{"id":183,"date":"2018-12-04T07:16:31","date_gmt":"2018-12-04T07:16:31","guid":{"rendered":"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=183"},"modified":"2018-12-04T10:35:26","modified_gmt":"2018-12-04T10:35:26","slug":"lattice-vibrations-and-thermal-properties-3","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/chapter\/lattice-vibrations-and-thermal-properties-3\/","title":{"rendered":"Lattice Vibrations and Thermal Properties 3"},"content":{"raw":"<div>\r\n<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/yC-JyT-qyF0\" target=\"_blank\" rel=\"noopener\"><img src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a>\r\n<\/span><\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nThe objective of the module is to understand the following\r\n<ul>\r\n \t<li>Reviewing the concept of specific heat<\/li>\r\n \t<li>Concept<span style=\"text-align: initial;font-size: 1em\"> of lattice specific heat in <\/span>preview<span style=\"text-align: initial;font-size: 1em\"> of lattice dynamics<\/span><\/li>\r\n \t<li>Reviewing the explanation of specific heat on the basis of classical<span style=\"text-align: initial;font-size: 1em\"> model given by Dulong and\u00a0<\/span>Petit\u2019 law.<\/li>\r\n \t<li>Concept<span style=\"text-align: initial;font-size: 1em\"> of quantization of elastic waves and phonons<\/span><\/li>\r\n \t<li>Understanding of<span style=\"text-align: initial;font-size: 1em\"> the shortfall of classical theory and the need of the other.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a01.Lattice Specific Heat : An Overview<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Since heat is not a thermodynamic quantity, <\/span>however<span style=\"text-align: initial;font-size: 1em\"> it becomes so under the constraints of constant volume and constant pressure i.e,<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">dQ<\/span><strong style=\"text-align: initial;font-size: 1em\">|<\/strong><strong style=\"text-align: initial;font-size: 1em\">p<\/strong><span style=\"text-align: initial;font-size: 1em\"> = dH<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">and<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">dQ<\/span><strong style=\"text-align: initial;font-size: 1em\">|<\/strong><span style=\"text-align: initial;font-size: 1em\">v = dE<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">where H and E are <\/span>enthalpy<span style=\"text-align: initial;font-size: 1em\"> and internal energy respectively.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The specific heat is defined as the amount of heat energy required to raise the temperature of a unit mass of solid by one degree. <\/span>Also<span style=\"text-align: initial;font-size: 1em\"> it is the heat capacity per unit mass. Consequently<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(\u2202H\/\u2202T)p = Cp and (\u2202E\/\u2202T)v = Cv are then the expressions for heat capacity at constant pressure and heat capacity at constant volume respectively.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">For solids and liquids, Cp ~ Cv, especially at low temperatures (~0-20K), but even at higher temperatures (~300K), the difference is not more than 2%.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">With the supply of heat energy to a solid, there is an increase in its internal energy. The increase in the internal energy is manifested as an increase in the vibrations of the atoms about their mean positions and also as an increase in the kinetic energies of the free electrons. The specific heat corresponding to lattice energy is called lattice specific heat. <\/span>Table<span style=\"text-align: initial;font-size: 1em\"> below shows specific heats of a few materials.<\/span><\/p>\r\n\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-187\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-42.png\" alt=\"\" width=\"782\" height=\"288\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">There are various theories of lattice specific heat. In all the theories the vibrational energy of a crystal containing N number of atoms is considered equivalent to the energy of a 3N harmonic oscillator. The distinction in various theories is the difference in <\/span>proposed<span style=\"text-align: initial;font-size: 1em\"> frequency spectrum of the oscillators and the problem regarding calculations of wavelengths and frequencies of the possible modes of vibration in the crystal remains the centric concern.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In order to understand the lattice dynamics and the phenomenon that occur when heat or thermal energy is supplied to a solid and is consequently raised to a temperature (it occurs even at absolute zero), it will be easier to begin with the first law of thermodynamics which states that whenever some amount of heat dQ is given to a system, it results in the increase in energy dE of the system plus the amount of work done, i.e<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">dQ<span style=\"text-align: initial;font-size: 1em\"> = dE+ pdV ---------------(1)<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Considering that the work done by the system is of a mechanical nature only.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Now, E is determined uniquely by the temperature and volume of the system. Hence<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"size-full wp-image-188 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-43.png\" alt=\"\" width=\"662\" height=\"177\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\ndQ\/dT is the general expression for specific heat. Our interest remains in Specific heat at constant volume C<em>v<\/em> and specific heat at constant pressure C<em>p<\/em>.