{"id":220,"date":"2018-12-04T09:37:48","date_gmt":"2018-12-04T09:37:48","guid":{"rendered":"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=220"},"modified":"2018-12-04T10:33:24","modified_gmt":"2018-12-04T10:33:24","slug":"lattice-vibrations-and-thermal-properties-5","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/chapter\/lattice-vibrations-and-thermal-properties-5\/","title":{"rendered":"Lattice Vibrations and Thermal Properties 5"},"content":{"raw":"<div>\r\n<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/1rFIIpIoWiY\" 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\nThe objective of the module is to\r\n<ul>\r\n \t<li>Understand the importance of neutron scattering in order to study phonon interaction in detail.<\/li>\r\n \t<li>Have an insight of importance<span style=\"text-align: initial;font-size: 1em\"> of anharmonicity in crystal interactions.<\/span><\/li>\r\n \t<li>Correlate thermal expansion and thermal conduction in context<span style=\"text-align: initial;font-size: 1em\"> of <\/span>anhamonic<span style=\"text-align: initial;font-size: 1em\"> crystal interaction.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a01.\u00a0<\/strong><strong>Neutron scattering:-<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The neutrons have a great advantage to be used as a probe in condensed matter physics. Scattering of free neutrons with the matter is used to measure details which are of prime importance in material research. Elastic scattering (neutron diffraction) is used in determining structures whereas inelastic scattering provides valuable information on atomic vibrations and excitations. The interaction of phonons is assumed as scattering collisions.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">A neutron sees the crystal lattice by interacting with the nuclei of the atoms and exchange energy which is used to measure details of atomic and molecular motions by the use of inelastic neutron scattering (INS). When a crystal gets scattered by a crystal, it results either in absorption or emission of energy of an amount equal to quantum of phonon (h\u03bd). The wave vectors before and after the interaction gives information about the phonon energies.<\/p>\r\n&nbsp;\r\n\r\nHere we study the neutron scattering and phonon interaction dynamics in detail.\r\n\r\n&nbsp;\r\n\r\n<strong>1.1 Elastic scattering<\/strong><strong>:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In elastic scattering of X-rays in a crystal (Bragg\u2019s diffraction) the magnitude of wave vector (or the frequency ) remains unchanged i.e,<\/p>\r\n&nbsp;\r\n\r\n\u03c9 = \u03ce and\r\n\r\n&nbsp;\r\n\r\n?+G = \u1e31\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">where k and \u1e31 are wave vectors of incident and scattered photons, which obey selection rules. G is one of the reciprocal lattice vectors.<\/p>\r\n&nbsp;\r\n\r\nAlso the energy and momenta are conserved i.e,\r\n\r\n&nbsp;\r\n\r\n\u0127\u03c9 = \u0127\u03ce\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 and\r\n\r\n&nbsp;\r\n\r\n\u0127\u00a0\u00a0 + \u0127G = \u0127\u1e31\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">As the frequencies of incident and scattered photons are unchanged the crystal as a whole recoils with momentum -\u0127G . This process is called N-Process.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">1.2 Inelastic scattering:<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">But in case the scattering is inelastic, then the process leads to either emission or absorption of a phonon of wave vector K. This phonon will interact with photons, neutrons and electrons as if having a momentum <\/span>\u0127k ,however<span style=\"text-align: initial;font-size: 1em\"> a phonon carries no physical momentum on lattices as the phonon <\/span>co ordinates<span style=\"text-align: initial;font-size: 1em\"> (other <\/span>than k<span style=\"text-align: initial;font-size: 1em\"> = 0) are linked with the relative <\/span>co ordinates<span style=\"text-align: initial;font-size: 1em\"> of the constituents of the lattice. But for the sake of practical purpose, a phonon is considered to have its momentum, more precisely referred to as crystal momentum.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The selection rule so obeyed is as under<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u03c9 = \u03ce +?<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">here <span style=\"text-align: initial;font-size: 1em\">?<\/span>\u00a0 is<span style=\"text-align: initial;font-size: 1em\"> the frequency of the emitted <\/span>phonon<span style=\"text-align: initial;font-size: 1em\">.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">And energy and momentum conservation laws become<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u0127\u03c9 = \u0127\u03ce + \u0127? and<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u0127? + \u0127G = \u0127\u1e31 + \u0127K<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The conservation of the selection rules require that<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">K + G = \u1e31 <\/strong><span style=\"text-align: initial;font-size: 1em\">\u00b1<\/span><strong style=\"text-align: initial;font-size: 1em\"> K<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">It is found that the frequency of <\/span>incident<span style=\"text-align: initial;font-size: 1em\"> photon and <\/span>scattered<span style=\"text-align: initial;font-size: 1em\"> photon is not same i.e. <\/span><strong style=\"text-align: initial;font-size: 1em\">\u03c9<\/strong><span style=\"text-align: initial;font-size: 1em\"> \u2260 <\/span><strong style=\"text-align: initial;font-size: 1em\">\u03ce<\/strong><span style=\"text-align: initial;font-size: 1em\"> and the momentum is transferred to the whole crystal<\/span><strong style=\"text-align: initial;font-size: 1em\">.