses from all sources, I II a +j I r j I r a jωavε 0ε r j ( j tanδ ) I r d The above equation suggest that the lossy dielectric can be described with the aid of a complex relative permittivity ε * r given by ε * r ε r (-j tanδ) or ε * r ε r -jε r ε r ε r * C R δ φ ε r where ε tanδ ε " r ' r Therefore the product (ε r tan δ) is known as the loss factor. A lossy dielectric is represented by a resistance parallel to the capacitor.

129 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY Lorentz field Figure 6 The local field in a three dimensional solid is determined by the structure of the solid. Let us consider a dielectric slab kept in a uniform electric field, E (Figure 5). Let a molecule be at the point O and is surrounded by a spherical cavity of radius r. Let r be arbitrary but sufficiently large compared to molecular dimensions and sufficiently small compared to the dimensions of the dielectric slab. The spherical cavity contains many molecules within it. The molecule at O experiences three electric fields acting on it. i) The external electric field E. ii) The field E due to induced charges on the surface of the spherical cavity. iii) The field E due to the molecular dipoles present in the spherical cavity. Therefore, the total internal field intensity, E i is given by E i E+ E +E To calculate E, let us imagine that the dielectric is removed from the sphere. For the actual pattern of the electric field not to be distorted, a surface electric charge should be placed on the spherical surface. At each point of the sphere, the surface charge density is given by σp cos θ where θ is the angle between radius vector r and the direction E. The charge on the element ds of the surface of the sphere will be dq σds P cos θ ds This charge will produce electric field intensity de at the centre of the sphere dq P de cosθ ds 4πε0r 4πε0r This electric field can be resolved into two components: one component de cos θ parallel to the direction of E and the other de sinθ perpendicular to the direction of E. P de cosθ cos θ ds 4πε0r P de cosθ cosθ sinθ ds 4πε r 0

130 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY It is obvious that the perpendicular components of the upper and lower half of the sphere cancel each other and only the parallel components contribute to the total intensity E. E is obtained by integrating de over the whole surface area of the sphere. Thus, E But dsπr sinθ dθ. Therefore, π 0 P E ε P de cos θ ds θ ds Πε r cos 4 π 0 0 cos θ sinθ dθ Let cosθ x and therefore, -sin θ dθ dx. when θ0, cos θ and θπ, cos π- 0 π 0 P P x P P P P E - ε x dx 0 ε 0 ε 0 ε 0 6ε 0 ε 0 As there exists symmetrical distribution of molecular dipoles around the molecules at O within the cavity, their contribution cancel each other. Therefore E 0. Hence the total internal field is given by E i E+ E P E i E+ ε 0 The field given by the above equation is called Lorentz field or local field. 4

131 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY THERMAL CONDUCTIVITY Introduction: Matter exists in solid, liquid or gaseous states. The particular state of a substance depends on its temperature. According to kinetic theory, the molecules of a substance in the solid state have less degrees of freedom than a substance in the liquid or gaseous state. The molecules are free to move in the gaseous state. The change of state can be brought about by supplying or withdrawing heat from the substance. Ice is the solid state of water. By supplying heat to ice, it can be changed into water. Similarly by supplying heat to water, it can be converted into the gaseous state (steam). The reverse processes occur when heat is withdrawn from steam. This is true for all substances. Even permanent gases like oxygen, nitrogen, hydrogen etc. can be liquefied at low temperature. For a given substance, the change of state takes place at a fixed temperature and pressure. When the substance changes from solid to liquid state, the heat supplied at constant temperature (called melting point) is used in overcoming the forces of intermolecular attraction. The mean molecular distance increases and the molecules are more free to move. Similarly, when the substance changes from the liquid to gaseous state at a fixed temperature (called the boiling point) and pressure, the heat supplied is used in increasing the mean molecular distance. The molecules become free to move about in the whole space available to them. Heat can be transferred from one place to the other in three different ways i.e., Conduction, Convection and Radiation. Conduction is the process in which heat is transmitted from one point to the other through the substance without the actual motion of the particles. When one end of a metal bar is heated, the molecules at the other end vibrate with higher amplitude (kinetic energy) and transmit the heat energy from one particle to the next and so on. However, the particles remain in their mean positions of equilibrium. This process of conduction is prominent in the case of solids. Metals are good conductors of heat. The space between the two walls of a thermo flask is evacuated because vacuum is a poor conductor of heat. The air enclosed in the woollen fabric helps in protecting us from cold because air is a poor conductor of heat. Convection is the process in which heat is transmitted from one place to the other by the actual movement of the particles. It is prominent in the case of liquids and gases. Land and sea breezes are formed due to convection. Radiation is the process in which heat is transferred from one place to other directly without the necessity of the intervening medium. We get heat radiations directly from the sun without affecting the intervening medium. Example is Radiation from sun. Linear flow of heat through a long bar; the steady state Pioneer work on conduction was done by Fourier and followed by Ohm. In his arrangement one end of a uniform rod is heated with a heat source as shown in figure. 5

132 UNIT- V: DIELECTRICS AND THERMAL CODUCTIVITY The rod has equally spaced holes along its length. A number of thermometers are inserted in them. It was found that the thermometer nearest to the hot end first registered a rise in temperature, followed by other in succession, as the heat travelled from the hotter to the colder end of the rod. Such a rise in temperature along the rod actually depends upon the quantity of heat conducted, the specific heat and density of the material of the rod. With the temperature thus changing along it, the rod is said to be in variable state of temperature. Figure The rod has equally spaced holes along its length. A number of thermometers are inserted in them. It was found that the thermometer nearest to the hot end first registered a rise in temperature, followed by other in succession, as the heat travelled from the hotter to the colder end of the rod. Such a rise in temperature along the rod actually depends upon the quantity of heat conducted, the specific heat and density of the material of the rod. With the temperature thus changing along it, the rod is said to be in variable state of temperature. Of the total heat entering any element or section of the rod, a part of it is absorbed or retained by the element to raise its own temperature, another part is conducted to the adjoining element and the rest of the heat is lost to the surroundings by radiation from the surface of the rod. With the passage of time, as the element gets hotter, the proportion of heat absorbed by it increases and attains a constant value, with the result that each segment of the rod attains a constant temperature. The excess of heat entering an element over that absorbed by it and conducted is lost by radiation. This state is called a steady state or stationary state. The temperature of any element of the rod in the steady state is purely a function of its distance from the hotter end and is independent of time. Since no change of temperature is involved, the flow of heat and the distribution of temperature along the rod now cease to be dependent on the density or specific heat of its material. Figure During the steady state, the quantity of heat flowing across the two opposite faces of an element (figure ) is directly proportional to (i) the difference of temperature (θ - θ ) between 6