{"id":368,"date":"2018-12-03T11:16:16","date_gmt":"2018-12-03T11:16:16","guid":{"rendered":"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=368"},"modified":"2018-12-05T06:33:07","modified_gmt":"2018-12-05T06:33:07","slug":"motion-of-electrons-in-periodic-potential","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp06\/chapter\/motion-of-electrons-in-periodic-potential\/","title":{"rendered":"Motion of electrons in periodic potential"},"content":{"raw":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/H_V2xP923iQ\" 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\r\n<div>\r\n\r\n<strong>TABLE OF CONTENTS<\/strong>\r\n\r\n&nbsp;\r\n\r\n23.1 Motion of electrons in a periodic potential.\r\n\r\n23.2 Significance of Translational periodicity.\r\n\r\n23. 3 Periodic Boundary Conditions and Wave vector.\r\n\r\n23.4 Bloch Theorem.\r\n\r\n23.5 Kronig-Penney Model.\r\n\r\n23.6 Brillouin Zones.\r\n\r\n23.7 Significance of Brillouin Zones.\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>LEARNING OBJECTIVES<\/strong>\r\n<ul>\r\n \t<li>In this module motion of electrons in a periodic potential is described and discussed.<\/li>\r\n \t<li>Significance of translational periodicity in a crystalline solid is explained.<\/li>\r\n \t<li>Somerfield's model where electron is assumed to travel in a potential which is constant everywhere\u00a0 inside<\/li>\r\n \t<li>the metal is described.<\/li>\r\n \t<li>The concept of De Broglie wave or Somerfield waves or Bloch waves are given<\/li>\r\n \t<li>The kinetic energy of the electron is shown to be proportional to the square of wave number and the<\/li>\r\n \t<li>model leads to a relationship between energy and wave vector for a free electron that is parabolic.<\/li>\r\n \t<li>The Bloch theorem is explained.<\/li>\r\n \t<li>Kronig Penney model describing the motion of electrons in a periodic potential is explained.<\/li>\r\n \t<li>The model is shown to lead to energy versus wave vector for electron in a periodic potential as being<\/li>\r\n \t<li>discontinuous at k = +n\u03c0\/a and k = -n\u03c0\/a.<\/li>\r\n \t<li>The concept of Brillioun zones is given and their significance in crystallography is discussed.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>23.1<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0 <strong>Motion of electrons in a perio dic potential<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Atoms in a crystalline solid are positioned at regular intervals because of wh ich it exhib its periodic ity in the arrangement of atoms in all d imensions. On account of this periodicity, it exh ibits various types symmet ries and as such the atoms or molecules composing the solid are always related by some symmet ry elements. Crystalline solids exh ibit various types of physical properties which are structure sensitive. So, the physical behaviour of crystalline solids cannot be explained unless we have a sound background of crystallography. The symmetry elements have already been discussed in the previous sections. It is essential to describe the significance of symmetry in crystalline solids in so far as its relevance to the development of theory in the understanding of their physical properties is concerned. One of the physical properties is that of conduction in crystalline solids. Metals have been considered as being composed of atoms with their valence electrons roaming freely inside the solid. The classical theory considered these electrons as moving freely inside the solids like molecules of a gas in an enclosure. Drude and Lorentz theory assumed that these electrons behave like classical part icles, move in a constant potential field andas such applied Maxwell Bolt zmann statistics tounderstand and exp lain the behaviour of solids. However, it was modified by Sommerfeldby applying quantum mechanical concepts which could explain some of the physical properties of solids. The free elect ron theory, however, could not explain as to why some solids are good conductors like metals, so me semiconductors or insulators. It is in situations of this type that crystallography in general and period icity of crystals in particu lar beco mes significant in the develop ment o f solid state theory.<\/p>\r\n&nbsp;\r\n\r\n<strong>23.2\u00a0 Sig nificance of translati onal peri o dici ty.<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">We may select some function say f(r) in the direct lattice space. It is well known that atoms exhib it translational periodicity in a crystalline solid and because of this fact, this function isalso expected to be periodic in the sense that it should be the same atphysically equivalent points in the crystal latt ice . The period icity of the funct ion f(r) could be expressed as:<\/p>\r\n&nbsp;\r\n\r\nf( r + R ) = f ( r )\u2026\u2026\u2026\u2026\u2026\u2026.23.1\r\n\r\n&nbsp;\r\n\r\nfor all points located at r . The vecto r R by virtue o f t ranslat ional period icity is represented by:\r\n\r\n&nbsp;\r\n\r\nR = l a + m b + n c,\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">where l, m, n are integers and a,b,c are translational period icit ies along x, y and z directions . Let us start with a one-dimensional case for the sake of convenience and simplicity of mathematical treat ment and represent equat ion 23.1 fo r one-d imensional case as:<\/p>\r\n&nbsp;\r\n\r\nf( x + a ) = f(x)\u2026...\u2026\u2026\u2026\u2026 23.2\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Since the function of equation 23.2 is periodic, it may be exp ressed in the form o f Fourier series.<\/p>\r\nThe exp ression of equation 23.2, if expressed in the Fourier series form can be written as:\r\n\r\n&nbsp;\r\n\r\nf(x) = \u2211 Fn2\u03c0inx\/a\u2026\u2026\u2026........23.3\r\n\r\nn\r\n\r\n&nbsp;\r\n\r\nwhere n is an integer, a is period of the function.\r\n\r\n&nbsp;\r\n\r\nWe may exp ress equation 23.3 as:\r\n\r\n&nbsp;\r\n\r\nf(x) = \u2211 A\u0444ei\u0444x , where\u00a0 \u2502 \u0444n \u2502\u00a0\u00a0 = n.2\u03c0\/a \u2026\u2026\u2026\u2026\u2026\u2026\u202623.4\r\n\r\n\u0444\r\n\r\n&nbsp;\r\n\r\nThe \u0444\u2019s represent the reciprocal lattice vectors for the linear lattice in one-dimension:\r\n\r\n&nbsp;\r\n\r\nThe Fourier coefficients A\u0444in equation 23.4 are given by:\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\nA\u0444 = 1\/a\u222b f(x) e\u00a0 \u2500i\u0444xd x\u2026\u2026\u2026\u2026\u2026\u2026\u2026.23.5\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nIn terms of the Fouriercoefficient the L.H.S. of equation 23.2may be expressed as:\r\n\r\n&nbsp;\r\n\r\nf (x+a ) = \u2211 A\u0444ei\u0444(x+a) \u2026\u2026\u2026\u2026\u2026\u2026\u2026.23.6\r\n\r\n<img class=\"aligncenter size-full wp-image-372\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-23.png\" alt=\"\" width=\"492\" height=\"158\" \/>\r\n\r\nIt shows that the function f(x) = \u2211 A\u0444e<sup>i\u0444x<\/sup>is a function in the d irect lattice space.