{"id":288,"date":"2018-12-05T09:34:43","date_gmt":"2018-12-05T09:34:43","guid":{"rendered":"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=288"},"modified":"2018-12-05T09:52:44","modified_gmt":"2018-12-05T09:52:44","slug":"surface-physics-i","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/chapter\/surface-physics-i\/","title":{"rendered":"Surface Physics I"},"content":{"raw":"<div>\r\n<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/rpStE-TNQVE\" 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<\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div><span style=\"text-align: initial;font-size: 1em\">The objective of the module is to<\/span><\/div>\r\n<ul>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Acquire understanding of the basic Einstein\u2019s quantum model.<\/span><\/li>\r\n \t<li>Observe the acceptance of Einstein\u2019s theory over Classical theory.<\/li>\r\n \t<li>Understand the need of<span style=\"text-align: initial;font-size: 1em\"> further modifications in the basic quantum theory.<\/span><\/li>\r\n \t<li>Get an understanding of concept<span style=\"text-align: initial;font-size: 1em\"> of Density of States and modes of vibration.<\/span><\/li>\r\n \t<li style=\"text-align: justify\">Appreciate the success of Debye\u2019s theory over the shortfall of Einstein\u2019s theory Mathematically arrive at Debye\u2019s <sup>T3<\/sup> law at low temperatures.<\/li>\r\n<\/ul>\r\n<div>\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 1.Introduction<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">At low temperatures, the specific heat Cv falls below the limiting value of 3NR, where R is universal gas constant. It was assumed that some of the oscillators stop vibrating as the temperature is lowered, and Einstein supported the logic on the basis of Plank\u2019s hypothesis on quantization of energy. As a consequence, <\/span>the there<span style=\"text-align: initial;font-size: 1em\"> is a decrease in the number of oscillators oscillating at lower temperatures (below room temperatures).<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">An oscillator must possess at least one quantum of energy (phonon) to oscillate at all. But the oscillators which do not have even single quantum will not oscillate and have zero energy according to Plank\u2019s hypothesis.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">But Einstein visualized some ingenuity in this and rather assumed a rigid lattice made of identical oscillators and all of them to be oscillating with <\/span>same<span style=\"font-size: 1em\"> frequency.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">2.\u00a0 <\/strong><strong style=\"text-align: initial;font-size: 1em\">Einstein\u2019s Theory<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">To understand the dip in specific heat curve at low temperatures Einstein employed a physical model (although oversimplified to arrive at desired results), but it indicated that the problem had its solution in quantum approach.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The suggested model assumes the following<\/span><\/p>\r\n\r\n<ul>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">lattice containing N atoms is equivalent to 3N harmonic oscillators <\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">atoms vibrate independently of each other<\/span><\/li>\r\n \t<li style=\"text-align: justify\">all atoms vibrate with same<span style=\"text-align: initial;font-size: 1em\"> frequency ( as all have <\/span>same<span style=\"text-align: initial;font-size: 1em\"> environment)<\/span><\/li>\r\n \t<li style=\"text-align: justify\">harmonic oscillators have discrete quantized energy levels represented as<\/li>\r\n<\/ul>\r\n<div>\r\n\r\n<img class=\"size-full wp-image-207 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-54.png\" alt=\"\" width=\"680\" height=\"370\" \/>\r\n\r\n<\/div>\r\n<\/div>\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-208 alignleft\" style=\"text-align: initial;font-size: 1em\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-55.png\" alt=\"\" width=\"771\" height=\"522\" \/>\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&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-209\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-56.png\" alt=\"\" width=\"683\" height=\"360\" \/>\r\n\r\n<img class=\"size-full wp-image-210 alignleft\" style=\"text-align: initial;font-size: 1em\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-57.png\" alt=\"\" width=\"682\" height=\"36\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Now for T=0, \u00a0h\u03bd \/2 and not zero, so the first term in equation (8) is referred to as \u2018zero point energy\u2019 and according to quantum mechanics the atoms possess vibrational energy even at absolute zero.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Recalling the question that why did the Dulong and Petit\u2019s law fail at lower temperatures?<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">The energy h\u03bd of the oscillator indicates the difference between the allowed energy states of the given oscillator. In case the difference is small, ie <strong><em>h\u03bd \/kT<\/em><\/strong> is negligible in comparison with unity then Cv will be equal to 3R at all finite temperatures.