<\/strong><span style=\"text-align: initial;font-size: 1em\"> This process is known as U-Process.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In neutron elastic scattering process a neutron interacts with the crystal by interacting with the nuclei of the atom. It is used for determination of phonon dispersion relation.<\/span><\/p>\r\n\r\n<\/div>\r\n<img class=\"aligncenter size-full wp-image-225\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-66.png\" alt=\"\" width=\"832\" height=\"354\" \/>\r\n<div>\r\n\r\n<img class=\"size-full wp-image-226 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-67.png\" alt=\"\" width=\"510\" height=\"237\" \/>\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<strong>2 Anharmonic Interactions in Crystals<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The study of lattice dynamics in this chapter has so far been dealt only with potential energies to quadratic terms and the higher terms have been neglected. The lattice vibrations and atomic motions are studied within the harmonic approximations, under which the consequences are that the lattice waves never interact and the wave form never changes or decays with time. To deal with real time phenomenon of thermal expansion, thermal\u00a0<span style=\"text-align: initial;font-size: 1em\">conductivity, <\/span>temperature<span style=\"text-align: initial;font-size: 1em\"> dependence of elastic constants and the problem of increase in heat capacity at T &gt; <\/span>\u03b8 ,the<span style=\"text-align: initial;font-size: 1em\"> anharmonic effects are to be taken into account.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Let for an atom in any position obeying Hook\u2019s law, the energy expression is<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u0404r = \u0404ro + A(r-<\/span>ro<span style=\"text-align: initial;font-size: 1em\">)<sup>2<\/sup><\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In such a case, the phonon interaction becomes impossible.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">One cannot neglect anharmonicity which is a source of coupling between lattice waves and makes <\/span>scattering<span style=\"text-align: initial;font-size: 1em\"> of waves possible.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Therefore energy expression has to be of the form<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u0404r = \u0404ro + A(r-ro)<sup>2<\/sup> +B(r-ro)<sup>3<\/sup> + _ _ _ +_ _<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The higher order anharmonic terms makes phonons to collide with perfect crystal by making mean free path of finite dimensions.<\/p>\r\n&nbsp;\r\n\r\n<strong>3 Thermal Expansion in Solids<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The general answer to why do solids expand on increasing temperature is that an increase in energy results in an increase in equilibrium spacing of atomic bonds.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Thermal expansion is measured by the coefficient of linear thermal expansion defined as the increase in length per unit rise in temperature. The thermal expansion in solids is a direct consequence of an harmonicity of the atomic interactions. With the increase in temperature the amplitude of lattice vibrations increases and the equilibrium position shifts as the atoms spend time at greater than original spacing due as the repulsion at short distances is greater than the corresponding attraction at farther distances.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">In case of a one dimensional solid if the vibrations are considered harmonic , the potential energy will be<\/p>\r\n<img class=\"size-full wp-image-227 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-68.png\" alt=\"\" width=\"92\" height=\"32\" \/>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<strong>\u00a0<\/strong>\r\n\r\n<img class=\"aligncenter size-full wp-image-228\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-69.png\" alt=\"\" width=\"573\" height=\"432\" \/>\r\n<div>\r\n<p style=\"text-align: center\"><strong>Fig (2) showing the change in equilibrium position of vibrations with respect to energy<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">And it will be a parabola with xo as the equilibrium position, then the harmonic potential is a perfectly parabola function and the mean position does not shift with rise in temperature.<\/p>\r\n&nbsp;\r\n\r\nThe mean displacement <strong>,<\/strong>\r\n\r\n<img class=\"size-full wp-image-229 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-70.png\" alt=\"\" width=\"538\" height=\"62\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">But the atomic oscillators differ from perfectly harmonic oscillators and essentially have harmonicity in the potential with comes into play with higher order terms in \u2018x\u2019<\/p>\r\n&nbsp;\r\n\r\nTherefore for\u00a0 anharmonic potential,\r\n\r\n<img class=\"size-full wp-image-230 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-71.png\" alt=\"\" width=\"363\" height=\"114\" \/>\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-231 alignleft\" style=\"text-align: initial;font-size: 1em\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-72.png\" alt=\"\" width=\"597\" height=\"114\" \/>\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\">The above equation suggests a prominent anharmonicity or a departure from harmonic behavior and the curve starts to deviate. The mean positions start shifting with increasing temperature corresponding to thermal expansion in physical behavior as demonstrated in figure (2).