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">We may extend the expressions 23.2, 23.3&amp;23.4 to the more general case by considering lattice periodicity along the three orthogonal axes which may be represented by \u2502a\u2502 ,\u2502 b \u2502 and \u2502 c \u2502. In that case, we have on using equation 23.1,<\/p>\r\n&nbsp;\r\n\r\nf ( r + R )\u00a0 = f ( r + l a + m b + n c\u00a0 ) = f ( r )\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202623.8\r\n\r\n&nbsp;\r\n\r\nAs in the case of one-dimensions, the exp ression for three-dimensions\u00a0 may be written as:\r\n\r\n&nbsp;\r\n\r\nf\u00a0 (r ) = \u2211 B (\u0444 1, \u0444 2 , \u0444 3 ) ei (\u0444 1 x + \u0444 2 y + \u0444 3 z )\u00a0 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.23.9\r\n\r\n&nbsp;\r\n\r\n\u0444 1 \u0444 2 \u0444 3\r\n\r\n&nbsp;\r\n\r\nEquation 23.9 may also be written in the form:\r\n\r\n&nbsp;\r\n\r\nf(r) = \u2211 B \u0444 ei A r\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202623.10\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nwhere \u0444\u00a0 is the recip rocal\u00a0\u00a0 lattice vector with co mponents\u00a0 \u0444 1\u00a0 , \u0444 2\u00a0 and\u00a0 \u0444 3 along the x,y and z axes.\r\n\r\nAs in the one-dimensional case above, here also:\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-373\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-24.png\" alt=\"\" width=\"442\" height=\"284\" \/>\r\n\r\nWhich is in agreement with equation no. 23.1\r\n\r\n&nbsp;\r\n\r\n<strong>23.3\u00a0\u00a0\u00a0 Perio dic Boun dary Con di tions &amp; Wave\u00a0 Vector .<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In the So mmerfeld\u2019s model, the free electrons are assumed\u00a0 to travel in a potential wh ich is constant everywhere inside the metal. The electron is considered bound inside a cubical crystal of side say \u2018l\u2019.The potential inside the crystal is assumed to be zero and the potential outside the crystal is infinity. The motion of the electrons is calculated by solving the Schrodinger equation :<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">2\u03a8+( 8\u03c02 m\/h2 )E \u03a8 = 0<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Instead of assuming that the electron wave function vanishes at x=0 and at x= l, it is considered more realistic to assume that the electron wave function is periodic with a period l. That means there must be integral number o f wavelengths of the electron wave (de Broglie wave) in the distance \u2018l\u2019 of the crystal. It can be exp ressed mathematically as:<\/p>\r\n&nbsp;\r\n\r\n\u03a8\u00a0 [ ( x+l), y , z ] = \u03a8 (x,y,z) ,\r\n\r\n&nbsp;\r\n\r\nwith similar exp ressions for y and z coo rd inates.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">This is popularlyknown as periodic boundarycondition, also known as cyclic boundary condition .It is an important concept in solid state physics.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Considering the fact that atoms in a crystal are arranged in an orderly manner consequently leading to periodicity of the lattice, one can express the periodic boundary condition for the electron wave function as:<\/p>\r\n&nbsp;\r\n<p style=\"text-align: center\">\u03a8 ( x + a ) =\u00a0 \u03a8 (x ) ,<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\nwhere a is the lattice constant \u03a8 ( x + a ) = \u03a8 (x ) ,\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The relation between the wave vector k and the energy E is given by : K = 2\u03c0 (2mE) 1\/2 h The De Broglie waves, also known as Sommerfeld waves or Bloch waves in so far as motion of electrons in a crystalline solids is concerned, are expressed in terms of the wave vector. We know<\/p>\r\n&nbsp;\r\n\r\n.E =p 2 \/2m, Since p = h\/\u03bb, E = h 2 \/2\u03bb 2m\r\n\r\n&nbsp;\r\n\r\nSubstituting this expression for E in the above equation, we get\r\n\r\n&nbsp;\r\n\r\nK = 2\u03c0\/\u03bb\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Therefore, k is a vector whose magnitude is 2\u03c0\/\u03bband whosedirection is the same as the direction of propagation of electron wave or the De Broglie wave (So mmerfeld waves or Bloch waves). It is also called wave vector or wave number. The kinetic energy of the electron is proportional to the square of the wave number and the relationship between E and k for a free electron is parabolic as shown in the figure 23.1<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-374\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-25.png\" alt=\"\" width=\"362\" height=\"285\" \/>\r\n<p style=\"text-align: center\">Figure 23.1: Plot of E Vs. wave vector k fo r a free electron<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">23.4 Bloch Theorem<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">What has been discussed above is regarding general functions with the periodicity of the lattice. It is general case of any such function. We shall now consider here a specific function of periodic potential. We know that the atoms orions in a crystalline solid are positioned in regular periodic arrays and assuch theelectrostatic potential exhib ited by them is also periodic which may be exp ressed as:<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">V ( r + R ) =\u00a0 V ( r ) \u2026\u2026\u2026\u2026\u2026\u202623.11<\/span><\/p>\r\n\r\n<div>\r\n\r\n&nbsp;\r\n\r\nfor all Bravais lattice\u00a0 vectors r .\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">This potential assumes significance when we deal with an understanding of motion ofelectrons in a crystalline solid. The periodicity of the electrostatic potential V is of the order of 1\u01fawhich is almost\u00a0<span style=\"text-align: initial;text-indent: 1em;font-size: 1em\">crystal is free fro m any imperfection wh ich is rather an ideal concept. The real crystalline solid is not free fro m any imperfection because of which the periodicity breaks down in the region around imperfection. The imperfections influence the physical properties of crystalline solids. However, it is necessary to assume that we are dealing with an ideal crystalline solid in order to be able to develop the theory regarding physical properties of solids in the background of symmetries and periodicity in crystals. It is in this ideal crystal that we shall consider the motion of electrons as represented by one-electron potential V(r) which fo llo ws the expression no. 23.11 for an ideal crystal, i.e .,<\/span><\/p>\r\n&nbsp;\r\n\r\nV (r + R) = V (r)\r\n\r\n&nbsp;\r\n\r\nThe motion of electrons is described by Schrodinger equation:\r\n\r\n&nbsp;\r\n\r\nH\u03a8 = (- \u01272\/2m 2\u00a0 +V (r)\u00a0 ) \u03a8 = E \u03a8 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202623.12\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Where H stands for Hamiltonian, m is the effective mass of the electron;\u0127= h\/2\u03c0 (where h represents Planck\u2019s constant , \u03a8 is the wave function , 2is the operator \u22022\/\u2202r2 and E is the total energy). In order to describe the motion of the electrons in crystalline solid , equation 23.12 is required to be solved with the periodic potential of an ideal crystal lattice V (r + R ) = V ( r ). The solution of this equation is given by :<\/p>\r\n&nbsp;\r\n\r\n\u03a8k( r )\u00a0 = Vk (r) eik. r ,\u2026\u2026\u2026\u2026\u2026\u2026.23.13\r\n\r\n&nbsp;\r\n\r\nWhere\u00a0\u00a0\u00a0\u00a0 Vk( r + R ) = Vk(r) \u2026\u2026\u2026\u2026.23.14\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Functions of the type of equation23.13 are called Bloch functions,k the wave vector (= \u03bb\/2\u03c0). It was Bloch who in 1928 showed that the solutions are of the form as given by equation 23.13.