<\/p>\r\n&nbsp;\r\n\r\nHowever at low temperatures, unity is negligible as compared to <strong><em>h\u03bd \/kT<\/em><\/strong> and Cv becomes proportional to.\r\n\r\ne<sup>-hv\/KT<\/sup>\r\n\r\nHeat capacity decreases exponentially at low temperatures.\r\n\r\n&nbsp;\r\n\r\nThis is in fact a indirect evidence of quantization of energy\u00a0 as proposed by Plank\u2019s hypothesis. At kT&gt;&gt;h\u03bd, equation (8) reduces to classical result value, however at lower temperatures specific heat decreases.\r\n\r\n&nbsp;\r\n\r\nIt is better understood by rewriting equation (6) in terms of \u03b8E, (Einstein temperature) as\r\n\r\n<img class=\"size-full wp-image-211 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-58.png\" alt=\"\" width=\"283\" height=\"53\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Where FE\u00a0 is Einstein\u2019s function determining the ratio of specific heat at temperature T and classical value 3R.<\/p>\r\n&nbsp;\r\n\r\nFor T &gt;&gt; \u03b8E,\r\n\r\n&nbsp;\r\n\r\nCv =3R,\r\n\r\n&nbsp;\r\n\r\nie Dulong Petit law is obeyed.\r\n\r\n&nbsp;\r\n\r\nFor T &lt;&lt; \u03b8E,\r\n\r\n&nbsp;\r\n\r\n,\u00a0 Cv = 3R(\u03b8E\/)2\r\n\r\n&nbsp;\r\n\r\nthe specific heat is proportional to \u03b8E\/T .\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">This is in good agreement with experimental results at higher temperatures but fails to give exact values at very low temperatures. The reason for this discrepancy must be sought in the oversimplification of the model assumed by Einstein considering the vibrations to be independent of each other.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">A modification suggesting a solution is made by Debye known as Debye\u2019s approximation theory.<\/p>\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">3.\u00a0 <\/strong><strong style=\"text-align: initial;font-size: 1em\">Debye\u2019s Theory<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Although Einstein\u2019s assumption that the atoms vibrate independently of each other in the lattice with a constant frequency had been very successful in explaining the exponential decrease in heat capacity at low temperatures <\/span>but<span style=\"text-align: initial;font-size: 1em\"> the predicted decrease was much sharper than the experimentally observed values. <\/span>Further<span style=\"text-align: initial;font-size: 1em\"> Einstein\u2019s model failed to explain and fit T3 dependence<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">To explain the variation of specific heat of solids with temperature, Debye relied on the following assumptions<\/span><\/p>\r\n\r\n<ul>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">the vibrations of all atoms are coupled together <\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">solid acts as an isotropic elastic continuum<\/span><\/li>\r\n \t<li style=\"text-align: justify\">atomic vibrations produce a continuous spectrum of frequencies<\/li>\r\n \t<li style=\"text-align: justify\">the frequency is same<span style=\"text-align: initial;font-size: 1em\"> for longitudinal and transverse vibrations produced in solids <\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">beyond a certain frequency VD (Debye frequency), no vibrations are produced<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<div>\r\n\r\n\u00a0 \u00a0 Consider an elastic wave <em>u(x, y, z, t)<\/em> propagating through a medium. Writing its wave equation as\r\n\r\n<img class=\"size-full wp-image-212 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-59.png\" alt=\"\" width=\"728\" height=\"419\" \/>\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\nEqn (12) shows that the different values of frequencies are quantized.\r\n\r\nDifferent no. of modes of vibration can be determined using this equation.\r\n\r\n&nbsp;\r\n\r\nLet Z(\u03bd) is the number of modes of vibration in frequency interval \u03bd and \u03bd+ d\u03bd\r\n\r\nTherefore, the number of modes of vibration is\r\n\r\n<img class=\"size-full wp-image-213 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-60.png\" alt=\"\" width=\"701\" height=\"145\" \/>\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<p style=\"text-align: justify\">Equation (13) gives the possible modes of vibration also known as <em>Density of states<\/em> of elastic continuous medium.<\/p>\r\n<p style=\"text-align: justify\">In deriving equation (12) it has been assumed that the velocity v is irrespective of longitudinal or transverse nature of the wave propagating in the medium. But actually the frequency is associated with one longitudinal mode and two transverse modes; hence equation (13) gets modified as<\/p>\r\n<img class=\"size-full wp-image-214 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-61.png\" alt=\"\" width=\"719\" height=\"51\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nEquation (14) gives the total number of modes of vibration in elastic continuum medium.