{"id":162,"date":"2018-12-04T05:56:41","date_gmt":"2018-12-04T05:56:41","guid":{"rendered":"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=162"},"modified":"2018-12-04T10:37:19","modified_gmt":"2018-12-04T10:37:19","slug":"lattice-vibrations-and-thermal-properties-2","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/chapter\/lattice-vibrations-and-thermal-properties-2\/","title":{"rendered":"Lattice Vibrations and Thermal Properties 2"},"content":{"raw":"<div>\r\n<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/w89t53jfV5Y\" 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<strong>\u00a0 \u00a0 <\/strong>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong>Learning Outcomes<\/strong>\r\n\r\n&nbsp;\r\n\r\nThe objective of the module is to\r\n<ul>\r\n \t<li>Visualise the lattice dynamics with the help of simple lattice model<\/li>\r\n \t<li>Understand the basic phenomenon of lattice vibrations and extending the study to mono atomic<span style=\"text-align: initial;font-size: 1em\"> and diatomic lattice chains.<\/span><\/li>\r\n \t<li>Mathematically analyse<span style=\"text-align: initial;font-size: 1em\"> the waves for displacements and frequency response.<\/span><\/li>\r\n \t<li>Relate the idea of dispersive and non dispersive<span style=\"text-align: initial;font-size: 1em\"> media and consequence as <\/span>acoustic<span style=\"text-align: initial;font-size: 1em\"> and optical branch.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 1.\u00a0<\/strong><strong style=\"text-align: initial;font-size: 1em\">Lattice vibrations :-<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">To understand crystal lattice dynamics, it is assumed that a crystal consists of a regular arrangement of atoms (or molecules) which are fixed at equilibrium positions at absolute zero. But when there is a rise in temperature these particles execute simple harmonic motion and vibrate at mean positions. The atoms are connected with each other by elastic springs such that each atom exerts a force on its neighboring atoms and the vibrations so produced in the whole crystal are termed as lattice vibrations.<\/span><\/p>\r\n\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-165\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-25.png\" alt=\"\" width=\"633\" height=\"481\" \/>\r\n\r\n&nbsp;\r\n\r\n<img class=\"size-full wp-image-166 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-26.png\" alt=\"\" width=\"578\" height=\"173\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nIt is accompanied by either emission or absorption of <em>quantized<\/em><em>thermal energy <\/em>called<em> phonon<\/em>.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">So the lattice vibrations are thermally excited phonons that are the smallest energy of vibration, which require a medium to propagate and exhibit elastic wave behavior. Phonons behave like a mass less particle with energy h\u03bd, it does not carry any physical momentum, but it appears to interact as if carrying a momentum \u0127\u00a0 \u00a0more precisely called the crystal momentum, where \u20d7 is the wave vector of phonon.<\/p>\r\n&nbsp;\r\n\r\n<strong>1.1 Lattice vibrations in one \u2013 dimensional mono atomic crystal:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">To understand the dynamics of lattice behavior, a simple model is proposed to make the relevant studies. It is assumed that<\/p>\r\n\r\n<ol>\r\n \t<li style=\"text-align: justify\">Lattice is a periodic arrangement of particles (atoms, ions or molecules) in three dimensions connected to each other through mass less springs executing simple harmonic motion.<\/li>\r\n \t<li style=\"text-align: justify\">Particles are identical spheres of same mass m and there is no interaction between these particles except for nearest neighbours.<\/li>\r\n \t<li style=\"text-align: justify\">The system obeys Hooke\u2019s law essentially and for the sake of simplicity, the system is worked within harmonic approximation.<\/li>\r\n<\/ol>\r\n<p style=\"text-align: justify\">\u00a0 It is known fact that atoms in solids vibrate because of their thermal energies. Let us consider a one dimensional chain of atoms forming a ring so that all atoms have similar environment. With the vibration of one atom, the disturbance travels through the chain and say the displacement of <em>nth<\/em> atom from equilibrium position be Xn. and Xn-1 &amp; Xn+1 be the displacements of <em>(n-1)th<\/em> atom and <em>(n+1)th<\/em> atom respectively from their mean positions.<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-167\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-27.png\" alt=\"\" width=\"794\" height=\"287\" \/>\r\n\r\n<\/div>\r\n<img class=\"size-full wp-image-168 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-28.png\" alt=\"\" width=\"745\" height=\"543\" \/>\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&nbsp;\r\n\r\n&nbsp;\r\n\r\n<img class=\"size-full wp-image-169 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-29.png\" alt=\"\" width=\"716\" height=\"307\" \/>\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<\/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<div>\r\n\r\n<img class=\"size-full wp-image-170 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-30.png\" alt=\"\" width=\"746\" height=\"423\" \/>\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-171\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-31.png\" alt=\"\" width=\"690\" height=\"373\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>1.2 Lattice vibrations in one dimensional diatomic crystal:-<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In case of a one dimensional diatomic ( two atoms per primitive cell) linear chain, the underlying principle for describing vibrational modes is more or less same as that of monoatomic lattice but the only difference is the different masses of the two atoms, see figure (4).