\r\n\r\n&nbsp;\r\n\r\nFrom equation (2)\r\n\r\n<strong><img class=\"size-full wp-image-189 alignleft\" style=\"font-size: 1em\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-44.png\" alt=\"\" width=\"661\" height=\"57\" \/><\/strong>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nThe second law of thermodynamics is stated as relation between C<sub>p<\/sub> and C<sub>v<\/sub> in the form\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<img class=\"size-full wp-image-190 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-45.png\" alt=\"\" width=\"686\" height=\"412\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-192\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-46.png\" alt=\"\" width=\"602\" height=\"432\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">As depicted from <\/span><strong style=\"text-align: initial;font-size: 1em\">fig (1)<\/strong><span style=\"text-align: initial;font-size: 1em\"> at low temperatures their difference becomes very small and both C<\/span><em style=\"text-align: initial;font-size: 1em\">p<\/em><span style=\"text-align: initial;font-size: 1em\"> &amp; C<\/span><em style=\"text-align: initial;font-size: 1em\">v<\/em><span style=\"text-align: initial;font-size: 1em\"> go to zero at <\/span><em style=\"text-align: initial;font-size: 1em\">T<\/em><span style=\"text-align: initial;font-size: 1em\"> = 0.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In general, the variation of C<\/span><em style=\"text-align: initial;font-size: 1em\">v<\/em><span style=\"text-align: initial;font-size: 1em\"> with time is studied. We assume that a change in volume is not much with a small increase in temperature.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">As the temperature increases from absolute zero, there is a rapid increase in specific heat and it finally levels off at a nearly constant value (6 <\/span>cal<span style=\"font-size: 1em\">\/mole) at high temperature as shown in figure(2) below. This is the classical approach put forward as in Dulong Petit\u2019s Law.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">2. Dulong and Petit Law (Classical overview)<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The physical properties of the solids <\/span>is<span style=\"text-align: initial;font-size: 1em\"> roughly assumed to be<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n<ul>\r\n \t<li>due to the contributions resulting from the atomic vibrations (as in crystals)<\/li>\r\n \t<li style=\"text-align: justify\">due to additional<span style=\"text-align: initial;font-size: 1em\"> contribution to specific heat from <\/span>electronic<span style=\"text-align: initial;font-size: 1em\"> system (as in metals and semiconductors). Although this contribution is relatively small to that of lattice vibrations.<\/span><\/li>\r\n<\/ul>\r\n<p style=\"text-align: justify\">\u00a0 \u00a0 Taking these assumptions into consideration and assuming the behaviour of the atoms of the crystal as to be independent classical harmonic oscillator, the heat capacity of the solid crystal can be calculated simply by finding average thermal energy of a single oscillator and then multiplying it by total number of oscillators, N.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">In comparison to the translational motion of the ideal gas molecules, the constituent atoms in a crystal , have almost fixed positions which vibrate about their mean positions executing simple harmonic motion. The total energy of the oscillator at an instant is a composition of its instantaneous potential energy and kinetic energy. Classically a harmonic oscillator vibrating with its natural frequency \u03c9 has energy expression as<\/p>\r\n&nbsp;\r\n\r\nE = P<sup>2<\/sup>\/2m + \u00bd m\u03c92(x<sup>2<\/sup>+ y<sup>2<\/sup> + z<sup>2<\/sup>)\r\n\r\nE = P<sup>2<\/sup>\/2m +m\u03c9<sup>2<\/sup>q<sup>2<\/sup>\/2 -----------------------(6)\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">where p is the momentum in kinetic energy term and q is the displacement from equilibrium position in potential energy term .<\/p>\r\n&nbsp;\r\n\r\nThe average energy of each harmonic oscillator as given by Plank\u2019s distribution law is\r\n\r\n<img class=\"size-full wp-image-193 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-47.png\" alt=\"\" width=\"698\" height=\"514\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<img class=\"size-full wp-image-194 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-48.png\" alt=\"\" width=\"418\" height=\"349\" \/>\r\n\r\n<\/div>\r\n<strong>\u00a0<\/strong>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">This gives the vibrational energy of one harmonic oscillator. To obtain total vibrational energy of the crystal whish has N independent harmonic oscillators in three dimensions, we have<\/p>\r\n<img class=\"size-full wp-image-195 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-49.png\" alt=\"\" width=\"551\" height=\"120\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nThis is called Dulong Petit\u2019s classical law of specific heat.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">This turns out to be that the heat capacity is independent of temperature and has a constant value at room temparature.