<\/p>\r\n&nbsp;\r\n\r\nFor small anharmonicity the first term of the numerator takes the form as under\r\n\r\n<img class=\"size-full wp-image-232 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-73.png\" alt=\"\" width=\"554\" height=\"189\" \/>\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\">We find that there appears a non zero mean displacement in the presence of the anharmonicity which contributes to thermal expansion.<\/p>\r\n&nbsp;\r\n\r\n<strong>4. Thermal Conductivity<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">As it has been discussed in the previous sections how heat energy gets transmitted through annihilation and creation of phonons in a crystal. An attempt is made to understand the heat conductivity and energy transmission in solids taking phonon factor into account.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">For a long rod let a steady heat flows along its length with temperature gradient <strong><em>dT\/dx<\/em><\/strong>. It is assumed that the temperature variation is very small such that the average phonon number is definite. As the neighboring regions have slightly varying temperatures, the phonon number behaves as a function of position. The energy transmitted across unit area per unit time called the flux of thermal energy is given as<\/p>\r\n&nbsp;\r\n\r\n<strong>E<\/strong><strong>th<\/strong><strong> = - K dT\/dx<\/strong>\r\n\r\n&nbsp;\r\n\r\nWhere K is the coefficient of thermal conductivity in solids.\r\n\r\n&nbsp;\r\n\r\nConsider phonon gas analogous to molecular gas then on the basis of kinetic theory of gases,\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">K = 1\/3 C<\/strong><sub><strong style=\"text-align: initial\">v<\/strong><\/sub><strong style=\"text-align: initial;font-size: 1em\">\u03bd\u03bb<\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Cv is the lattice specific heat which is a measure of phonon density.<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">\u03bd is the average phonon velocity and \u03bb is the mean free path of the phonons.<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Hence the expression for thermal flux reduces to<\/span>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">E<\/strong><sub style=\"text-align: initial\"><strong>th<\/strong><\/sub><strong style=\"text-align: initial;font-size: 1em\"> = - 1\/3 C<\/strong><sub style=\"text-align: initial\"><strong>v<\/strong><\/sub><strong style=\"text-align: initial;font-size: 1em\">\u03bd\u03bb dT\/dx<\/strong>\r\n\r\n&nbsp;\r\n\r\nActually<span style=\"text-align: initial;font-size: 1em\"> total thermal conductivity receives <\/span>contribution<span style=\"text-align: initial;font-size: 1em\"> from conductivity due to electrons and phonons each, hence<\/span>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">K<\/strong><sub><strong style=\"text-align: initial\">total<\/strong><\/sub><strong style=\"text-align: initial;font-size: 1em\"> = K<\/strong><sub><strong style=\"text-align: initial\">electron<\/strong><\/sub><strong style=\"text-align: initial;font-size: 1em\"> + K<\/strong><sub><strong style=\"text-align: initial\">phonon<\/strong><\/sub>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">The expression for thermal conductivity due to electrons is deduced and is<\/span>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">K<\/strong><sub><strong style=\"text-align: initial\">electron<\/strong><\/sub><strong style=\"text-align: initial;font-size: 1em\"> = [\u03c0<\/strong><sup><strong style=\"text-align: initial\">2<\/strong><\/sup><strong style=\"text-align: initial;font-size: 1em\">Nk<\/strong><sup><strong style=\"text-align: initial\">2<\/strong><\/sup><strong style=\"text-align: initial;font-size: 1em\">\u03c4\/3m]T<\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Here N is the number of free electrons per mole and \u03c4 is the mean free time of the electron before <\/span>collision<span style=\"text-align: initial;font-size: 1em\"> with the positive ion where it gives its entire thermal energy.<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">It is observed that in the expression, Kelectron is almost independent of temperature as the mean free time \u03c4 varies as T-1 above Debye\u2019s temperature \u03b8D.<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">So the phonons carry most of the heat energy while electrons stay immobile. We have studied in detail the lattice specific heat as given by Debye\u2019s law as<\/span>\r\n\r\n&nbsp;\r\n\r\n<strong><span style=\"text-align: initial;font-size: 1em\">C<sub>v<\/sub> = \u03b1T<sup>3<\/sup> for T \u02c2 \u03b8D.<\/span><\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong><span style=\"text-align: initial;font-size: 1em\">And C<sub>v<\/sub> = 3Nk for T \u02c3 \u03b8D.<\/span><\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Hence the corresponding thermal conductivity expressions are<\/span>\r\n\r\n&nbsp;\r\n\r\n<strong><span style=\"text-align: initial;font-size: 1em\">K<sub>phonon<\/sub> = 1\/3 \u03bd\u03bb \u03b1T<sup>3<\/sup> for T \u02c2 \u03b8D<\/span><\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong><span style=\"text-align: initial;font-size: 1em\">K<sub>phonon<\/sub> = 3 \u03bd\u03bb Nk for T \u02c3 \u03b8D.<\/span><\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The above expressions clearly indicate that thermal conductivity of insulators is constant at high temperatures but is proportional to T3 at lower temperatures and this variation in K arises due to <\/span>phonon \u2013 phonon<span style=\"text-align: initial;font-size: 1em\"> interaction or as we said <\/span>anharmonocity<span style=\"text-align: initial;font-size: 1em\">. The phonon mean free path \u03bb is found to be inversely proportional to absolute temperature so consequently the expression for thermal conduction due to phonons becomes<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">K<\/strong><sub><strong style=\"text-align: initial\">phonon<\/strong><\/sub><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u03b1 T<\/strong><sup><strong style=\"text-align: initial\">-1<\/strong><\/sup><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">So thermal conductivity of an insulator is proportional to T-1\u00a0 at high temperatures.