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Debye proposed that a solid is a continuously vibrating medium giving rise to a spectrum of frequencies (wavelengths comparable to inter-atomic separations) called Debye\u2019s continuum. Then the total energy expression becomes<\/p>\r\n<img class=\"size-full wp-image-215 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-62.png\" alt=\"\" width=\"125\" height=\"32\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">As there are N atoms therefore the frequency spectrum has to comply with 3N modes of vibration i.e there must be a maximum frequency VD beyond which no frequencies are possible such that the internal energy expression becomes,<\/p>\r\n<img class=\"size-full wp-image-216 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-63.png\" alt=\"\" width=\"705\" height=\"357\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\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\nThe following conclusions are drawn\r\n\r\n&nbsp;\r\n\r\n1)\u00a0 for high temperature range T&gt;&gt;\u03b8D, equation (16) reduces to classical limit i.e,\r\n\r\n<img class=\"size-full wp-image-217 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-64.png\" alt=\"\" width=\"691\" height=\"264\" \/>\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 is Debye\u2019s T3 law which holds for sufficiently low temperatures for long wavelength excitations, i.e the modes that may be treated in elastic continuum with macroscopic elastic constants.<\/p>\r\n&nbsp;\r\n\r\nExcept for the extreme situation of temperatures, Debye\u2019s approximation falls back to classical model.\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-218\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-65.png\" alt=\"\" width=\"595\" height=\"521\" \/>\r\n\r\n<\/div>\r\n<\/div>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>you can view video on Lattice Vibrations and Thermal Energy 4<\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/rpStE-TNQVE\" 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<strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 4. Summary:<\/strong>\r\n<div>\r\n\r\n\u00a0 \u00a0After the completion of this module we are able to\r\n<ul>\r\n \t<li style=\"text-align: justify\">Understand the lapse in explaining dip in specific heats at low temperatures which is not explained as in classical theory.<\/li>\r\n \t<li style=\"text-align: justify\">Study a basic quantum model designed by Einstein in accordance with Plank\u2019s hypothesis.<\/li>\r\n \t<li style=\"text-align: justify\">Relate lattice behavior to independent harmonic oscillators vibrating with same<span style=\"text-align: initial;font-size: 1em\"> frequency.<\/span><\/li>\r\n \t<li style=\"text-align: justify\">How Einstein\u2019s model successfully arrived at experimental results at higher temperatures but at the same time failed to explain the same at low temperatures. We have come across the limitations of the most accepted theories.<\/li>\r\n \t<li style=\"text-align: justify\">Understand the need of modifications in the basic quantum model to explain dip<span style=\"text-align: initial;font-size: 1em\"> in specific heats at low temperatures.<\/span><\/li>\r\n \t<li style=\"text-align: justify\">How Debye\u2019s modified model with assumption<span style=\"text-align: initial;font-size: 1em\"> of the coupled vibrations of\u00a0 lattice over a\u00a0<\/span>Continuous<span style=\"text-align: initial;font-size: 1em\"> spectrum of frequencies removed this discrepancy.<\/span><\/li>\r\n<\/ul>\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>Debye\u2019s approximations leading to accurate measurements in the low temperature region still showed deviations from theoretical predictions. As we observed that Debye\u2019s theory suggests thst T<\/strong><strong>3<\/strong><strong> law should hold in low temperature regions i.e, T\u22640.1 \u03b8<\/strong><strong>D<\/strong><strong> .<\/strong><\/p>\r\n&nbsp;\r\n\r\n<strong>But it deviated from actual results from the data produced by Blachman Paper.<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>It seems that these deviations still point out doubts at continuum approximations. The deficiencies in the results were further taken up by Blackman and Kellermann that expected T<\/strong><strong>3<\/strong><strong> law to hold for temperature region T\u2264 \u03b8<\/strong><strong>D<\/strong><strong>\/50, i.e, at considerably lower temperature than predicted by Debye in his approximation.<\/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<strong>Charles Kittel, Introduction to Solid State Physics.<\/strong>\r\n\r\n<strong>James D Patterson Bernard C Bailey, Solid State Physics, Introduction To Theory<\/strong>\r\n\r\n<\/div>\r\n<strong>\u00a0 \u00a0 Glossary:<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>Elastic vibrations:<\/strong>\r\n\r\n&nbsp;\r\n\r\nWaves when require a material medium to travel(unlike e.m waves), are called elastic vibrations.\r\n\r\n&nbsp;\r\n\r\nThese can have both longitudinal and transverse modes of vibration.\r\n\r\n&nbsp;\r\n\r\n<strong>Density of modes:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">It\u2019s the number of modes of vibration per unit interval. The density of modes per unit volume is a constant independent of the magnitude or the shape of periodicity of the volume.<\/p>\r\n&nbsp;\r\n\r\n<strong>Debye frequency:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">It is a critical frequency chosen so that the density of normal modes is 3N. Debye frequency is dependent only on the velocity of sound in solid.