<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-196\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-50.png\" alt=\"\" width=\"594\" height=\"381\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong>3. Shortfall of Classical Theory:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The classical theory gives a good idea of energy contributions to the total energy of the system as long as the volume remains constant, which is given in equation (9).<\/p>\r\n&nbsp;\r\n\r\nThe specific heat at constant volume as obtained for a solid containing N number of atoms is given as\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\"><img class=\"size-full wp-image-197 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-51.png\" alt=\"\" width=\"311\" height=\"52\" \/>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nThis is the level off value of the experimental results at high temperature.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Equation (8) is a frequency independent expression and also it depends only on temperature. The results obtained here are in quantitative agreement with those obtained from experiments but at higher temperatures only.<\/p>\r\n&nbsp;\r\n\r\nThis does not explain the decrease of specific heat at low temperatures as observed for all solids.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The discrepancy to a certain extent is removed by using quantum theory but before that it is important to understand the concept of quantization of elastic waves and the concept of phonons.<\/p>\r\n&nbsp;\r\n\r\n<strong>4. Quantization of Elastic Waves:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">As is a well understood concept that the atoms in solids vibrate about their equilibrium positions and these lattice vibrations can be expressed in the form of waves, like repetitive and systematic sequence of atomic displacements that can be longitudinal or transverse or even a combination of both.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">The energy of a lattice vibration is quantized and the quantum of energy is called \u2018Phonon\u2019, analogous to quanta of electromagnetic energy \u2013 photon.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">We have assumed vibrations of a linear lattice connected through springs and their particle motion can be quantized in a similar way as that for a harmonic oscillator or a combination of coupled harmonic oscillators. The Hamiltonian for a harmonic oscillator is<\/p>\r\n&nbsp;\r\n\r\nH = (1\/2M) p<sup>2<\/sup> + \u00bd Cx<sup>2<\/sup>\r\n\r\n&nbsp;\r\n\r\nWith energy eigen values for n = 1, 2 , 3 ,\u2026.\r\n\r\n&nbsp;\r\n\r\nHence average thermal energy of the oscillator is\r\n\r\n&nbsp;\r\n\r\n\u0404n= (n + \u00bd)\u0127\u03c9\r\n\r\n&nbsp;\r\n\r\nWhere \u03c9 is the angular frequency.\r\n\r\n&nbsp;\r\n\r\nThe term \u00bd \u0127\u03c9 is called zero point energy of the mode\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-198\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-52.png\" alt=\"\" width=\"506\" height=\"370\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Thus thermal lattice vibrations are thermally excited phonons. Thermal conduction in non metallic crystals is a consequence of annihilation or creation of a phonon. The energy of phonons is ~0.1 ev.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">We know that if the particles have zero spin then these follow Bose \u2013 Einstein statistics and the probability of such particles having energy E is given as<\/p>\r\n<img class=\"size-full wp-image-199 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-53.png\" alt=\"\" width=\"363\" height=\"184\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The following observations confirm the experimental evidence for the quantization of lattice vibrational energy:<\/p>\r\n\r\n<ul>\r\n \t<li>at absolute zero the lattice contribution of heat energy always approaches zero<\/li>\r\n \t<li>the neutrons and X rays are scattered inelastically by crystals and the changes in energy and momentum correspond to creation or absorption of phonons<span style=\"text-align: initial;font-size: 1em\">.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<div>\r\n\r\n\u00a0 \u00a0 (Which is discussed in detail in the later section of the modules.)\r\n\r\n&nbsp;\r\n\r\n<strong>5.<\/strong>\u00a0<strong>Summary:<\/strong>\r\n\r\n&nbsp;\r\n\r\nAfter the completion of this module we are able to understand and correlate the following\r\n<ul>\r\n \t<li>Heat capacity, a thermodynamic concept as a function of temperature.<\/li>\r\n \t<li>Classical<span style=\"text-align: initial;font-size: 1em\"> theory of specific heat capacity and discrepancy with the experimental results obtained at lower temperatures.<\/span><\/li>\r\n \t<li>Shortfall<span style=\"text-align: initial;font-size: 1em\"> of the Dulong Petit\u2019s classical theory and need of new (quantum) approach. <\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Quantization of elastic waves, its physical significance <\/span>and<span style=\"text-align: initial;font-size: 1em\"> mathematical treatment. <\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Introduction of <\/span>concept<span style=\"text-align: initial;font-size: 1em\"> of phonons and its relevance in lattice dynamics.<\/span><\/li>\r\n<\/ul>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>you can view video on Lattice Vibrations and Thermal Properties 3<\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/yC-JyT-qyF0\" target=\"_blank\" rel=\"noopener\"><img class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 Value Addition:<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>Do You Know?