<\/span><\/p>\r\n\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 Properties 5<\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/1rFIIpIoWiY\" 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&nbsp;\r\n\r\n<strong>Summary:<\/strong>\r\n\r\n&nbsp;\r\n\r\nAfter the completion of this module we are able to understand the following\r\n<ul>\r\n \t<li style=\"text-align: justify\">Scattering of neutrons is a probe to study collisions in order to obtain information regarding lattice dynamics through phonon energies.<\/li>\r\n \t<li style=\"text-align: justify\">Inelastic scattering and the concept wave vector linked with absorption or emission of phonons is important to determine phonon dispersion relations.<\/li>\r\n \t<li style=\"text-align: justify\">Have an insight of importance<span style=\"text-align: initial;font-size: 1em\"> of anharmonicity in crystal interactions Correlate thermal expansion as a direct consequence of anharmonicity.<\/span><\/li>\r\n \t<li style=\"text-align: justify\">Visualize thermal conduction in context<span style=\"text-align: initial;font-size: 1em\"> of <\/span>anhamonic<span style=\"text-align: initial;font-size: 1em\"> crystal interaction.<\/span><\/li>\r\n<\/ul>\r\n<strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 Value Addition:<\/strong>\r\n<div>\r\n<p style=\"text-align: justify\"><strong>\u00a0 \u00a0 The phonon interaction with photons, electrons, neutrons , magnons and excitons are of prime importance in studying and understanding the physical properties of solids.<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>As in weak coupling cases, scattering caused by phonons puts a limitation on the mean free path of electrons and its importance is well appreciated in understanding the conductivity of metals and mobility of carriers in semiconductors.<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>The electron \u2013 phonon interactions have gained significant interest and importance in solid state physics. The transition metals and their compounds are the area of interest in this concern. Interestingly the electron \u2013 phonon interactions are thought to be responsible for high temperature superconductivity in compounds like V<\/strong><strong>3<\/strong><strong>Si and NbC.<\/strong><\/p>\r\n&nbsp;\r\n\r\n<strong>Suggested reading<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>Physics through The 1990\u2019s Condensed Matter Physics , National Academy Press, Washington D.C., 1986<\/strong>\r\n\r\n<\/div>\r\n<div><\/div>\r\n<strong>\u00a0 \u00a0 Glossary:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>Phonon interaction: <\/strong>it is assumed as a scattering collision during which at a time only a single phonon is absorbed or emitted. Also the energy of the lattice vibrations involved is an integral multiple of \u0127\u03c9.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>Total energy of phonons<\/strong>: it is actually the thermal energy of the solidbecause at a given temperature the solid is assumed to be completely filled with phonons.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>Temperature gradient<\/strong>: it is the variation in temperature with respect to distance from the hot end to the cold one.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>Mean phonon free path<\/strong>:\u00a0\u00a0\u00a0 the average of all the lengths travelled by the phonon between collisions.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>Excitons <\/strong>: these are electron \u2013hole pairs bound by coulombs electrostatic force that is responsible for transport of energy without transporting net charge.<\/p>","rendered":"<div>\n<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/1rFIIpIoWiY\" 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>The objective of the module is to<\/p>\n<ul>\n<li>Understand the importance of neutron scattering in order to study phonon interaction in detail.<\/li>\n<li>Have an insight of importance<span style=\"text-align: initial;font-size: 1em\"> of anharmonicity in crystal interactions.<\/span><\/li>\n<li>Correlate thermal expansion and thermal conduction in context<span style=\"text-align: initial;font-size: 1em\"> of <\/span>anhamonic<span style=\"text-align: initial;font-size: 1em\"> crystal interaction.<\/span><\/li>\n<\/ul>\n<\/div>\n<div>\n<p><strong>\u00a0 \u00a01.\u00a0<\/strong><strong>Neutron scattering:-<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The neutrons have a great advantage to be used as a probe in condensed matter physics. Scattering of free neutrons with the matter is used to measure details which are of prime importance in material research. Elastic scattering (neutron diffraction) is used in determining structures whereas inelastic scattering provides valuable information on atomic vibrations and excitations. The interaction of phonons is assumed as scattering collisions.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">A neutron sees the crystal lattice by interacting with the nuclei of the atoms and exchange energy which is used to measure details of atomic and molecular motions by the use of inelastic neutron scattering (INS). When a crystal gets scattered by a crystal, it results either in absorption or emission of energy of an amount equal to quantum of phonon (h\u03bd). The wave vectors before and after the interaction gives information about the phonon energies.