<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-172\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-32.png\" alt=\"\" width=\"487\" height=\"156\" \/>\r\n<p style=\"text-align: center\"><strong>Fig (4)<\/strong>\u00a0\u00a0\u00a0\u00a0 One dimensional diatomic linear crystal chain<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Let the two masses be m and M, m be the mass of odd numbered atoms and M the mass of even numbered atoms, such that mM and \u2018a\u2019 is the distance between immediate neighbors.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">The distance between immediate atoms is same as in linear monoatomic lattice i.e \u2018a\u2019 but it is assumed that the distance between similar atoms is \u20182a\u2019, see figure (5).<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><img class=\"aligncenter size-full wp-image-173\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-33.png\" alt=\"\" width=\"635\" height=\"294\" \/><strong style=\"text-align: initial;font-size: 1em\">Fig (5) <\/strong><span style=\"text-align: initial;font-size: 1em\">Displacement of atoms in <\/span>linear<span style=\"text-align: initial;font-size: 1em\"> diatomic lattice. <\/span>Distance<span style=\"text-align: initial;font-size: 1em\"> between immediate neighboring atoms is \u2018a\u2019 while that between atoms of <\/span>same<span style=\"text-align: initial;font-size: 1em\"> kind is \u20182a'.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Assuming immediate neighbor interaction alone, the equation of motion will be<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"size-full wp-image-174 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-34.png\" alt=\"\" width=\"731\" height=\"441\" \/>\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\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 dispersion relation for one dimensional diatomic lattice. It is observed that unlike mono atomic lattice , the frequency here obtained has two components + and - For ka \u226a 1 (k=0),<\/p>\r\n<img class=\"size-full wp-image-175 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-35.png\" alt=\"\" width=\"579\" height=\"149\" \/>\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 solutions obtained from above equations are plotted in fig (6)\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-176\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-36.png\" alt=\"\" width=\"708\" height=\"402\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The allowed frequencies are split into two branches as shown in fig (6). The lower branch is called acoustical branch and the upper one is termed as optical branch.<\/p>\r\n<img class=\"size-full wp-image-177 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-37.png\" alt=\"\" width=\"634\" height=\"80\" \/>\r\n\r\n<\/div>\r\n<strong>\u00a0<\/strong>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nFor m=M , ?\/? = -1\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Indicating that the two masses move opposite to each other as in transverse <\/span>waves ,<span style=\"text-align: initial;font-size: 1em\"> hence termed optical branch.<\/span><\/p>\r\n\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-178\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-38.png\" alt=\"\" width=\"488\" height=\"322\" \/>\r\n\r\n&nbsp;\r\n\r\nFig (7) Two modes of vibration of masses (M and m) corresponding to acoustical branch and optical branch\r\n\r\n&nbsp;\r\n\r\nIn acoustical waves for longer wavelengths k0, implies that <em>coska<\/em>1- (k2a2)\/2, hence the ratio\r\n\r\n<img class=\"size-full wp-image-179 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-39.png\" alt=\"\" width=\"253\" height=\"117\" \/>\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\">Indicating that the neighbouring atoms vibrate in same direction as in longitudinal waves, so for this reason this branch is termed acoustical branch<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Also we notice that between two branches there exists a band gap (forbidden gap) where there are no possible solutions in this frequency range as shown in fig (8). The width of this band depends on the ratio of two masses such that larger the ratio, wider is the forbidden gap.<\/p>\r\n&nbsp;\r\n\r\nFor M = m branches become degenerate as band disappears 11\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-180\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-40.png\" alt=\"\" width=\"501\" height=\"385\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: center\"><strong style=\"text-align: initial;font-size: 1em\">Fig (8) <\/strong><span style=\"text-align: initial;font-size: 1em\">Forbidden band between acoustical and optical frequencies corresponding to which no waves can propagate<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>1.3 Dispersion Relation for 3-Dimensional Crystal.