<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>The relation between specific heat at constant pressure and at constant volume in terms of coefficient of volume expansion \u03b1<\/strong><strong>v<\/strong><strong> and the compressibility K is given as<\/strong><\/p>\r\n&nbsp;\r\n\r\n<strong>Cp \u2013 Cv = \u03b1<\/strong><strong>v<\/strong><strong>2<\/strong><strong>TV\/K,<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>Cp \u2013 Cv \u2265 0<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <strong>\u03b1<\/strong><strong>v<\/strong><strong> and K being positive quantities,<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>And in general the relationship employed is<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>Cp = Cv (1+ \u03b3\u03b1<\/strong><strong>v<\/strong><strong>T)<\/strong>\r\n\r\n&nbsp;\r\n\r\n\u03b3\u00a0\u00a0 <strong>= \u03b1<\/strong><strong>v<\/strong><strong> V \/ KCv, is Gr\u00fcneisen constant and is independent of temperature.<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>By calculation of \u03b3 at any arbitrary temperature, an approximation for Cv is obtained.<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>As already discussed that a part of energy given to a system results in the increase in its internal energy which is associated with more vigorous notion of the atoms. In metals and semiconductors the remaining contribution in the specific heat comes from the electronic system, although this contribution is very small as compared to that of the lattice.<\/strong><\/p>\r\n&nbsp;\r\n\r\n<strong>C<\/strong><strong>solid<\/strong><strong> =C<\/strong><strong>lattice +<\/strong><strong> C<\/strong><strong>electronic<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>Anomalies have been observed in specific heat curves for ferromagnetic metals and e.g, nickel, iron and cobalt. A prominent peak appears near Curie temperature. This is correlated with transition from ordered state to disordered state. The alloys show more peaks in specific heat curves that exhibit similar transitions.<\/strong><\/p>\r\n&nbsp;\r\n\r\n<strong>For More Details ( on this topic and other related topics ) See<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>Adrianus J Dekker, Solid State Physics<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Glossary:<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Elastic vibrations:<\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Waves when require a material medium to travel (unlike e.m waves<\/span>),<span style=\"text-align: initial;font-size: 1em\"> are called elastic vibrations.<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">These can have both longitudinal and transverse modes of vibration.<\/span>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Crystal momentum:<\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Phonons do not carry any momentum on the lattice but a phonon wave vector k interacts with the crystal constituents and other particles as if its momentum is \u0127k, more precisely called Crystal momentum<\/span>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/div>","rendered":"<div>\n<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/yC-JyT-qyF0\" target=\"_blank\" rel=\"noopener\"><img decoding=\"async\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a><br \/>\n<\/span><\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>The objective of the module is to understand the following<\/p>\n<ul>\n<li>Reviewing the concept of specific heat<\/li>\n<li>Concept<span style=\"text-align: initial;font-size: 1em\"> of lattice specific heat in <\/span>preview<span style=\"text-align: initial;font-size: 1em\"> of lattice dynamics<\/span><\/li>\n<li>Reviewing the explanation of specific heat on the basis of classical<span style=\"text-align: initial;font-size: 1em\"> model given by Dulong and\u00a0<\/span>Petit\u2019 law.<\/li>\n<li>Concept<span style=\"text-align: initial;font-size: 1em\"> of quantization of elastic waves and phonons<\/span><\/li>\n<li>Understanding of<span style=\"text-align: initial;font-size: 1em\"> the shortfall of classical theory and the need of the other.<\/span><\/li>\n<\/ul>\n<\/div>\n<p><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a01.Lattice Specific Heat : An Overview<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Since heat is not a thermodynamic quantity, <\/span>however<span style=\"text-align: initial;font-size: 1em\"> it becomes so under the constraints of constant volume and constant pressure i.e,<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">dQ<\/span><strong style=\"text-align: initial;font-size: 1em\">|<\/strong><strong style=\"text-align: initial;font-size: 1em\">p<\/strong><span style=\"text-align: initial;font-size: 1em\"> = dH<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">and<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">dQ<\/span><strong style=\"text-align: initial;font-size: 1em\">|<\/strong><span style=\"text-align: initial;font-size: 1em\">v = dE<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">where H and E are <\/span>enthalpy<span style=\"text-align: initial;font-size: 1em\"> and internal energy respectively.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The specific heat is defined as the amount of heat energy required to raise the temperature of a unit mass of solid by one degree. <\/span>Also<span style=\"text-align: initial;font-size: 1em\"> it is the heat capacity per unit mass. Consequently<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(\u2202H\/\u2202T)p = Cp and (\u2202E\/\u2202T)v = Cv are then the expressions for heat capacity at constant pressure and heat capacity at constant volume respectively.