<\/p>\r\n&nbsp;\r\n\r\nFro m equations 23.13, 23.14, we have:\r\n\r\n&nbsp;\r\n\r\n\u03a8k( r + R ) = e ik.R . \u03a8k\u00a0 (r)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.23.15\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Bloch\u00a0 function\u00a0 can\u00a0 also\u00a0 be exp ressed\u00a0 by the following\u00a0 equation by selecting<\/p>\r\n&nbsp;\r\n\r\nEigen states of H in such a way\u00a0 that corresponding to each \u03a8 there is a wave vector k so that:\r\n\r\n&nbsp;\r\n\r\n\u03a8\u00a0 ( r + R ) = e ik.R \u03a8 (r) \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..23.16 for all possible Bravais lattice vectors R .\r\n\r\n&nbsp;\r\n\r\nReferring\u00a0 to Schrodinger equation no. 23.12, its solution is given by:\r\n\r\n&nbsp;\r\n\r\n<span style=\"font-size: 1em;text-align: initial\">\u03a8( r )\u00a0 = f ( r ) u ( r )\u2026\u2026\u2026\u2026\u2026\u2026\u2026.23.17<\/span>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">where u ( r ) is a periodic function . Since the potential u(r) is periodic, all the quantities associated with the electron are also periodic. The quantity \u2502 \u03a8 (r)\u25022has also to be periodic provided f(r) satisfies the condition :<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n\u2502\u00a0 f ( r + R ) \u25022\u00a0 = \u2502 f ( r )\u25022 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202623.18\r\n\r\n&nbsp;\r\n\r\nThe solution of this equation is expressed to be of this form:\r\n\r\n&nbsp;\r\n\r\n\u03a8\u00a0\u00a0 (r) = u (r) e ikr \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..23.19 This equation represents Bloch theorem.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">If one considers the periodicity of atoms in the crystalline solid, it is obvious that the electron moves in the field of periodic potential. For example, figure 23.2 indicates the qualitative form of the electrostatic potential energy of conduction electrons in the field of positive ion cores of monatomic linear lattice. The ion cores left out by conduction electrons are positively charged. The potential energy of an electron in the field of these positive ion cores is negative.<\/p>\r\n<img class=\"aligncenter size-full wp-image-375\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-26.png\" alt=\"\" width=\"403\" height=\"233\" \/>\r\n<p style=\"text-align: center\">Figure 23.2: Variat ion\u00a0 of electrostatic potential energy of electrons\u00a0 in the field of ions mo no ato mic linear\u00a0 lattice<\/p>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong>23.5<\/strong>\u00a0<strong>Kronig -Penney Mo del<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Here, we consider the motion of electrons in a simple one-dimensional periodic potential and use Bloch theorem to understand about the allowed and forbidden electronic energy bands in a onedimensional lattice. For this purpose we shall consider Kronig-Penney model in which the potential energy variation is assumed to be of the form as shown in figure 23.3.It shows an ideal periodic square well potential used in Kronig-Penney Model to illustrate the general characteristics of the quantum behaviour of the electrons in periodic lattices.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">The essential features of the behaviour of electrons in a periodic potential may be studied by considering the model first discussed by Kronig and Penney (1931) and so is named after them. In this model, it is assumed that the potential energy of an electron moving in onedimensional perfect crystal lattice is represented in the form of a periodic array of rectangular wells as indicated in the figure; the period of potential being (a+b).In regions ranging 0 &lt; x &lt; a potential energy is assumed to be zero whereas in the regions ranging \u2500 b &lt; x &lt; 0 , the potential energy is V0 . Each of the potential energy may be treated as a rough approximation for the potential in the vicinity of an atom. With this infinite one-dimensional rectangular well potential, it is possible to obtain an exact solution of the Schrodinger equation.<\/p>\r\n<img class=\"aligncenter size-full wp-image-376\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-27.png\" alt=\"\" width=\"421\" height=\"247\" \/>\r\n<p style=\"text-align: center\">Figure 23.3: A one-dimensional periodic potential of periodicity (a+b) for the Kronig-Penney\r\nmodel<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Although it is somewhat an idealized periodic potential wh ich in reality is just a crude approximation towhat in reality exists in the real crystal, it is nevertheless very useful because it serves the purpose of illustrating the most explicit way with the help of which several important characteristic features of the quantum behaviour of electrons in a periodic lattice can be described.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-377\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-28.png\" alt=\"\" width=\"389\" height=\"65\" \/>\r\n<p style=\"text-align: center\">And\u00a0\u00a0\u00a0\u00a0 V (x + a + b) = V (x)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202623.21<\/p>\r\n&nbsp;\r\n\r\nSo, the periodicity of the potential as represented\u00a0 by the figure is (a + b). The Schrodinger\u00a0 equations\u00a0for the two regions are given by:\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-378\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-29.png\" alt=\"\" width=\"475\" height=\"69\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Here, it may be assumed that the energy E of the electrons is smaller than V0 i.e., E &lt;V0 . We also put two real quantities \u03b1 and \u03b2 by:<\/span><img class=\"aligncenter size-full wp-image-379\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-30.png\" alt=\"\" width=\"477\" height=\"58\" \/>\r\n\r\n&nbsp;\r\n\r\nE &lt; V0, because \u03b1 and \u03b2 are real quant ities.\r\n\r\nIf E &gt;V0 then \u03b2 will be imaginary.\r\n\r\n&nbsp;\r\n\r\nThe Schrodinger equation 23.22 and 23.23 may be written as :\r\n\r\n<img class=\"aligncenter size-full wp-image-380\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-31.png\" alt=\"\" width=\"467\" height=\"55\" \/>\r\n\r\n<\/div>\r\n<div><\/div>\r\n<div style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">From Bloch theorem we know that the solution of wave equation for a period ic potential will be of the form o f a plane wave modulated with the periodicity of the lattice o f the form:<\/span><\/div>\r\n<div style=\"text-align: justify\">\r\n\r\n<span style=\"text-align: initial;font-size: 1em\"><img class=\"aligncenter size-full wp-image-381\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-32.png\" alt=\"\" width=\"413\" height=\"41\" \/><\/span>Where uk(x) is the periodic function in x with the period (a+b).