<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The properties of dispersion relations(monoatomic and di atomic) corresponding to one dimensional cases are also applicable to three dimensional system.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">However the only additional feature in three dimensional crystal is the polarization of lattice waves. But the basic difficulty in developing the lattice dynamics of 3-D space lattice involves very complex mathematics.<\/p>\r\n&nbsp;\r\n\r\nBut qualitatively for a simple cubic lattice, the normal mode solution may be written as\r\n\r\n&nbsp;\r\n\r\nUn\u00a0 =Aexpi(k.r-\u03c9t)\r\n\r\n&nbsp;\r\n\r\nwhere the wave vector k specify both, the wavelength and direction of propagation.\r\n\r\n&nbsp;\r\n\r\nA represents amplitude as well as the direction of vibrations of the atoms.\r\n\r\n&nbsp;\r\n\r\nr is the position vector.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Correspondingly three dispersion relations are obtained for acoustic and optical branches each as shown in figure.<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-181\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-41.png\" alt=\"\" width=\"566\" height=\"470\" \/>\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>you can view video on Lattice vibration and Thermal properties-2<\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/w89t53jfV5Y\" 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 style=\"text-align: initial;font-size: 1em\">2.Summary:<\/strong>\r\n\r\n<\/div>\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\">The study of solids cannot be done on an isolated system of particle, but the entire crystal determines its behaviour.<\/li>\r\n \t<li style=\"text-align: justify\">The concept of harmonic motion of atoms at their mean positions and lattice vibrations in mono-atomic and di-atomic crystal structures is also studied in detail.<\/li>\r\n \t<li style=\"text-align: justify\">Concept<span style=\"font-size: 1em\"> of <\/span>continuum<span style=\"font-size: 1em\"> and <\/span>dispersive<span style=\"font-size: 1em\"> medium is understood in context to lattice dynamics and the consequence of frequency responses giving <\/span>idea<span style=\"font-size: 1em\"> of acoustic and optic modes of vibrations is clear.<\/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>Atoms in solids execute oscillations about their mean positions called lattice vibrations. The basic lattice mechanical behavior formalism was originally formulated by Born and Von-Karman<\/strong><\/p>\r\n&nbsp;\r\n\r\n<strong>Even at absolute zero the vibrations do not cease but exist and these are called zero point vibrations.<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>The behavior of solids cannot be understood if we consider a single atom or molecule. The elements form solids because for certain range of temperatures and pressure the solids have less free energy than other states of matter.<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>These vibrations are primarily responsible for the characteristic properties of solids such as electrical and thermal conductivity, specific heat, phase change momentum, dielectric and electrical properties etc.<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>For More Details ( on this topic and other topics discussed in Text Module) See <\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>J. A. Riessland, The Physics of Phonons, Wiley-Interscience Publication<\/strong>,<\/p>\r\n<p style=\"text-align: justify\"><strong>James D Patterson Bernard C Bailey, Solid State Physics, Introduction To Theory <\/strong><\/p>\r\n<p style=\"text-align: justify\"><strong>Adrianus J Dekker,Solid State Physics<\/strong><\/p>\r\n<strong>Charles Kittel, Introduction to Solid State Physics.<\/strong>\r\n\r\n<strong>N. W. Ashcroft, D. Mermin, Solid State Physics, Holt, Rinehart and Winston,<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Glossary:<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Phonon;<\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">It is <\/span>ksnown<span style=\"text-align: initial;font-size: 1em\"> as the quanta of sound which is analogous to quanta of light \u2013 photon.<\/span>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Optical Branch:<\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">The upper branch of the allowed wave propagation (in the long wavelength limit) in a crystal structure.<\/span>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">In <\/span>this<span style=\"text-align: initial;font-size: 1em\"> the two atoms vibrate <\/span>opposite<span style=\"text-align: initial;font-size: 1em\"> to each other with lighter mass having greater amplitude.<\/span>\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div>\n<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/w89t53jfV5Y\" 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><strong>\u00a0 \u00a0 <\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Learning Outcomes<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>The objective of the module is to<\/p>\n<ul>\n<li>Visualise the lattice dynamics with the help of simple lattice model<\/li>\n<li>Understand the basic phenomenon of lattice vibrations and extending the study to mono atomic<span style=\"text-align: initial;font-size: 1em\"> and diatomic lattice chains.<\/span><\/li>\n<li>Mathematically analyse<span style=\"text-align: initial;font-size: 1em\"> the waves for displacements and frequency response.