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">For solids and liquids, Cp ~ Cv, especially at low temperatures (~0-20K), but even at higher temperatures (~300K), the difference is not more than 2%.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">With the supply of heat energy to a solid, there is an increase in its internal energy. The increase in the internal energy is manifested as an increase in the vibrations of the atoms about their mean positions and also as an increase in the kinetic energies of the free electrons. The specific heat corresponding to lattice energy is called lattice specific heat. <\/span>Table<span style=\"text-align: initial;font-size: 1em\"> below shows specific heats of a few materials.<\/span><\/p>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-187\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-42.png\" alt=\"\" width=\"782\" height=\"288\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-42.png 782w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-42-300x110.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-42-768x283.png 768w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-42-65x24.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-42-225x83.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-42-350x129.png 350w\" sizes=\"auto, (max-width: 782px) 100vw, 782px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">There are various theories of lattice specific heat. In all the theories the vibrational energy of a crystal containing N number of atoms is considered equivalent to the energy of a 3N harmonic oscillator. The distinction in various theories is the difference in <\/span>proposed<span style=\"text-align: initial;font-size: 1em\"> frequency spectrum of the oscillators and the problem regarding calculations of wavelengths and frequencies of the possible modes of vibration in the crystal remains the centric concern.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In order to understand the lattice dynamics and the phenomenon that occur when heat or thermal energy is supplied to a solid and is consequently raised to a temperature (it occurs even at absolute zero), it will be easier to begin with the first law of thermodynamics which states that whenever some amount of heat dQ is given to a system, it results in the increase in energy dE of the system plus the amount of work done, i.e<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">dQ<span style=\"text-align: initial;font-size: 1em\"> = dE+ pdV &#8212;&#8212;&#8212;&#8212;&#8212;(1)<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Considering that the work done by the system is of a mechanical nature only.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Now, E is determined uniquely by the temperature and volume of the system. Hence<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-188 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-43.png\" alt=\"\" width=\"662\" height=\"177\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-43.png 662w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-43-300x80.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-43-65x17.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-43-225x60.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-43-350x94.png 350w\" sizes=\"auto, (max-width: 662px) 100vw, 662px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>dQ\/dT is the general expression for specific heat. Our interest remains in Specific heat at constant volume C<em>v<\/em> and specific heat at constant pressure C<em>p<\/em>.<\/p>\n<p>&nbsp;<\/p>\n<p>From equation (2)<\/p>\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-189 alignleft\" style=\"font-size: 1em\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-44.png\" alt=\"\" width=\"661\" height=\"57\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-44.png 661w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-44-300x26.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-44-65x6.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-44-225x19.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-44-350x30.png 350w\" sizes=\"auto, (max-width: 661px) 100vw, 661px\" \/><\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>The second law of thermodynamics is stated as relation between C<sub>p<\/sub> and C<sub>v<\/sub> in the form<\/p>\n<div>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-190 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-45.png\" alt=\"\" width=\"686\" height=\"412\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-45.png 686w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-45-300x180.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-45-65x39.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-45-225x135.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-45-350x210.png 350w\" sizes=\"auto, (max-width: 686px) 100vw, 686px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-192\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-46.png\" alt=\"\" width=\"602\" height=\"432\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-46.png 602w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-46-300x215.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-46-65x47.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-46-225x161.