<\/p>\n<p>&nbsp;<\/p>\n<p>Here we study the neutron scattering and phonon interaction dynamics in detail.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>1.1 Elastic scattering<\/strong><strong>:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In elastic scattering of X-rays in a crystal (Bragg\u2019s diffraction) the magnitude of wave vector (or the frequency ) remains unchanged i.e,<\/p>\n<p>&nbsp;<\/p>\n<p>\u03c9 = \u03ce and<\/p>\n<p>&nbsp;<\/p>\n<p>?+G = \u1e31<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">where k and \u1e31 are wave vectors of incident and scattered photons, which obey selection rules. G is one of the reciprocal lattice vectors.<\/p>\n<p>&nbsp;<\/p>\n<p>Also the energy and momenta are conserved i.e,<\/p>\n<p>&nbsp;<\/p>\n<p>\u0127\u03c9 = \u0127\u03ce\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 and<\/p>\n<p>&nbsp;<\/p>\n<p>\u0127\u00a0\u00a0 + \u0127G = \u0127\u1e31<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">As the frequencies of incident and scattered photons are unchanged the crystal as a whole recoils with momentum -\u0127G . This process is called N-Process.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">1.2 Inelastic scattering:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">But in case the scattering is inelastic, then the process leads to either emission or absorption of a phonon of wave vector K. This phonon will interact with photons, neutrons and electrons as if having a momentum <\/span>\u0127k ,however<span style=\"text-align: initial;font-size: 1em\"> a phonon carries no physical momentum on lattices as the phonon <\/span>co ordinates<span style=\"text-align: initial;font-size: 1em\"> (other <\/span>than k<span style=\"text-align: initial;font-size: 1em\"> = 0) are linked with the relative <\/span>co ordinates<span style=\"text-align: initial;font-size: 1em\"> of the constituents of the lattice. But for the sake of practical purpose, a phonon is considered to have its momentum, more precisely referred to as crystal momentum.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The selection rule so obeyed is as under<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u03c9 = \u03ce +?<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">here <span style=\"text-align: initial;font-size: 1em\">?<\/span>\u00a0 is<span style=\"text-align: initial;font-size: 1em\"> the frequency of the emitted <\/span>phonon<span style=\"text-align: initial;font-size: 1em\">.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">And energy and momentum conservation laws become<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u0127\u03c9 = \u0127\u03ce + \u0127? and<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u0127? + \u0127G = \u0127\u1e31 + \u0127K<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The conservation of the selection rules require that<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">K + G = \u1e31 <\/strong><span style=\"text-align: initial;font-size: 1em\">\u00b1<\/span><strong style=\"text-align: initial;font-size: 1em\"> K<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">It is found that the frequency of <\/span>incident<span style=\"text-align: initial;font-size: 1em\"> photon and <\/span>scattered<span style=\"text-align: initial;font-size: 1em\"> photon is not same i.e. <\/span><strong style=\"text-align: initial;font-size: 1em\">\u03c9<\/strong><span style=\"text-align: initial;font-size: 1em\"> \u2260 <\/span><strong style=\"text-align: initial;font-size: 1em\">\u03ce<\/strong><span style=\"text-align: initial;font-size: 1em\"> and the momentum is transferred to the whole crystal<\/span><strong style=\"text-align: initial;font-size: 1em\">.<\/strong><span style=\"text-align: initial;font-size: 1em\"> This process is known as U-Process.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In neutron elastic scattering process a neutron interacts with the crystal by interacting with the nuclei of the atom. It is used for determination of phonon dispersion relation.<\/span><\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-225\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-66.png\" alt=\"\" width=\"832\" height=\"354\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-66.png 832w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-66-300x128.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-66-768x327.png 768w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-66-65x28.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-66-225x96.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-66-350x149.png 350w\" sizes=\"auto, (max-width: 832px) 100vw, 832px\" \/><\/p>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-226 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-67.png\" alt=\"\" width=\"510\" height=\"237\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-67.png 510w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-67-300x139.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-67-65x30.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-67-225x105.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-67-350x163.png 350w\" sizes=\"auto, (max-width: 510px) 100vw, 510px\" \/><\/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><strong>2 Anharmonic Interactions in Crystals<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The study of lattice dynamics in this chapter has so far been dealt only with potential energies to quadratic terms and the higher terms have been neglected. The lattice vibrations and atomic motions are studied within the harmonic approximations, under which the consequences are that the lattice waves never interact and the wave form never changes or decays with time. To deal with real time phenomenon of thermal expansion, thermal\u00a0<span style=\"text-align: initial;font-size: 1em\">conductivity, <\/span>temperature<span style=\"text-align: initial;font-size: 1em\"> dependence of elastic constants and the problem of increase in heat capacity at T &gt; <\/span>\u03b8 ,the<span style=\"text-align: initial;font-size: 1em\"> anharmonic effects are to be taken into account.