<\/span><\/li>\n<li>Relate the idea of dispersive and non dispersive<span style=\"text-align: initial;font-size: 1em\"> media and consequence as <\/span>acoustic<span style=\"text-align: initial;font-size: 1em\"> and optical branch.<\/span><\/li>\n<\/ul>\n<\/div>\n<p><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 1.\u00a0<\/strong><strong style=\"text-align: initial;font-size: 1em\">Lattice vibrations :-<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">To understand crystal lattice dynamics, it is assumed that a crystal consists of a regular arrangement of atoms (or molecules) which are fixed at equilibrium positions at absolute zero. But when there is a rise in temperature these particles execute simple harmonic motion and vibrate at mean positions. The atoms are connected with each other by elastic springs such that each atom exerts a force on its neighboring atoms and the vibrations so produced in the whole crystal are termed as lattice vibrations.<\/span><\/p>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-165\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-25.png\" alt=\"\" width=\"633\" height=\"481\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-25.png 633w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-25-300x228.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-25-65x49.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-25-225x171.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-25-350x266.png 350w\" sizes=\"auto, (max-width: 633px) 100vw, 633px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-166 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-26.png\" alt=\"\" width=\"578\" height=\"173\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-26.png 578w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-26-300x90.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-26-65x19.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-26-225x67.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-26-350x105.png 350w\" sizes=\"auto, (max-width: 578px) 100vw, 578px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>It is accompanied by either emission or absorption of <em>quantized<\/em><em>thermal energy <\/em>called<em> phonon<\/em>.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">So the lattice vibrations are thermally excited phonons that are the smallest energy of vibration, which require a medium to propagate and exhibit elastic wave behavior. Phonons behave like a mass less particle with energy h\u03bd, it does not carry any physical momentum, but it appears to interact as if carrying a momentum \u0127\u00a0 \u00a0more precisely called the crystal momentum, where \u20d7 is the wave vector of phonon.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>1.1 Lattice vibrations in one \u2013 dimensional mono atomic crystal:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">To understand the dynamics of lattice behavior, a simple model is proposed to make the relevant studies. It is assumed that<\/p>\n<ol>\n<li style=\"text-align: justify\">Lattice is a periodic arrangement of particles (atoms, ions or molecules) in three dimensions connected to each other through mass less springs executing simple harmonic motion.<\/li>\n<li style=\"text-align: justify\">Particles are identical spheres of same mass m and there is no interaction between these particles except for nearest neighbours.<\/li>\n<li style=\"text-align: justify\">The system obeys Hooke\u2019s law essentially and for the sake of simplicity, the system is worked within harmonic approximation.<\/li>\n<\/ol>\n<p style=\"text-align: justify\">\u00a0 It is known fact that atoms in solids vibrate because of their thermal energies. Let us consider a one dimensional chain of atoms forming a ring so that all atoms have similar environment. With the vibration of one atom, the disturbance travels through the chain and say the displacement of <em>nth<\/em> atom from equilibrium position be Xn. and Xn-1 &amp; Xn+1 be the displacements of <em>(n-1)th<\/em> atom and <em>(n+1)th<\/em> atom respectively from their mean positions.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-167\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-27.png\" alt=\"\" width=\"794\" height=\"287\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-27.png 794w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-27-300x108.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-27-768x278.png 768w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-27-65x23.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-27-225x81.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-27-350x127.png 350w\" sizes=\"auto, (max-width: 794px) 100vw, 794px\" \/><\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-168 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-28.png\" alt=\"\" width=\"745\" height=\"543\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-28.png 745w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-28-300x219.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-28-65x47.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-28-225x164.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-28-350x255.png 350w\" sizes=\"auto, (max-width: 745px) 100vw, 745px\" \/><\/p>\n<div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&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-169 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-29.png\" alt=\"\" width=\"716\" height=\"307\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-29.png 716w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-29-300x129.