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-46-350x251.png 350w\" sizes=\"auto, (max-width: 602px) 100vw, 602px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">As depicted from <\/span><strong style=\"text-align: initial;font-size: 1em\">fig (1)<\/strong><span style=\"text-align: initial;font-size: 1em\"> at low temperatures their difference becomes very small and both C<\/span><em style=\"text-align: initial;font-size: 1em\">p<\/em><span style=\"text-align: initial;font-size: 1em\"> &amp; C<\/span><em style=\"text-align: initial;font-size: 1em\">v<\/em><span style=\"text-align: initial;font-size: 1em\"> go to zero at <\/span><em style=\"text-align: initial;font-size: 1em\">T<\/em><span style=\"text-align: initial;font-size: 1em\"> = 0.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In general, the variation of C<\/span><em style=\"text-align: initial;font-size: 1em\">v<\/em><span style=\"text-align: initial;font-size: 1em\"> with time is studied. We assume that a change in volume is not much with a small increase in temperature.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">As the temperature increases from absolute zero, there is a rapid increase in specific heat and it finally levels off at a nearly constant value (6 <\/span>cal<span style=\"font-size: 1em\">\/mole) at high temperature as shown in figure(2) below. This is the classical approach put forward as in Dulong Petit\u2019s Law.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">2. Dulong and Petit Law (Classical overview)<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The physical properties of the solids <\/span>is<span style=\"text-align: initial;font-size: 1em\"> roughly assumed to be<\/span><\/p>\n<\/div>\n<div>\n<ul>\n<li>due to the contributions resulting from the atomic vibrations (as in crystals)<\/li>\n<li style=\"text-align: justify\">due to additional<span style=\"text-align: initial;font-size: 1em\"> contribution to specific heat from <\/span>electronic<span style=\"text-align: initial;font-size: 1em\"> system (as in metals and semiconductors). Although this contribution is relatively small to that of lattice vibrations.<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify\">\u00a0 \u00a0 Taking these assumptions into consideration and assuming the behaviour of the atoms of the crystal as to be independent classical harmonic oscillator, the heat capacity of the solid crystal can be calculated simply by finding average thermal energy of a single oscillator and then multiplying it by total number of oscillators, N.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In comparison to the translational motion of the ideal gas molecules, the constituent atoms in a crystal , have almost fixed positions which vibrate about their mean positions executing simple harmonic motion. The total energy of the oscillator at an instant is a composition of its instantaneous potential energy and kinetic energy. Classically a harmonic oscillator vibrating with its natural frequency \u03c9 has energy expression as<\/p>\n<p>&nbsp;<\/p>\n<p>E = P<sup>2<\/sup>\/2m + \u00bd m\u03c92(x<sup>2<\/sup>+ y<sup>2<\/sup> + z<sup>2<\/sup>)<\/p>\n<p>E = P<sup>2<\/sup>\/2m +m\u03c9<sup>2<\/sup>q<sup>2<\/sup>\/2 &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8211;(6)<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">where p is the momentum in kinetic energy term and q is the displacement from equilibrium position in potential energy term .<\/p>\n<p>&nbsp;<\/p>\n<p>The average energy of each harmonic oscillator as given by Plank\u2019s distribution law is<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-193 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-47.png\" alt=\"\" width=\"698\" height=\"514\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-47.png 698w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-47-300x221.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-47-65x48.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-47-225x166.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-47-350x258.png 350w\" sizes=\"auto, (max-width: 698px) 100vw, 698px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-194 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-48.png\" alt=\"\" width=\"418\" height=\"349\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-48.png 418w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-48-300x250.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-48-65x54.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-48-225x188.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-48-350x292.png 350w\" sizes=\"auto, (max-width: 418px) 100vw, 418px\" \/><\/p>\n<\/div>\n<p><strong>\u00a0<\/strong><\/p>\n<div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">This gives the vibrational energy of one harmonic oscillator. To obtain total vibrational energy of the crystal whish has N independent harmonic oscillators in three dimensions, we have<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-195 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-49.png\" alt=\"\" width=\"551\" height=\"120\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-49.png 551w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-49-300x65.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-49-65x14.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-49-225x49.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-49-350x76.png 350w\" sizes=\"auto, (max-width: 551px) 100vw, 551px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>This is called Dulong Petit\u2019s classical law of specific heat.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">This turns out to be that the heat capacity is independent of temperature and has a constant value at room temparature.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-196\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-50.png\" alt=\"\" width=\"594\" height=\"381\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-50.png 594w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-50-300x192.