\r\n\r\n<\/div>\r\n<img class=\"aligncenter size-full wp-image-382\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-33.png\" alt=\"\" width=\"433\" height=\"54\" \/>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-383\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-34.png\" alt=\"\" width=\"662\" height=\"97\" \/>\r\n\r\n&nbsp;\r\n\r\nSubstituting fro m equations 23.27 and 23.29 in equation no. 23.25 we get :\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-384\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-35.png\" alt=\"\" width=\"528\" height=\"36\" \/>\r\n\r\nFor the sake of convenience let us drop the subscripts and write the above expression as:\r\n\r\n<img class=\"aligncenter size-full wp-image-385\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-36.png\" alt=\"\" width=\"486\" height=\"47\" \/>\r\n\r\nSimilarly, substituting equations 23.27 and 23.29 in equation 23.26, we have:\r\n\r\n<img class=\"aligncenter size-full wp-image-386\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-37.png\" alt=\"\" width=\"649\" height=\"85\" \/>\r\n\r\nRepresenting the value of u(x) in the interval 0&lt; x &lt; a by u1(x) and in the interval -b &lt; x &lt; 0 by u2(x), equation 23.30 and 23.31 may be written as :\r\n\r\n<img class=\"aligncenter size-full wp-image-387\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-38.png\" alt=\"\" width=\"536\" height=\"60\" \/>\r\n\r\n&nbsp;\r\n\r\nThe differential equations 23.32 and 23.33 are easily solved by the standard procedure as follo ws:\r\n\r\nIn solving these differential equations, it is assumed that the solution is of the follo wing form\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 :\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-388\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-39.png\" alt=\"\" width=\"748\" height=\"561\" \/><img class=\"aligncenter size-full wp-image-389\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-40.png\" alt=\"\" width=\"721\" height=\"377\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The first two conditions under equations 23.37(a) and 23.37(b) are imposed because of the requirements of continuity of the wave functions. Because of continuity at x=0, the two wave functions and their derivatives must have the same value at x=0. The other two conditions viz., 23.38(a) &amp; 23.38(b) are required because of the periodicity of uk(x). Thus, the function uk(x) should have the same value at x=a and x= \u2500 b.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Imposition of the first two conditions 23.37(a) &amp; 23.37 (b) in equations 23.34 and 23.36 leads to:<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-390\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-41.png\" alt=\"\" width=\"585\" height=\"67\" \/>\r\n\r\n&nbsp;\r\n\r\nThe derivation of equation no.23.40 is given as under:\r\n\r\nFro m equation 23.35, we have:\r\n\r\n<img class=\"aligncenter size-full wp-image-391\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-42.png\" alt=\"\" width=\"407\" height=\"44\" \/>\r\n\r\nSimilarily, from equation 23.36, we have:\r\n\r\n<img class=\"aligncenter size-full wp-image-392\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-43.png\" alt=\"\" width=\"433\" height=\"81\" \/>\r\n\r\nEquation 23.40 is thus derived here. Moving forward we have:\r\n\r\nImposition of the next two conditions 23.38 (a &amp; b ) in equations 23.34 and 23.36, we have:\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-393\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-44.png\" alt=\"\" width=\"559\" height=\"86\" \/>\r\n\r\n<img class=\"aligncenter size-full wp-image-394\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-45.png\" alt=\"\" width=\"675\" height=\"23\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Thus application of 23.37(a ,b) on equation no. 23.34 and 23.36 leads to four linear homogeneous equations in the constants A, B , C and D. The coefficients A, B, C and D can thus be determined as the solution of a set of these four simu ltaneous linear homogeneous equations in these quantities. There is no solution other than A =B =C =D =0, unless the determinant of the coefficients vanishes. That means these four equations 23.39 to 23.42 have a solution only if the determinant of the coefficients of A,B,C, and D van ishes.<\/p>\r\n\r\n<\/div>\r\n<div><\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-395\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-46.png\" alt=\"\" width=\"673\" height=\"119\" \/>\r\n\r\nOn expansion of this determinant, one can show after rigorous and straight forward algebra that equation 23.43 can be expressed as:\r\n\r\n<img class=\"aligncenter size-full wp-image-396\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-47.png\" alt=\"\" width=\"571\" height=\"43\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In order to obtain a more convenient equation, Kronig and Penney considers the case for which the potential barriers become delta functions i.e., V0 \u2192 \u221e &amp; b \u2192 0 but the product<\/p>\r\n&nbsp;\r\n\r\nV<sub>0b<\/sub> or \u03b2<sub>2b<\/sub> remains finite. Such a function is known as delta function. Under these circu mstances equation 23.44 reduces to:\r\n\r\n<img class=\"aligncenter size-full wp-image-397\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-48.png\" alt=\"\" width=\"403\" height=\"49\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">It is because when b \u2192 0 sinh \u03b2b \u2192 \u03b2b and cosh \u03b2b \u2192 1 and because E &lt; V0 and V0 \u2192 \u221e so E is very small, \u03b12 may be neglected but \u03b2 cannot be neglected on account of involvement of the term (V0 \u2500 E).<\/p>\r\n&nbsp;\r\n\r\nEquation 23.45 can be further exp ressed as:\r\n\r\n<img class=\"aligncenter size-full wp-image-398\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-49.png\" alt=\"\" width=\"209\" height=\"54\" \/>\r\n\r\nLet us put \u03b22ab\/2 = P in the above equation, we get:\r\n\r\n<img class=\"aligncenter size-full wp-image-399\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-50.png\" alt=\"\" width=\"221\" height=\"57\" \/>\r\n\r\nThe quantity P is defined by the exp ression:\r\n\r\n<img class=\"aligncenter size-full wp-image-400\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-51.png\" alt=\"\" width=\"221\" height=\"54\" \/>\r\n\r\npotential barrier. The physical significance of this quantity is that if P is increased, the area of the potential barrier is increased and so a given electron is bound more strongly to a particular potential well. When P \u2192 0, the potential barrier beco mes very weak which in other words means that electrons are free electrons\r\n\r\n&nbsp;\r\n\r\nWhen P \u2192 0 i.e., for free electrons, expression at 23.46 can be written as:\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-401\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-52.png\" alt=\"\" width=\"660\" height=\"151\" \/>\r\n\r\nThe expression at 23.47 resembles the result as was obtained by considering the So mmerfeld model of\r\n\r\nmetals.\r\n\r\nA\u00a0\u00a0 plot of the function (Psin\u03b1a\/\u03b1a + cos\u03b1a) when drawn against \u03b1a for P = 3\u03c0\/2 appears as shown in figure 23.4.