<\/p>\r\n&nbsp;\r\n\r\n<strong>Debye temperature:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">When the temperature greatly exceeds <strong>\u03b8<\/strong><strong>D<\/strong> <strong>=h\u03bd\/k ,(the debye temperature)<\/strong>, Dulong petit law is recovered.<\/p>\r\n\r\n<\/div>","rendered":"<div>\n<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/rpStE-TNQVE\" 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<\/div>\n<div><\/div>\n<div><\/div>\n<div><\/div>\n<div><\/div>\n<div><span style=\"text-align: initial;font-size: 1em\">The objective of the module is to<\/span><\/div>\n<ul>\n<li><span style=\"text-align: initial;font-size: 1em\">Acquire understanding of the basic Einstein\u2019s quantum model.<\/span><\/li>\n<li>Observe the acceptance of Einstein\u2019s theory over Classical theory.<\/li>\n<li>Understand the need of<span style=\"text-align: initial;font-size: 1em\"> further modifications in the basic quantum theory.<\/span><\/li>\n<li>Get an understanding of concept<span style=\"text-align: initial;font-size: 1em\"> of Density of States and modes of vibration.<\/span><\/li>\n<li style=\"text-align: justify\">Appreciate the success of Debye\u2019s theory over the shortfall of Einstein\u2019s theory Mathematically arrive at Debye\u2019s <sup>T3<\/sup> law at low temperatures.<\/li>\n<\/ul>\n<div>\n<p><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 1.Introduction<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">At low temperatures, the specific heat Cv falls below the limiting value of 3NR, where R is universal gas constant. It was assumed that some of the oscillators stop vibrating as the temperature is lowered, and Einstein supported the logic on the basis of Plank\u2019s hypothesis on quantization of energy. As a consequence, <\/span>the there<span style=\"text-align: initial;font-size: 1em\"> is a decrease in the number of oscillators oscillating at lower temperatures (below room temperatures).<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">An oscillator must possess at least one quantum of energy (phonon) to oscillate at all. But the oscillators which do not have even single quantum will not oscillate and have zero energy according to Plank\u2019s hypothesis.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">But Einstein visualized some ingenuity in this and rather assumed a rigid lattice made of identical oscillators and all of them to be oscillating with <\/span>same<span style=\"font-size: 1em\"> frequency.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">2.\u00a0 <\/strong><strong style=\"text-align: initial;font-size: 1em\">Einstein\u2019s Theory<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">To understand the dip in specific heat curve at low temperatures Einstein employed a physical model (although oversimplified to arrive at desired results), but it indicated that the problem had its solution in quantum approach.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The suggested model assumes the following<\/span><\/p>\n<ul>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">lattice containing N atoms is equivalent to 3N harmonic oscillators <\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">atoms vibrate independently of each other<\/span><\/li>\n<li style=\"text-align: justify\">all atoms vibrate with same<span style=\"text-align: initial;font-size: 1em\"> frequency ( as all have <\/span>same<span style=\"text-align: initial;font-size: 1em\"> environment)<\/span><\/li>\n<li style=\"text-align: justify\">harmonic oscillators have discrete quantized energy levels represented as<\/li>\n<\/ul>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-207 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-54.png\" alt=\"\" width=\"680\" height=\"370\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-54.png 680w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-54-300x163.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-54-65x35.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-54-225x122.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-54-350x190.png 350w\" sizes=\"auto, (max-width: 680px) 100vw, 680px\" \/><\/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><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-208 alignleft\" style=\"text-align: initial;font-size: 1em\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-55.png\" alt=\"\" width=\"771\" height=\"522\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-55.png 771w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-55-300x203.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-55-768x520.png 768w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-55-65x44.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-55-225x152.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-55-350x237.png 350w\" sizes=\"auto, (max-width: 771px) 100vw, 771px\" \/><\/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>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-209\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-56.png\" alt=\"\" width=\"683\" height=\"360\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-56.png 683w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-56-300x158.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-56-65x34.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-56-225x119.