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Let for an atom in any position obeying Hook\u2019s law, the energy expression is<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u0404r = \u0404ro + A(r-<\/span>ro<span style=\"text-align: initial;font-size: 1em\">)<sup>2<\/sup><\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In such a case, the phonon interaction becomes impossible.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">One cannot neglect anharmonicity which is a source of coupling between lattice waves and makes <\/span>scattering<span style=\"text-align: initial;font-size: 1em\"> of waves possible.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Therefore energy expression has to be of the form<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u0404r = \u0404ro + A(r-ro)<sup>2<\/sup> +B(r-ro)<sup>3<\/sup> + _ _ _ +_ _<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The higher order anharmonic terms makes phonons to collide with perfect crystal by making mean free path of finite dimensions.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>3 Thermal Expansion in Solids<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The general answer to why do solids expand on increasing temperature is that an increase in energy results in an increase in equilibrium spacing of atomic bonds.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Thermal expansion is measured by the coefficient of linear thermal expansion defined as the increase in length per unit rise in temperature. The thermal expansion in solids is a direct consequence of an harmonicity of the atomic interactions. With the increase in temperature the amplitude of lattice vibrations increases and the equilibrium position shifts as the atoms spend time at greater than original spacing due as the repulsion at short distances is greater than the corresponding attraction at farther distances.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In case of a one dimensional solid if the vibrations are considered harmonic , the potential energy will be<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-227 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-68.png\" alt=\"\" width=\"92\" height=\"32\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-68.png 92w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-68-65x23.png 65w\" sizes=\"auto, (max-width: 92px) 100vw, 92px\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p><strong>\u00a0<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-228\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-69.png\" alt=\"\" width=\"573\" height=\"432\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-69.png 573w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-69-300x226.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-69-65x49.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-69-225x170.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-69-350x264.png 350w\" sizes=\"auto, (max-width: 573px) 100vw, 573px\" \/><\/p>\n<div>\n<p style=\"text-align: center\"><strong>Fig (2) showing the change in equilibrium position of vibrations with respect to energy<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">And it will be a parabola with xo as the equilibrium position, then the harmonic potential is a perfectly parabola function and the mean position does not shift with rise in temperature.<\/p>\n<p>&nbsp;<\/p>\n<p>The mean displacement <strong>,<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-229 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-70.png\" alt=\"\" width=\"538\" height=\"62\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-70.png 538w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-70-300x35.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-70-65x7.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-70-225x26.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-70-350x40.png 350w\" sizes=\"auto, (max-width: 538px) 100vw, 538px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">But the atomic oscillators differ from perfectly harmonic oscillators and essentially have harmonicity in the potential with comes into play with higher order terms in \u2018x\u2019<\/p>\n<p>&nbsp;<\/p>\n<p>Therefore for\u00a0 anharmonic potential,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-230 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-71.png\" alt=\"\" width=\"363\" height=\"114\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-71.png 363w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-71-300x94.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-71-65x20.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-71-225x71.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-71-350x110.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><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-231 alignleft\" style=\"text-align: initial;font-size: 1em\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-72.png\" alt=\"\" width=\"597\" height=\"114\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-72.png 597w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-72-300x57.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-72-65x12.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-72-225x43.