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-29-65x28.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-29-225x96.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-29-350x150.png 350w\" sizes=\"auto, (max-width: 716px) 100vw, 716px\" \/><\/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<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-170 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-30.png\" alt=\"\" width=\"746\" height=\"423\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-30.png 746w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-30-300x170.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-30-65x37.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-30-225x128.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-30-350x198.png 350w\" sizes=\"auto, (max-width: 746px) 100vw, 746px\" \/><\/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-171\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-31.png\" alt=\"\" width=\"690\" height=\"373\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-31.png 690w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-31-300x162.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-31-65x35.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-31-225x122.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-31-350x189.png 350w\" sizes=\"auto, (max-width: 690px) 100vw, 690px\" \/><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>1.2 Lattice vibrations in one dimensional diatomic crystal:-<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In case of a one dimensional diatomic ( two atoms per primitive cell) linear chain, the underlying principle for describing vibrational modes is more or less same as that of monoatomic lattice but the only difference is the different masses of the two atoms, see figure (4).<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-172\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-32.png\" alt=\"\" width=\"487\" height=\"156\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-32.png 487w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-32-300x96.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-32-65x21.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-32-225x72.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-32-350x112.png 350w\" sizes=\"auto, (max-width: 487px) 100vw, 487px\" \/><\/p>\n<p style=\"text-align: center\"><strong>Fig (4)<\/strong>\u00a0\u00a0\u00a0\u00a0 One dimensional diatomic linear crystal chain<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Let the two masses be m and M, m be the mass of odd numbered atoms and M the mass of even numbered atoms, such that mM and \u2018a\u2019 is the distance between immediate neighbors.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The distance between immediate atoms is same as in linear monoatomic lattice i.e \u2018a\u2019 but it is assumed that the distance between similar atoms is \u20182a\u2019, see figure (5).<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-173\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-33.png\" alt=\"\" width=\"635\" height=\"294\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-33.png 635w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-33-300x139.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-33-65x30.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-33-225x104.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-33-350x162.png 350w\" sizes=\"auto, (max-width: 635px) 100vw, 635px\" \/><strong style=\"text-align: initial;font-size: 1em\">Fig (5) <\/strong><span style=\"text-align: initial;font-size: 1em\">Displacement of atoms in <\/span>linear<span style=\"text-align: initial;font-size: 1em\"> diatomic lattice. <\/span>Distance<span style=\"text-align: initial;font-size: 1em\"> between immediate neighboring atoms is \u2018a\u2019 while that between atoms of <\/span>same<span style=\"text-align: initial;font-size: 1em\"> kind is \u20182a&#8217;.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Assuming immediate neighbor interaction alone, the equation of motion will be<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-174 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-34.png\" alt=\"\" width=\"731\" height=\"441\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-34.png 731w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-34-300x181.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-34-65x39.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-34-225x136.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-34-350x211.png 350w\" sizes=\"auto, (max-width: 731px) 100vw, 731px\" \/><\/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>&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 dispersion relation for one dimensional diatomic lattice. It is observed that unlike mono atomic lattice , the frequency here obtained has two components + and &#8211; For ka \u226a 1 (k=0),<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-175 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-35.png\" alt=\"\" width=\"579\" height=\"149\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-35.png 579w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-35-300x77.