\r\n\r\n<\/div>\r\n<img class=\"aligncenter size-full wp-image-402\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-53.png\" alt=\"\" width=\"736\" height=\"301\" \/>\r\n<div>\r\n<p style=\"text-align: center\">Figure 23.4: Variation of function (P s in \u03b1a\/ \u03b1a + cos\u03b1a) with \u03b1a for P = 3\u03c0\/2.The allowed values of energy are given by those ranges of \u03b1 =( 2mE\/\u01272)1\/2 for wh ich the function lies between \u2500 1 and\u00a0 + 1<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">We know that \u03b1 = (2mE\/\u01272)1\/2\u00a0 which suggests\u00a0 that \u03b12\u00a0 is proportional to energy E. The abscissa is a\u00a0measure of energy and in order to calculate the energy represented by the function at a point, the value of \u03b1a corresponding to that point is determined. The values of \u03b1a which satisfy equation 23.46are found out by drawing a line parallel to \u03b1a axis at a distance coska from it. If ka is continuously varied from 0 to \u03c0, i.e., coska from +1 to \u2500 1, one is able to find all possible values of \u03b1a and hence that of energy. It is important to realize that the right hand side of equation 23.46 can accept only values between \u2500 1 and + 1 (i.e., ka continuously varying from 0 to \u03c0 and coska from +1 to \u2500 1) as indicated by the horizontal lines in the figure. Therefore, the condition as imposed by equation 23.46 can be met only for values of \u03b1a for which the L.H.S. lies between \u00b11.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;text-indent: 1em;font-size: 1em\">In figure 23.4, the function: f(\u03b1a) = Psin \u03b1a\/\u03b1a + cos \u03b1a is plotted against \u03b1a. Obviously, for k to be real \u2502f (\u03b1a) \u2502should be less than 1. Those values of E for which \u2502 (\u03b1a) \u2502&gt; 1 will be forbidden.Those values of E for which f(\u03b1a) \u2264 1 will correspond to allowed values of energy. It may also be noted that if k is replaced by k+2n\u03c0\/a (where n= \u00b11, \u00b12, \u00b13\u2026\u2026), the right hand side of equation 23.46 will remain the same. That means one cannot uniquely determine the value of k. It is usually restricted to the domain:<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">which is known as the first Brillouin zone.<\/p>\r\n<p style=\"text-align: justify\">\u2500 \u03c0\/a\u2264 k\u2264 + \u03c0\/a<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">The E \u2013 k diagram showing the allowed and forbidden energy bands are shown in figures 23.5 &amp; 23.6. At the zone boundaries ( k=\u00b1 \u03c0\/a ) the Bloch wave function satisfies the Bragg condition and the group velocity [=( 1\/\u0127)dE\/dk] is zero which corresponds to a standing wave.<\/p>\r\n\r\n<\/div>\r\n<div><\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-403\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-54.png\" alt=\"\" width=\"714\" height=\"449\" \/>\r\n<p style=\"text-align: center\">Figure 23.5: E Vs. wave vector K for electron in a periodic potential<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-404\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-55.png\" alt=\"\" width=\"709\" height=\"438\" \/>\r\n<p style=\"text-align: center\">Figure 23.6: Full curve of E Vs. wave vector k showing d iscontin uit ies at k=\u00b1n \u03c0\/ a (where n=1, 2, 3\u2026)<\/p>\r\n&nbsp;\r\n\r\nLet us analyse the details of what may be concluded fro m the figure drawn on the basis of equation 23.46.\r\n\r\n&nbsp;\r\n\r\n(i) There are infinite nu mbers of allo wed energy bands separated by intervals in which there are no energy\r\n\r\nlevels. Such regions are known as forbidden regions.\r\n\r\n&nbsp;\r\n\r\nThe boundaries of the allowed ranges of \u03b1a correspond to the values for which coska=\u00b11\r\n\r\n&nbsp;\r\n\r\n<span style=\"font-size: 1em;text-align: initial\">That happens if ka = n\u03c0, or k = n\u03c0\/a, where n = 1, 2, 3\u2026\u2026.<\/span>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">(ii) The first term in the expression for f (\u03b1a) i.e ., P.sin\u03b1a\/\u03b1a decreases on an average with increas ing \u03b1a. It suggests that the width of the allowed energy bands increases and the forbidden regions get narrower as \u03b1a increases or as energy increases.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">(iii) As P increases, the width of the allowed band decreases. Increase in the value of P means the increase in \u201cbinding energy\u201d of the electrons. In the extreme case, when P is infinite, the allowed energy bands get infinitely narrower, consequently the energy spectrum becomes line spectrum and are independent of k.<\/p>\r\nIf P\u2192 \u221e, the allowed energy ranges of \u03b1a reduce to points given by:\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-405\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-56.png\" alt=\"\" width=\"478\" height=\"131\" \/>\r\n\r\nIt shows that E is independent of k.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The energy levels in this case are discrete and the electron is completely bound. It is trapped within the potential wells and moves only in one cell of width \u2018a \u2019.<\/p>\r\n&nbsp;\r\n\r\nFro m the discussion on analysis of equation 23.46, we co me to the conclusion that in the one-dimensional problem for the limit ing case, the spectrum of energy values that are permitted is found to be consisting of continuous regions separated by finite intervals. By vary ing the quantity\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">P (i.e., V<sub>0<\/sub> b) fro m zero to infinity, we move fro m the case of free electrons to that of bound electrons and so are able to study the changes in the allowed and forbidden ranges of energy and the wave function.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">We may also use equation 23.46 to study the variation of energy E with the wave number k.The variation is as shown in figures23.5 &amp; 23.6.Fro m these figures one arrives at the follo wing conclusions:<\/p>\r\n&nbsp;\r\n\r\n(i) The energy spectrum of the electrons consists of a number of allowed energy bands separated by forbidden regions.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">(ii) The width of the allowed energy bands increases with increase of energy values.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">(iii) If V0 is made zero, the E versus k curve becomes a continuous parabola, the same as that of free electrons (see figure 23.1)<\/p>\r\n&nbsp;\r\n\r\n(iv) There are d iscontinuities in the E versus k curve which occur for k = n\u03c0\/a, where n = \u00b11, \u00b12, \u00b13.The range of allowed values of k between \u2500 \u03c0\/a and + \u03c0\/a constitutes the first Brillouin zone. The\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">range of values of k between \u2500\u03c0\/a and \u2500 2\u03c0\/a and between + \u03c0\/a and + 2\u03c0\/a constitutes the second Brillouin zone.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">The energy E is represented as function of k for P = 3\u03c0\/2.Analyzing the curve of E against k further regarding the discontinuities in the E versus k at k = n\u03c0\/a, where n= 1, 2, 3\u2026\u2026.The k-values define the boundaries of the first (I), second (II), third (III) and so on Brillouin zones. Figure 23.5 shown here provides only half of the E-k curve; the first zone in fact extends from k= \u2500 \u03c0\/a to + \u03c0\/a.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Similarly, the second Brillouin zone is composed of two parts; the one extending from +\u03c0\/a to + 2\u03c0\/a as shown in the figure and the other part extending fro m \u2500 \u03c0\/a to \u2500 2\u03c0\/a. Figure 23.6 represents the full E-k curve. We may call each position of the curve as a band. The curve of E-k has the follo wing characteristics:<\/p>\r\n&nbsp;\r\n\r\n1.