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-56-350x184.png 350w\" sizes=\"auto, (max-width: 683px) 100vw, 683px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-210 alignleft\" style=\"text-align: initial;font-size: 1em\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-57.png\" alt=\"\" width=\"682\" height=\"36\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-57.png 682w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-57-300x16.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-57-65x3.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-57-225x12.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-57-350x18.png 350w\" sizes=\"auto, (max-width: 682px) 100vw, 682px\" \/><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Now for T=0, \u00a0h\u03bd \/2 and not zero, so the first term in equation (8) is referred to as \u2018zero point energy\u2019 and according to quantum mechanics the atoms possess vibrational energy even at absolute zero.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Recalling the question that why did the Dulong and Petit\u2019s law fail at lower temperatures?<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The energy h\u03bd of the oscillator indicates the difference between the allowed energy states of the given oscillator. In case the difference is small, ie <strong><em>h\u03bd \/kT<\/em><\/strong> is negligible in comparison with unity then Cv will be equal to 3R at all finite temperatures.<\/p>\n<p>&nbsp;<\/p>\n<p>However at low temperatures, unity is negligible as compared to <strong><em>h\u03bd \/kT<\/em><\/strong> and Cv becomes proportional to.<\/p>\n<p>e<sup>-hv\/KT<\/sup><\/p>\n<p>Heat capacity decreases exponentially at low temperatures.<\/p>\n<p>&nbsp;<\/p>\n<p>This is in fact a indirect evidence of quantization of energy\u00a0 as proposed by Plank\u2019s hypothesis. At kT&gt;&gt;h\u03bd, equation (8) reduces to classical result value, however at lower temperatures specific heat decreases.<\/p>\n<p>&nbsp;<\/p>\n<p>It is better understood by rewriting equation (6) in terms of \u03b8E, (Einstein temperature) as<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-211 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-58.png\" alt=\"\" width=\"283\" height=\"53\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-58.png 283w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-58-65x12.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-58-225x42.png 225w\" sizes=\"auto, (max-width: 283px) 100vw, 283px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Where FE\u00a0 is Einstein\u2019s function determining the ratio of specific heat at temperature T and classical value 3R.<\/p>\n<p>&nbsp;<\/p>\n<p>For T &gt;&gt; \u03b8E,<\/p>\n<p>&nbsp;<\/p>\n<p>Cv =3R,<\/p>\n<p>&nbsp;<\/p>\n<p>ie Dulong Petit law is obeyed.<\/p>\n<p>&nbsp;<\/p>\n<p>For T &lt;&lt; \u03b8E,<\/p>\n<p>&nbsp;<\/p>\n<p>,\u00a0 Cv = 3R(\u03b8E\/)2<\/p>\n<p>&nbsp;<\/p>\n<p>the specific heat is proportional to \u03b8E\/T .<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">This is in good agreement with experimental results at higher temperatures but fails to give exact values at very low temperatures. The reason for this discrepancy must be sought in the oversimplification of the model assumed by Einstein considering the vibrations to be independent of each other.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">A modification suggesting a solution is made by Debye known as Debye\u2019s approximation theory.<\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">3.\u00a0 <\/strong><strong style=\"text-align: initial;font-size: 1em\">Debye\u2019s Theory<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Although Einstein\u2019s assumption that the atoms vibrate independently of each other in the lattice with a constant frequency had been very successful in explaining the exponential decrease in heat capacity at low temperatures <\/span>but<span style=\"text-align: initial;font-size: 1em\"> the predicted decrease was much sharper than the experimentally observed values. <\/span>Further<span style=\"text-align: initial;font-size: 1em\"> Einstein\u2019s model failed to explain and fit T3 dependence<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">To explain the variation of specific heat of solids with temperature, Debye relied on the following assumptions<\/span><\/p>\n<ul>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">the vibrations of all atoms are coupled together <\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">solid acts as an isotropic elastic continuum<\/span><\/li>\n<li style=\"text-align: justify\">atomic vibrations produce a continuous spectrum of frequencies<\/li>\n<li style=\"text-align: justify\">the frequency is same<span style=\"text-align: initial;font-size: 1em\"> for longitudinal and transverse vibrations produced in solids <\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">beyond a certain frequency VD (Debye frequency), no vibrations are produced<\/span><\/li>\n<\/ul>\n<\/div>\n<div>\n<p>\u00a0 \u00a0 Consider an elastic wave <em>u(x, y, z, t)<\/em> propagating through a medium. Writing its wave equation as<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-212 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-59.png\" alt=\"\" width=\"728\" height=\"419\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-59.png 728w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-59-300x173.