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-50-65x42.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-50-225x144.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-50-350x224.png 350w\" sizes=\"auto, (max-width: 594px) 100vw, 594px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>3. Shortfall of Classical Theory:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The classical theory gives a good idea of energy contributions to the total energy of the system as long as the volume remains constant, which is given in equation (9).<\/p>\n<p>&nbsp;<\/p>\n<p>The specific heat at constant volume as obtained for a solid containing N number of atoms is given as<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-197 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-51.png\" alt=\"\" width=\"311\" height=\"52\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-51.png 311w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-51-300x50.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-51-65x11.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-51-225x38.png 225w\" sizes=\"auto, (max-width: 311px) 100vw, 311px\" \/>\u00a0<\/strong><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>This is the level off value of the experimental results at high temperature.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Equation (8) is a frequency independent expression and also it depends only on temperature. The results obtained here are in quantitative agreement with those obtained from experiments but at higher temperatures only.<\/p>\n<p>&nbsp;<\/p>\n<p>This does not explain the decrease of specific heat at low temperatures as observed for all solids.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The discrepancy to a certain extent is removed by using quantum theory but before that it is important to understand the concept of quantization of elastic waves and the concept of phonons.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>4. Quantization of Elastic Waves:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">As is a well understood concept that the atoms in solids vibrate about their equilibrium positions and these lattice vibrations can be expressed in the form of waves, like repetitive and systematic sequence of atomic displacements that can be longitudinal or transverse or even a combination of both.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The energy of a lattice vibration is quantized and the quantum of energy is called \u2018Phonon\u2019, analogous to quanta of electromagnetic energy \u2013 photon.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">We have assumed vibrations of a linear lattice connected through springs and their particle motion can be quantized in a similar way as that for a harmonic oscillator or a combination of coupled harmonic oscillators. The Hamiltonian for a harmonic oscillator is<\/p>\n<p>&nbsp;<\/p>\n<p>H = (1\/2M) p<sup>2<\/sup> + \u00bd Cx<sup>2<\/sup><\/p>\n<p>&nbsp;<\/p>\n<p>With energy eigen values for n = 1, 2 , 3 ,\u2026.<\/p>\n<p>&nbsp;<\/p>\n<p>Hence average thermal energy of the oscillator is<\/p>\n<p>&nbsp;<\/p>\n<p>\u0404n= (n + \u00bd)\u0127\u03c9<\/p>\n<p>&nbsp;<\/p>\n<p>Where \u03c9 is the angular frequency.<\/p>\n<p>&nbsp;<\/p>\n<p>The term \u00bd \u0127\u03c9 is called zero point energy of the mode<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-198\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-52.png\" alt=\"\" width=\"506\" height=\"370\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-52.png 506w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-52-300x219.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-52-65x48.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-52-225x165.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-52-350x256.png 350w\" sizes=\"auto, (max-width: 506px) 100vw, 506px\" \/><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Thus thermal lattice vibrations are thermally excited phonons. Thermal conduction in non metallic crystals is a consequence of annihilation or creation of a phonon. The energy of phonons is ~0.1 ev.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">We know that if the particles have zero spin then these follow Bose \u2013 Einstein statistics and the probability of such particles having energy E is given as<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-199 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-53.png\" alt=\"\" width=\"363\" height=\"184\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-53.png 363w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-53-300x152.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-53-65x33.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-53-225x114.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-53-350x177.png 350w\" sizes=\"auto, (max-width: 363px) 100vw, 363px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The following observations confirm the experimental evidence for the quantization of lattice vibrational energy:<\/p>\n<ul>\n<li>at absolute zero the lattice contribution of heat energy always approaches zero<\/li>\n<li>the neutrons and X rays are scattered inelastically by crystals and the changes in energy and momentum correspond to creation or absorption of phonons<span style=\"text-align: initial;font-size: 1em\">.<\/span><\/li>\n<\/ul>\n<\/div>\n<div>\n<p>\u00a0 \u00a0 (Which is discussed in detail in the later section of the modules.)