\u00a0 The curves are horizontal at the top and bottom.\r\n\r\n2.\u00a0 The curves are parabolic near the top and the bottom having curvatures in opposite direct ions.\r\n\r\n3.\u00a0 d2E\/dk2is positive in the lower portion of the band and negative in the upper port ion of the band.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Let us consider a linear lattice of lattice constant a. According to free electron theory E versus wave vector k is continuous as shown in figure 23.1. The low energy part of the band structure is shown in figure 23.7 for electrons which are nearly free but with an energy gap at wave vector k= \u00b1\u03c0\/a resulting into creation of forbidden gap as a result of our assumption that electron is moving in a periodic potential (unlike in the case of free electron theory where electron is assumed to be moving\u00a0<span style=\"font-size: 1em;text-align: initial\">in a constant potential field). The curve of E versus wave vector k for an electron in a monato mic linear lattice with lattice constant a appears as shown in figure 23.7. The energy gap Eg is associated with the first Bragg reflection at k = \u00b1\u03c0\/a.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-406\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-57.png\" alt=\"\" width=\"618\" height=\"350\" \/>\r\n<p style=\"text-align: center\">Figu re 23.7: E as a function\u00a0 of wave vector k fo r an electron\u00a0 in a monatomic\u00a0 linear\u00a0 lattice o f latt ice constant\u00a0 a. The forb idden\u00a0 band associated with the energy gap Egis shown The Band\u00a0 gap is associated\u00a0 with the first Bragg\u00a0 reflection at k=\u00b1\u03c0\/a .<\/p>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nThe Bragg condition in terms of the reciprocal lattice imp lies that:\r\n\r\n<img class=\"aligncenter size-full wp-image-407\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-58.png\" alt=\"\" width=\"278\" height=\"39\" \/>\r\n\r\nfor diffraction of a wave vector k, if taken in one dimension becomes on expanding the dot product and simp lify ing\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nwhere G is 2\u03c0 times a vector fro m the origin to a lattice point of the reciprocal lattice and k is a vector of magnitude 2\u03c0\/\u03bb along the direction of the incident x=ray beam. Equation 23.49 is the vector form of the Bragg equation.\r\n\r\n&nbsp;\r\n\r\nIn one dimensions equation 23.49 beco mes:\r\n\r\n&nbsp;\r\n\r\nk = \u00b1 \u00bd G =\u00b1 n\u03c0\/a, \u2026\u2026\u2026\u2026\u2026\u2026\u202623.50 Where G = \u00b12n\u03c0\/a is the reciprocal lattice vector.\r\n\r\n&nbsp;\r\n\r\nThe first reflection\u00a0\u00a0\u00a0 and\u00a0 the first energy gap\u00a0 results\u00a0 at k = \u00b1\u03c0\/a;\u00a0 other energy gaps\u00a0 result\u00a0 for other\r\n\r\n&nbsp;\r\n\r\nvalues of integer in equation 23.50.\r\n\r\nThe reflection\u00a0 at the wave vector k =\u00b1\u03c0\/a takes\u00a0 place because the wave reflected\u00a0 fro m one atom in\r\n\r\n&nbsp;\r\n\r\nthe linear lattice\u00a0 meets\u00a0 constructive\u00a0 interference\u00a0 with\u00a0 the wave fro m a nearest neighbour\u00a0 atom. The\r\n\r\n&nbsp;\r\n\r\nphase difference between the two reflected waves is \u00b12\u03c0 for these two values of k. The region in the k-space between \u2500\u03c0\/a and +\u03c0\/a is called the first Brillouin zone .\r\n\r\n&nbsp;\r\n\r\n<strong>23.6<\/strong>\u00a0 \u00a0<strong>Brillouin Zo nes<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Kronig\u2013Penney model has shown that the energy discontinuities in the monatomic one dimensional lattice result when the wave vector satisfies the relation :<\/p>\r\n&nbsp;\r\n\r\nk = n\u03c0\/a , where n is an integer which may be positive\u00a0 or negative.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">In the one-dimensional monatomic lattice a line representing the value of k is divided up by the energy discontinuities into segments of length \u03c0\/a as shown in figure 23.8<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-409\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-59.png\" alt=\"\" width=\"600\" height=\"152\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: center\">Figure 23.8: Line rep resent ing the value of k for one -d imension al monatomic lattice is sho wn divided into segments of length \u00b1n\u03c0\/a (n=1, 2,3\u2026)<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">These line segments are known as Brillouin zones. The segment \u2500 \u03c0\/a &lt; k &lt; \u03c0\/a represents first Brillouin zone; the two segments \u2500 2\u03c0\/a &lt; k &lt; \u2500 \u03c0\/a and \u03c0\/a &lt; k &lt; 2\u03c0\/a form the second Brillouin zone and so on. Brillouin zones are characteristic of a particular crystal structure and as such each crystal structure form its own characteristic Brillouin zones.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Let us take the examp le of two dimensional simple square lattice as shown in figure 23.9. The first Brillouin zone for this lattice will be a square ABCD whose boundaries are defined by:<\/p>\r\n&nbsp;\r\n\r\nkx = \u00b1\u03c0\/a;\u00a0\u00a0 ky\u00a0 =\u00b1 \u03c0\/a\r\n\r\n&nbsp;\r\n\r\nSimilarly, the boundaries of the second Brillouin zone are defined by :\r\n\r\n\u00b1kx =\u00b1ky\u00a0 = \u03c0\/a as represented by the diagram EFGH.\r\n\r\n<img class=\"aligncenter size-full wp-image-410\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-60.png\" alt=\"\" width=\"576\" height=\"326\" \/>\r\n\r\nFigu re 23.9: Brillouin zones in two d imens ional simple square latt ice\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">One can apply the same principle for the three d imensional crystal structures. For a simp le cub ic lattice, the first Brillouin zone is a cube of edge 2\u03c0\/a.<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">23.7<\/strong><span style=\"text-align: initial;font-size: 1em\">\u00a0<\/span><strong style=\"text-align: initial;font-size: 1em\">Significance of Brillouin Zones<\/strong>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">(i) Looking at the curve shown in figure 23.6 one finds that electron energy increases continuously from zero until the value of k reaches \u03c0\/a. Thereafter, it gets stopped as if it meets an obstacle or a wall at that instant and consequently gets reflected.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Treating\u00a0 electrons\u00a0 as waves and the propagation\u00a0 of electrons\u00a0 through a crystal as analogous\u00a0 to the propagation\u00a0 of electromagnetic\u00a0 waves, we arrive at a very important\u00a0 conclusion.\u00a0 We know that x-ray will\u00a0 suffer<\/p>\r\n&nbsp;\r\n\r\nreflection\u00a0 if\u00a0 incident normal to a set of planes\u00a0 of interplanar\u00a0 spacing \u2018a\u2019 provided\u00a0 this equation is satisfied:\r\n\r\n&nbsp;\r\n\r\nn\u03bb = 2a sin90\u2070\u00a0 ( for x-ray is at right angles to the planes)\r\n\r\n&nbsp;\r\n\r\nn\u03bb = 2a\r\n\r\n&nbsp;\r\n\r\nFor electron k = n\u03c0\/a and k = 2\u03c0\/\u03bb\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">So,\u00a0 2\u03c0\/\u03bb = n\u03c0\/a, or n\u03bb = 2a, wh ich is the same condition as for Bragg reflect ion. Thus, we may conclude\u00a0 that electron\u00a0\u00a0 suffers\u00a0\u00a0\u00a0 Bragg\u00a0\u00a0\u00a0 reflection\u00a0\u00a0\u00a0 when\u00a0\u00a0\u00a0 n\u00a0\u00a0 =\u00b11,\u00b12,\u00b13\u00a0\u00a0\u00a0 \u2026\u2026 corresponding to\u00a0 first, second, third\u00a0 orderreflections\u00a0 and\u00a0 so\u00a0 on. The\u00a0 zones\u00a0\u00a0 between\u00a0 the values\u00a0\u00a0 of k =\u00a0 \u2500 \u03c0\/a\u00a0 and\u00a0 + \u03c0\/a constitutes the first Brillouin zone and so on.