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-72-350x67.png 350w\" sizes=\"auto, (max-width: 597px) 100vw, 597px\" \/><\/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\">The above equation suggests a prominent anharmonicity or a departure from harmonic behavior and the curve starts to deviate. The mean positions start shifting with increasing temperature corresponding to thermal expansion in physical behavior as demonstrated in figure (2).<\/p>\n<p>&nbsp;<\/p>\n<p>For small anharmonicity the first term of the numerator takes the form as under<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-232 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-73.png\" alt=\"\" width=\"554\" height=\"189\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-73.png 554w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-73-300x102.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-73-65x22.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-73-225x77.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-73-350x119.png 350w\" sizes=\"auto, (max-width: 554px) 100vw, 554px\" \/><\/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\">We find that there appears a non zero mean displacement in the presence of the anharmonicity which contributes to thermal expansion.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>4. Thermal Conductivity<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">As it has been discussed in the previous sections how heat energy gets transmitted through annihilation and creation of phonons in a crystal. An attempt is made to understand the heat conductivity and energy transmission in solids taking phonon factor into account.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">For a long rod let a steady heat flows along its length with temperature gradient <strong><em>dT\/dx<\/em><\/strong>. It is assumed that the temperature variation is very small such that the average phonon number is definite. As the neighboring regions have slightly varying temperatures, the phonon number behaves as a function of position. The energy transmitted across unit area per unit time called the flux of thermal energy is given as<\/p>\n<p>&nbsp;<\/p>\n<p><strong>E<\/strong><strong>th<\/strong><strong> = &#8211; K dT\/dx<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Where K is the coefficient of thermal conductivity in solids.<\/p>\n<p>&nbsp;<\/p>\n<p>Consider phonon gas analogous to molecular gas then on the basis of kinetic theory of gases,<\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">K = 1\/3 C<\/strong><sub><strong style=\"text-align: initial\">v<\/strong><\/sub><strong style=\"text-align: initial;font-size: 1em\">\u03bd\u03bb<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Cv is the lattice specific heat which is a measure of phonon density.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">\u03bd is the average phonon velocity and \u03bb is the mean free path of the phonons.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Hence the expression for thermal flux reduces to<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">E<\/strong><sub style=\"text-align: initial\"><strong>th<\/strong><\/sub><strong style=\"text-align: initial;font-size: 1em\"> = &#8211; 1\/3 C<\/strong><sub style=\"text-align: initial\"><strong>v<\/strong><\/sub><strong style=\"text-align: initial;font-size: 1em\">\u03bd\u03bb dT\/dx<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Actually<span style=\"text-align: initial;font-size: 1em\"> total thermal conductivity receives <\/span>contribution<span style=\"text-align: initial;font-size: 1em\"> from conductivity due to electrons and phonons each, hence<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">K<\/strong><sub><strong style=\"text-align: initial\">total<\/strong><\/sub><strong style=\"text-align: initial;font-size: 1em\"> = K<\/strong><sub><strong style=\"text-align: initial\">electron<\/strong><\/sub><strong style=\"text-align: initial;font-size: 1em\"> + K<\/strong><sub><strong style=\"text-align: initial\">phonon<\/strong><\/sub><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">The expression for thermal conductivity due to electrons is deduced and is<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">K<\/strong><sub><strong style=\"text-align: initial\">electron<\/strong><\/sub><strong style=\"text-align: initial;font-size: 1em\"> = [\u03c0<\/strong><sup><strong style=\"text-align: initial\">2<\/strong><\/sup><strong style=\"text-align: initial;font-size: 1em\">Nk<\/strong><sup><strong style=\"text-align: initial\">2<\/strong><\/sup><strong style=\"text-align: initial;font-size: 1em\">\u03c4\/3m]T<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Here N is the number of free electrons per mole and \u03c4 is the mean free time of the electron before <\/span>collision<span style=\"text-align: initial;font-size: 1em\"> with the positive ion where it gives its entire thermal energy.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">It is observed that in the expression, Kelectron is almost independent of temperature as the mean free time \u03c4 varies as T-1 above Debye\u2019s temperature \u03b8D.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">So the phonons carry most of the heat energy while electrons stay immobile. We have studied in detail the lattice specific heat as given by Debye\u2019s law as<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><strong><span style=\"text-align: initial;font-size: 1em\">C<sub>v<\/sub> = \u03b1T<sup>3<\/sup> for T \u02c2 \u03b8D.<\/span><\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong><span style=\"text-align: initial;font-size: 1em\">And C<sub>v<\/sub> = 3Nk for T \u02c3 \u03b8D.