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-35-65x17.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-35-225x58.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-35-350x90.png 350w\" sizes=\"auto, (max-width: 579px) 100vw, 579px\" \/><\/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 solutions obtained from above equations are plotted in fig (6)<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-176\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-36.png\" alt=\"\" width=\"708\" height=\"402\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-36.png 708w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-36-300x170.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-36-65x37.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-36-225x128.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-36-350x199.png 350w\" sizes=\"auto, (max-width: 708px) 100vw, 708px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The allowed frequencies are split into two branches as shown in fig (6). The lower branch is called acoustical branch and the upper one is termed as optical branch.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-177 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-37.png\" alt=\"\" width=\"634\" height=\"80\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-37.png 634w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-37-300x38.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-37-65x8.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-37-225x28.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-37-350x44.png 350w\" sizes=\"auto, (max-width: 634px) 100vw, 634px\" \/><\/p>\n<\/div>\n<p><strong>\u00a0<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>For m=M , ?\/? = -1<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Indicating that the two masses move opposite to each other as in transverse <\/span>waves ,<span style=\"text-align: initial;font-size: 1em\"> hence termed optical branch.<\/span><\/p>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-178\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-38.png\" alt=\"\" width=\"488\" height=\"322\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-38.png 488w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-38-300x198.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-38-65x43.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-38-225x148.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-38-350x231.png 350w\" sizes=\"auto, (max-width: 488px) 100vw, 488px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Fig (7) Two modes of vibration of masses (M and m) corresponding to acoustical branch and optical branch<\/p>\n<p>&nbsp;<\/p>\n<p>In acoustical waves for longer wavelengths k0, implies that <em>coska<\/em>1- (k2a2)\/2, hence the ratio<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-179 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-39.png\" alt=\"\" width=\"253\" height=\"117\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-39.png 253w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-39-65x30.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-39-225x104.png 225w\" sizes=\"auto, (max-width: 253px) 100vw, 253px\" \/><\/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\">Indicating that the neighbouring atoms vibrate in same direction as in longitudinal waves, so for this reason this branch is termed acoustical branch<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Also we notice that between two branches there exists a band gap (forbidden gap) where there are no possible solutions in this frequency range as shown in fig (8). The width of this band depends on the ratio of two masses such that larger the ratio, wider is the forbidden gap.<\/p>\n<p>&nbsp;<\/p>\n<p>For M = m branches become degenerate as band disappears 11<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-180\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-40.png\" alt=\"\" width=\"501\" height=\"385\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-40.png 501w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-40-300x231.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-40-65x50.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-40-225x173.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-40-350x269.png 350w\" sizes=\"auto, (max-width: 501px) 100vw, 501px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\"><strong style=\"text-align: initial;font-size: 1em\">Fig (8) <\/strong><span style=\"text-align: initial;font-size: 1em\">Forbidden band between acoustical and optical frequencies corresponding to which no waves can propagate<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>1.3 Dispersion Relation for 3-Dimensional Crystal.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The properties of dispersion relations(monoatomic and di atomic) corresponding to one dimensional cases are also applicable to three dimensional system.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">However the only additional feature in three dimensional crystal is the polarization of lattice waves. But the basic difficulty in developing the lattice dynamics of 3-D space lattice involves very complex mathematics.<\/p>\n<p>&nbsp;<\/p>\n<p>But qualitatively for a simple cubic lattice, the normal mode solution may be written as<\/p>\n<p>&nbsp;<\/p>\n<p>Un\u00a0 =Aexpi(k.r-\u03c9t)<\/p>\n<p>&nbsp;<\/p>\n<p>where the wave vector k specify both, the wavelength and direction of propagation.