<\/p>\r\n&nbsp;\r\n\r\n(ii) The zone boundaries represent the maximu m energies that the electron can have without developing any discontinuity.\r\n\r\n&nbsp;\r\n\r\n(iii) The energy gap at the zero boundary is called the Forbidden zone or band; electrons cannot have those energies.\r\n\r\n&nbsp;\r\n\r\n(iv) Considering velocity of electrons in a periodic potential (given vg = 2\u03c0.d\u03bd\/dk; E = h\u03bd and so vg\r\n\r\n&nbsp;\r\n\r\n=2\u03c0\/h.d E\/dk)\r\n\r\n&nbsp;\r\n\r\nSlope d E\/dk = 0\u00a0\u00a0\u00a0\u00a0 when k = 0,\r\n\r\n&nbsp;\r\n\r\n&amp;\u00a0\u00a0 d E\/ dk = 0\u00a0\u00a0\u00a0\u00a0\u00a0 when k=\u00a0\u00a0\u00a0 \u03c0\/a.\r\n\r\n&nbsp;\r\n\r\nTherefore, the velocity of electron is zero both at the bottom and at the top of the first Brillou in zone or band\r\n\r\nAt intermediate regions in the zone the electron velocity reaches the free electron velocity h k\/ 2\u03c0 m.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">(v) Let us now consider the motion\u00a0 of the electron\u00a0 in the first Brillouin zone,\u00a0 under the continued application\u00a0 of\u00a0a force F (either due to electric field or any other agency). Velocity of the electron at k = 0 will\r\nincrease but when it approaches the value of k close to + \u03c0\/a its velocity begins to decrease and at k = + \u03c0\/a\r\nthe velocity becomes zero, indicating that the electron wave packet suffers a Bragg reflection and begins to\r\ntravel in a direction opposite to the applied force. The propagation in the negative direction continues until\r\nit becomes equal to \u2500 \u03c0\/a. Once again the electron suffers Bragg reflection and the forward propagation\r\nstarts until value of k reaches +\u03c0\/a. In other words, the electron is shunted back and forth in the first\r\nBrillouin zone under the application of a constant unidirectional force. It means that a stationary electron\r\nwave is set up in the first Brillouin zone instead of a travelling wave<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;text-indent: 1em;font-size: 1em\">vi) An electron remain ing in one zone or band cannot cross over to another zone by continuous application\u00a0 of force.Two consecutive Brillouin zones are separated by a forbidden energy gap. So, unless and until the electronin the first zone absorbs an energy equal to that of the forbidden gap in a single dose, the electron\r\ncannot cross over to the next Brillouin zone irrespective of the duration of the applied force<\/span><\/p>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">vii) If the electrons continue to move back and forth in their respective Brillouin zones there is no change in the sum of the mo menta of all the electrons and, as a result, no net current is conducted by the crystal. It, obviously, means that the material is an insulator. Now suppose the forbidden gap is small enough or if impurity atoms provide some localized electronic states within the forb idden energy gap, one will find a semi conductor behaviour. The third situation is that there is no forbidden energy gap. In that case there is no restriction for increase in the value of k and so electric conduction can take place. The material in this case is a good conductor.<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>SUMMAR Y<\/strong>\r\n<ul>\r\n \t<li>We have discussed motion of electrons, both in a constant potential field (as assumed by Somerfield) as well\u00a0<span style=\"font-size: 1em;text-align: initial\">as in a period ic potential field as assumed in Kronig Penney model.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Significance of t ranslational period icity in a crystalline solid is explained.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">The concept of De Broglie wave or So merfield waves or Bloch waves are given.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">It is shown that Somerfield's model of motion of a free electrons in a constant potential field leads to a relation between energy and wave vector for a free electron that is parabolic.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Bloch theorem is explained.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">It is shown that if electron is assumed to travel in a periodic potential as proposed by Kronig Penney model it leads to energy versus wave vector tha shows discontinuities at<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">The concept of brillioun zones and their significance in crystallography is exp lained<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>you can view video on Motionof electronsinperiodicpotential<\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/H_V2xP923iQ\" 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\r\n<strong>References.<\/strong>\r\n\r\n&nbsp;\r\n\r\n1.Verma,A.R. &amp;Srivastava,O.N.\u201d Crystallography for Solid State Physics\u201d,Wiley Eastern Ltd., N.Delhi,1982.\r\n\r\n2. Seitz ,F.: \u201c Modern Theory of Solids\u201d, McGraw-Hill, N.Y.,1940.\r\n\r\n3. Omar,M.A. \u201c Elementary Solid State Physics\u201d, Addison-Wesley, Readin,1975.\r\n\r\n4. Kittel, C.:\u201dIntroduction to Solid State Physics\u201d,Wiley-Eastern Ltd.,N.Delhi,1985.\r\n\r\n5. Wannier, G.H.: \u201c Elements of Solid State Theory\u201d,Cambridge Univ. Press,1959.\r\n\r\n6. Ziman,J.M.: \u201c Principles of the Theory of Solids\u201d, Cambridge Univ. Press,1964.\r\n\r\n7. Clark,H.: \u201c Solid State Physics\u201d,Macmillan,London,1968.\r\n\r\n8. Pillai,S.O. : \u201cSolid State Physics\u201d, New Age Int.(P) Ltd.Publishers, N.Delhi,1997.\r\n\r\n&nbsp;\r\n\r\n<strong>Suggested Reading.<\/strong>\r\n<ol>\r\n \t<li>Madelung,O :\u201d Introduction to Solid State Theory\u201d, Springer-Verlag,N.Y.,1978.<\/li>\r\n \t<li>Ghatak, A.K. &amp;Kothari,L.S.: \u201c Introduction to Lattice Dynamics\u201d,Addison Wesley,Reading,1971.<\/li>\r\n \t<li>Ashcroft,N.W. &amp;Mermin,N.D.: \u201cSolid State Physics\u201d,New York: Holt,Rinchart and Winston,1976.<\/li>\r\n<\/ol>\r\n&nbsp;","rendered":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/H_V2xP923iQ\" target=\"_blank\" rel=\"noopener\"><img decoding=\"async\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a><br \/>\n<\/span><\/div>\n<div>\n<p><strong>TABLE OF CONTENTS<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>23.1 Motion of electrons in a periodic potential.<\/p>\n<p>23.2 Significance of Translational periodicity.<\/p>\n<p>23. 3 Periodic Boundary Conditions and Wave vector.<\/p>\n<p>23.4 Bloch Theorem.<\/p>\n<p>23.5 Kronig-Penney Model.<\/p>\n<p>23.6 Brillouin Zones.<\/p>\n<p>23.7 Significance of Brillouin Zones.<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>LEARNING OBJECTIVES<\/strong><\/p>\n<ul>\n<li>In this module motion of electrons in a periodic potential is described and discussed.<\/li>\n<li>Significance of translational periodicity in a crystalline solid is explained.