<\/p>\n<p>&nbsp;<\/p>\n<p><strong>5.<\/strong>\u00a0<strong>Summary:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>After the completion of this module we are able to understand and correlate the following<\/p>\n<ul>\n<li>Heat capacity, a thermodynamic concept as a function of temperature.<\/li>\n<li>Classical<span style=\"text-align: initial;font-size: 1em\"> theory of specific heat capacity and discrepancy with the experimental results obtained at lower temperatures.<\/span><\/li>\n<li>Shortfall<span style=\"text-align: initial;font-size: 1em\"> of the Dulong Petit\u2019s classical theory and need of new (quantum) approach. <\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Quantization of elastic waves, its physical significance <\/span>and<span style=\"text-align: initial;font-size: 1em\"> mathematical treatment. <\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Introduction of <\/span>concept<span style=\"text-align: initial;font-size: 1em\"> of phonons and its relevance in lattice dynamics.<\/span><\/li>\n<\/ul>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on Lattice Vibrations and Thermal Properties 3<\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/yC-JyT-qyF0\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div>\n<p><strong>\u00a0 \u00a0 Value Addition:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Do You Know?<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>The relation between specific heat at constant pressure and at constant volume in terms of coefficient of volume expansion \u03b1<\/strong><strong>v<\/strong><strong> and the compressibility K is given as<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Cp \u2013 Cv = \u03b1<\/strong><strong>v<\/strong><strong>2<\/strong><strong>TV\/K,<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Cp \u2013 Cv \u2265 0<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <strong>\u03b1<\/strong><strong>v<\/strong><strong> and K being positive quantities,<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>And in general the relationship employed is<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Cp = Cv (1+ \u03b3\u03b1<\/strong><strong>v<\/strong><strong>T)<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>\u03b3\u00a0\u00a0 <strong>= \u03b1<\/strong><strong>v<\/strong><strong> V \/ KCv, is Gr\u00fcneisen constant and is independent of temperature.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>By calculation of \u03b3 at any arbitrary temperature, an approximation for Cv is obtained.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>As already discussed that a part of energy given to a system results in the increase in its internal energy which is associated with more vigorous notion of the atoms. In metals and semiconductors the remaining contribution in the specific heat comes from the electronic system, although this contribution is very small as compared to that of the lattice.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>C<\/strong><strong>solid<\/strong><strong> =C<\/strong><strong>lattice +<\/strong><strong> C<\/strong><strong>electronic<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>Anomalies have been observed in specific heat curves for ferromagnetic metals and e.g, nickel, iron and cobalt. A prominent peak appears near Curie temperature. This is correlated with transition from ordered state to disordered state. The alloys show more peaks in specific heat curves that exhibit similar transitions.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>For More Details ( on this topic and other related topics ) See<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Adrianus J Dekker, Solid State Physics<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Glossary:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Elastic vibrations:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Waves when require a material medium to travel (unlike e.m waves<\/span>),<span style=\"text-align: initial;font-size: 1em\"> are called elastic vibrations.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">These can have both longitudinal and transverse modes of vibration.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Crystal momentum:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Phonons do not carry any momentum on the lattice but a phonon wave vector k interacts with the crystal constituents and other particles as if its momentum is \u0127k, more precisely called Crystal momentum<\/span><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n","protected":false},"author":3,"menu_order":12,"template":"","meta":{"_acf_changed":false,"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["prof-mahavir-singh"],"pb_section_license":""},"chapter-type":[],"contributor":[59],"license":[],"class_list":["post-183","chapter","type-chapter","status-publish","hentry","contributor-prof-mahavir-singh"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters\/183","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":8,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters\/183\/revisions"}],"predecessor-version":[{"id":239,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters\/183\/revisions\/239"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters\/183\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/media?parent=183"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapter-type?post=183"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/contributor?post=183"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/license?post=183"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}