<\/p>\n<p>&nbsp;<\/p>\n<p>A represents amplitude as well as the direction of vibrations of the atoms.<\/p>\n<p>&nbsp;<\/p>\n<p>r is the position vector.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Correspondingly three dispersion relations are obtained for acoustic and optical branches each as shown in figure.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-181\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-41.png\" alt=\"\" width=\"566\" height=\"470\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-41.png 566w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-41-300x249.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-41-65x54.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-41-225x187.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/12\/Untitled-41-350x291.png 350w\" sizes=\"auto, (max-width: 566px) 100vw, 566px\" \/><\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on Lattice vibration and Thermal properties-2<\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/w89t53jfV5Y\" 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 style=\"text-align: initial;font-size: 1em\">2.Summary:<\/strong><\/p>\n<\/div>\n<p>After the completion of this module we are able to understand the following<\/p>\n<ul>\n<li style=\"text-align: justify\">The study of solids cannot be done on an isolated system of particle, but the entire crystal determines its behaviour.<\/li>\n<li style=\"text-align: justify\">The concept of harmonic motion of atoms at their mean positions and lattice vibrations in mono-atomic and di-atomic crystal structures is also studied in detail.<\/li>\n<li style=\"text-align: justify\">Concept<span style=\"font-size: 1em\"> of <\/span>continuum<span style=\"font-size: 1em\"> and <\/span>dispersive<span style=\"font-size: 1em\"> medium is understood in context to lattice dynamics and the consequence of frequency responses giving <\/span>idea<span style=\"font-size: 1em\"> of acoustic and optic modes of vibrations is clear.<\/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>Atoms in solids execute oscillations about their mean positions called lattice vibrations. The basic lattice mechanical behavior formalism was originally formulated by Born and Von-Karman<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Even at absolute zero the vibrations do not cease but exist and these are called zero point vibrations.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>The behavior of solids cannot be understood if we consider a single atom or molecule. The elements form solids because for certain range of temperatures and pressure the solids have less free energy than other states of matter.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>These vibrations are primarily responsible for the characteristic properties of solids such as electrical and thermal conductivity, specific heat, phase change momentum, dielectric and electrical properties etc.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>For More Details ( on this topic and other topics discussed in Text Module) See <\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>J. A. Riessland, The Physics of Phonons, Wiley-Interscience Publication<\/strong>,<\/p>\n<p style=\"text-align: justify\"><strong>James D Patterson Bernard C Bailey, Solid State Physics, Introduction To Theory <\/strong><\/p>\n<p style=\"text-align: justify\"><strong>Adrianus J Dekker,Solid State Physics<\/strong><\/p>\n<p><strong>Charles Kittel, Introduction to Solid State Physics.<\/strong><\/p>\n<p><strong>N. W. Ashcroft, D. Mermin, Solid State Physics, Holt, Rinehart and Winston,<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Glossary:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Phonon;<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">It is <\/span>ksnown<span style=\"text-align: initial;font-size: 1em\"> as the quanta of sound which is analogous to quanta of light \u2013 photon.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Optical Branch:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">The upper branch of the allowed wave propagation (in the long wavelength limit) in a crystal structure.<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">In <\/span>this<span style=\"text-align: initial;font-size: 1em\"> the two atoms vibrate <\/span>opposite<span style=\"text-align: initial;font-size: 1em\"> to each other with lighter mass having greater amplitude.<\/span><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n","protected":false},"author":3,"menu_order":11,"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-162","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\/162","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":4,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters\/162\/revisions"}],"predecessor-version":[{"id":241,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters\/162\/revisions\/241"}],"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\/162\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/media?parent=162"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapter-type?post=162"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/contributor?post=162"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/license?post=162"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}