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-59-65x37.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-59-225x129.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-59-350x201.png 350w\" sizes=\"auto, (max-width: 728px) 100vw, 728px\" \/><\/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>Eqn (12) shows that the different values of frequencies are quantized.<\/p>\n<p>Different no. of modes of vibration can be determined using this equation.<\/p>\n<p>&nbsp;<\/p>\n<p>Let Z(\u03bd) is the number of modes of vibration in frequency interval \u03bd and \u03bd+ d\u03bd<\/p>\n<p>Therefore, the number of modes of vibration is<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-213 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-60.png\" alt=\"\" width=\"701\" height=\"145\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-60.png 701w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-60-300x62.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-60-65x13.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-60-225x47.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-60-350x72.png 350w\" sizes=\"auto, (max-width: 701px) 100vw, 701px\" \/><\/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 style=\"text-align: justify\">Equation (13) gives the possible modes of vibration also known as <em>Density of states<\/em> of elastic continuous medium.<\/p>\n<p style=\"text-align: justify\">In deriving equation (12) it has been assumed that the velocity v is irrespective of longitudinal or transverse nature of the wave propagating in the medium. But actually the frequency is associated with one longitudinal mode and two transverse modes; hence equation (13) gets modified as<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-214 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-61.png\" alt=\"\" width=\"719\" height=\"51\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-61.png 719w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-61-300x21.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-61-65x5.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-61-225x16.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-61-350x25.png 350w\" sizes=\"auto, (max-width: 719px) 100vw, 719px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>Equation (14) gives the total number of modes of vibration in elastic continuum medium.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Debye proposed that a solid is a continuously vibrating medium giving rise to a spectrum of frequencies (wavelengths comparable to inter-atomic separations) called Debye\u2019s continuum. Then the total energy expression becomes<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-215 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-62.png\" alt=\"\" width=\"125\" height=\"32\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-62.png 125w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-62-65x17.png 65w\" sizes=\"auto, (max-width: 125px) 100vw, 125px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">As there are N atoms therefore the frequency spectrum has to comply with 3N modes of vibration i.e there must be a maximum frequency VD beyond which no frequencies are possible such that the internal energy expression becomes,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-216 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-63.png\" alt=\"\" width=\"705\" height=\"357\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-63.png 705w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-63-300x152.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-63-65x33.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-63-225x114.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-63-350x177.png 350w\" sizes=\"auto, (max-width: 705px) 100vw, 705px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/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>The following conclusions are drawn<\/p>\n<p>&nbsp;<\/p>\n<p>1)\u00a0 for high temperature range T&gt;&gt;\u03b8D, equation (16) reduces to classical limit i.e,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-217 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-64.png\" alt=\"\" width=\"691\" height=\"264\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-64.png 691w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-64-300x115.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-64-65x25.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-64-225x86.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-64-350x134.png 350w\" sizes=\"auto, (max-width: 691px) 100vw, 691px\" \/><\/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 is Debye\u2019s T3 law which holds for sufficiently low temperatures for long wavelength excitations, i.e the modes that may be treated in elastic continuum with macroscopic elastic constants.<\/p>\n<p>&nbsp;<\/p>\n<p>Except for the extreme situation of temperatures, Debye\u2019s approximation falls back to classical model.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-218\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-65.png\" alt=\"\" width=\"595\" height=\"521\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-65.png 595w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-65-300x263.