<\/span><\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Hence the corresponding thermal conductivity expressions are<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><strong><span style=\"text-align: initial;font-size: 1em\">K<sub>phonon<\/sub> = 1\/3 \u03bd\u03bb \u03b1T<sup>3<\/sup> for T \u02c2 \u03b8D<\/span><\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong><span style=\"text-align: initial;font-size: 1em\">K<sub>phonon<\/sub> = 3 \u03bd\u03bb Nk for T \u02c3 \u03b8D.<\/span><\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The above expressions clearly indicate that thermal conductivity of insulators is constant at high temperatures but is proportional to T3 at lower temperatures and this variation in K arises due to <\/span>phonon \u2013 phonon<span style=\"text-align: initial;font-size: 1em\"> interaction or as we said <\/span>anharmonocity<span style=\"text-align: initial;font-size: 1em\">. The phonon mean free path \u03bb is found to be inversely proportional to absolute temperature so consequently the expression for thermal conduction due to phonons becomes<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">K<\/strong><sub><strong style=\"text-align: initial\">phonon<\/strong><\/sub><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u03b1 T<\/strong><sup><strong style=\"text-align: initial\">-1<\/strong><\/sup><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">So thermal conductivity of an insulator is proportional to T-1\u00a0 at high temperatures.<\/span><\/p>\n<\/div>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on Lattice Vibrations and Thermal Properties 5<\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/1rFIIpIoWiY\" 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>&nbsp;<\/p>\n<p><strong>Summary:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>After the completion of this module we are able to understand the following<\/p>\n<ul>\n<li style=\"text-align: justify\">Scattering of neutrons is a probe to study collisions in order to obtain information regarding lattice dynamics through phonon energies.<\/li>\n<li style=\"text-align: justify\">Inelastic scattering and the concept wave vector linked with absorption or emission of phonons is important to determine phonon dispersion relations.<\/li>\n<li style=\"text-align: justify\">Have an insight of importance<span style=\"text-align: initial;font-size: 1em\"> of anharmonicity in crystal interactions Correlate thermal expansion as a direct consequence of anharmonicity.<\/span><\/li>\n<li style=\"text-align: justify\">Visualize thermal conduction in context<span style=\"text-align: initial;font-size: 1em\"> of <\/span>anhamonic<span style=\"text-align: initial;font-size: 1em\"> crystal interaction.<\/span><\/li>\n<\/ul>\n<p><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 Value Addition:<\/strong><\/p>\n<div>\n<p style=\"text-align: justify\"><strong>\u00a0 \u00a0 The phonon interaction with photons, electrons, neutrons , magnons and excitons are of prime importance in studying and understanding the physical properties of solids.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>As in weak coupling cases, scattering caused by phonons puts a limitation on the mean free path of electrons and its importance is well appreciated in understanding the conductivity of metals and mobility of carriers in semiconductors.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>The electron \u2013 phonon interactions have gained significant interest and importance in solid state physics. The transition metals and their compounds are the area of interest in this concern. Interestingly the electron \u2013 phonon interactions are thought to be responsible for high temperature superconductivity in compounds like V<\/strong><strong>3<\/strong><strong>Si and NbC.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Suggested reading<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Physics through The 1990\u2019s Condensed Matter Physics , National Academy Press, Washington D.C., 1986<\/strong><\/p>\n<\/div>\n<div><\/div>\n<p><strong>\u00a0 \u00a0 Glossary:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>Phonon interaction: <\/strong>it is assumed as a scattering collision during which at a time only a single phonon is absorbed or emitted. Also the energy of the lattice vibrations involved is an integral multiple of \u0127\u03c9.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>Total energy of phonons<\/strong>: it is actually the thermal energy of the solidbecause at a given temperature the solid is assumed to be completely filled with phonons.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>Temperature gradient<\/strong>: it is the variation in temperature with respect to distance from the hot end to the cold one.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>Mean phonon free path<\/strong>:\u00a0\u00a0\u00a0 the average of all the lengths travelled by the phonon between collisions.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>Excitons <\/strong>: these are electron \u2013hole pairs bound by coulombs electrostatic force that is responsible for transport of energy without transporting net charge.<\/p>\n","protected":false},"author":3,"menu_order":14,"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-220","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\/220","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":7,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters\/220\/revisions"}],"predecessor-version":[{"id":237,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters\/220\/revisions\/237"}],"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\/220\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/media?parent=220"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapter-type?post=220"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/contributor?post=220"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/license?post=220"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}