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-65-65x57.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-65-225x197.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-65-350x306.png 350w\" sizes=\"auto, (max-width: 595px) 100vw, 595px\" \/><\/p>\n<\/div>\n<\/div>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on Lattice Vibrations and Thermal Energy 4<\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/rpStE-TNQVE\" 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<p><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 4. Summary:<\/strong><\/p>\n<div>\n<p>\u00a0 \u00a0After the completion of this module we are able to<\/p>\n<ul>\n<li style=\"text-align: justify\">Understand the lapse in explaining dip in specific heats at low temperatures which is not explained as in classical theory.<\/li>\n<li style=\"text-align: justify\">Study a basic quantum model designed by Einstein in accordance with Plank\u2019s hypothesis.<\/li>\n<li style=\"text-align: justify\">Relate lattice behavior to independent harmonic oscillators vibrating with same<span style=\"text-align: initial;font-size: 1em\"> frequency.<\/span><\/li>\n<li style=\"text-align: justify\">How Einstein\u2019s model successfully arrived at experimental results at higher temperatures but at the same time failed to explain the same at low temperatures. We have come across the limitations of the most accepted theories.<\/li>\n<li style=\"text-align: justify\">Understand the need of modifications in the basic quantum model to explain dip<span style=\"text-align: initial;font-size: 1em\"> in specific heats at low temperatures.<\/span><\/li>\n<li style=\"text-align: justify\">How Debye\u2019s modified model with assumption<span style=\"text-align: initial;font-size: 1em\"> of the coupled vibrations of\u00a0 lattice over a\u00a0<\/span>Continuous<span style=\"text-align: initial;font-size: 1em\"> spectrum of frequencies removed this discrepancy.<\/span><\/li>\n<\/ul>\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>Debye\u2019s approximations leading to accurate measurements in the low temperature region still showed deviations from theoretical predictions. As we observed that Debye\u2019s theory suggests thst T<\/strong><strong>3<\/strong><strong> law should hold in low temperature regions i.e, T\u22640.1 \u03b8<\/strong><strong>D<\/strong><strong> .<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>But it deviated from actual results from the data produced by Blachman Paper.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>It seems that these deviations still point out doubts at continuum approximations. The deficiencies in the results were further taken up by Blackman and Kellermann that expected T<\/strong><strong>3<\/strong><strong> law to hold for temperature region T\u2264 \u03b8<\/strong><strong>D<\/strong><strong>\/50, i.e, at considerably lower temperature than predicted by Debye in his approximation.<\/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><strong>Charles Kittel, Introduction to Solid State Physics.<\/strong><\/p>\n<p><strong>James D Patterson Bernard C Bailey, Solid State Physics, Introduction To Theory<\/strong><\/p>\n<\/div>\n<p><strong>\u00a0 \u00a0 Glossary:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Elastic vibrations:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Waves when require a material medium to travel(unlike e.m waves), are called elastic vibrations.<\/p>\n<p>&nbsp;<\/p>\n<p>These can have both longitudinal and transverse modes of vibration.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Density of modes:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">It\u2019s the number of modes of vibration per unit interval. The density of modes per unit volume is a constant independent of the magnitude or the shape of periodicity of the volume.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Debye frequency:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">It is a critical frequency chosen so that the density of normal modes is 3N. Debye frequency is dependent only on the velocity of sound in solid.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Debye temperature:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">When the temperature greatly exceeds <strong>\u03b8<\/strong><strong>D<\/strong> <strong>=h\u03bd\/k ,(the debye temperature)<\/strong>, Dulong petit law is recovered.<\/p>\n<\/div>\n","protected":false},"author":3,"menu_order":18,"template":"","meta":{"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-288","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\/288","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":5,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters\/288\/revisions"}],"predecessor-version":[{"id":293,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters\/288\/revisions\/293"}],"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\/288\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/media?parent=288"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapter-type?post=288"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/contributor?post=288"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/license?post=288"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}