<\/li>\n<li>Somerfield&#8217;s model where electron is assumed to travel in a potential which is constant everywhere\u00a0 inside<\/li>\n<li>the metal is described.<\/li>\n<li>The concept of De Broglie wave or Somerfield waves or Bloch waves are given<\/li>\n<li>The kinetic energy of the electron is shown to be proportional to the square of wave number and the<\/li>\n<li>model leads to a relationship between energy and wave vector for a free electron that is parabolic.<\/li>\n<li>The Bloch theorem is explained.<\/li>\n<li>Kronig Penney model describing the motion of electrons in a periodic potential is explained.<\/li>\n<li>The model is shown to lead to energy versus wave vector for electron in a periodic potential as being<\/li>\n<li>discontinuous at k = +n\u03c0\/a and k = -n\u03c0\/a.<\/li>\n<li>The concept of Brillioun zones is given and their significance in crystallography is discussed.<\/li>\n<\/ul>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>23.1<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0 <strong>Motion of electrons in a perio dic potential<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Atoms in a crystalline solid are positioned at regular intervals because of wh ich it exhib its periodic ity in the arrangement of atoms in all d imensions. On account of this periodicity, it exh ibits various types symmet ries and as such the atoms or molecules composing the solid are always related by some symmet ry elements. Crystalline solids exh ibit various types of physical properties which are structure sensitive. So, the physical behaviour of crystalline solids cannot be explained unless we have a sound background of crystallography. The symmetry elements have already been discussed in the previous sections. It is essential to describe the significance of symmetry in crystalline solids in so far as its relevance to the development of theory in the understanding of their physical properties is concerned. One of the physical properties is that of conduction in crystalline solids. Metals have been considered as being composed of atoms with their valence electrons roaming freely inside the solid. The classical theory considered these electrons as moving freely inside the solids like molecules of a gas in an enclosure. Drude and Lorentz theory assumed that these electrons behave like classical part icles, move in a constant potential field andas such applied Maxwell Bolt zmann statistics tounderstand and exp lain the behaviour of solids. However, it was modified by Sommerfeldby applying quantum mechanical concepts which could explain some of the physical properties of solids. The free elect ron theory, however, could not explain as to why some solids are good conductors like metals, so me semiconductors or insulators. It is in situations of this type that crystallography in general and period icity of crystals in particu lar beco mes significant in the develop ment o f solid state theory.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>23.2\u00a0 Sig nificance of translati onal peri o dici ty.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">We may select some function say f(r) in the direct lattice space. It is well known that atoms exhib it translational periodicity in a crystalline solid and because of this fact, this function isalso expected to be periodic in the sense that it should be the same atphysically equivalent points in the crystal latt ice . The period icity of the funct ion f(r) could be expressed as:<\/p>\n<p>&nbsp;<\/p>\n<p>f( r + R ) = f ( r )\u2026\u2026\u2026\u2026\u2026\u2026.23.1<\/p>\n<p>&nbsp;<\/p>\n<p>for all points located at r . The vecto r R by virtue o f t ranslat ional period icity is represented by:<\/p>\n<p>&nbsp;<\/p>\n<p>R = l a + m b + n c,<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">where l, m, n are integers and a,b,c are translational period icit ies along x, y and z directions . Let us start with a one-dimensional case for the sake of convenience and simplicity of mathematical treat ment and represent equat ion 23.1 fo r one-d imensional case as:<\/p>\n<p>&nbsp;<\/p>\n<p>f( x + a ) = f(x)\u2026&#8230;\u2026\u2026\u2026\u2026 23.2<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Since the function of equation 23.2 is periodic, it may be exp ressed in the form o f Fourier series.<\/p>\n<p>The exp ression of equation 23.2, if expressed in the Fourier series form can be written as:<\/p>\n<p>&nbsp;<\/p>\n<p>f(x) = \u2211 Fn2\u03c0inx\/a\u2026\u2026\u2026&#8230;&#8230;..23.3<\/p>\n<p>n<\/p>\n<p>&nbsp;<\/p>\n<p>where n is an integer, a is period of the function.<\/p>\n<p>&nbsp;<\/p>\n<p>We may exp ress equation 23.3 as:<\/p>\n<p>&nbsp;<\/p>\n<p>f(x) = \u2211 A\u0444ei\u0444x , where\u00a0 \u2502 \u0444n \u2502\u00a0\u00a0 = n.2\u03c0\/a \u2026\u2026\u2026\u2026\u2026\u2026\u202623.4<\/p>\n<p>\u0444<\/p>\n<p>&nbsp;<\/p>\n<p>The \u0444\u2019s represent the reciprocal lattice vectors for the linear lattice in one-dimension:<\/p>\n<p>&nbsp;<\/p>\n<p>The Fourier coefficients A\u0444in equation 23.4 are given by:<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>A\u0444 = 1\/a\u222b f(x) e\u00a0 \u2500i\u0444xd x\u2026\u2026\u2026\u2026\u2026\u2026\u2026.23.5<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>In terms of the Fouriercoefficient the L.H.S. of equation 23.2may be expressed as:<\/p>\n<p>&nbsp;<\/p>\n<p>f (x+a ) = \u2211 A\u0444ei\u0444(x+a) \u2026\u2026\u2026\u2026\u2026\u2026\u2026.23.6<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-372\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-23.png\" alt=\"\" width=\"492\" height=\"158\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-23.png 492w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-23-300x96.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-23-65x21.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-23-225x72.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-23-350x112.png 350w\" sizes=\"auto, (max-width: 492px) 100vw, 492px\" \/><\/p>\n<p>It shows that the function f(x) = \u2211 A\u0444e<sup>i\u0444x<\/sup>is a function in the d irect lattice space.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">We may extend the expressions 23.2, 23.3&amp;23.4 to the more general case by considering lattice periodicity along the three orthogonal axes which may be represented by \u2502a\u2502 ,\u2502 b \u2502 and \u2502 c \u2502. In that case, we have on using equation 23.1,<\/p>\n<p>&nbsp;<\/p>\n<p>f ( r + R )\u00a0 = f ( r + l a + m b + n c\u00a0 ) = f ( r )\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202623.8<\/p>\n<p>&nbsp;<\/p>\n<p>As in the case of one-dimensions, the exp ression for three-dimensions\u00a0 may be written as:<\/p>\n<p>&nbsp;<\/p>\n<p>f\u00a0 (r ) = \u2211 B (\u0444 1, \u0444 2 , \u0444 3 ) ei (\u0444 1 x + \u0444 2 y + \u0444 3 z )\u00a0 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.23.9<\/p>\n<p>&nbsp;<\/p>\n<p>\u0444 1 \u0444 2 \u0444 3<\/p>\n<p>&nbsp;<\/p>\n<p>Equation 23.9 may also be written in the form:<\/p>\n<p>&nbsp;<\/p>\n<p>f(r) = \u2211 B \u0444 ei A r\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202623.10<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>where \u0444\u00a0 is the recip rocal\u00a0\u00a0 lattice vector with co mponents\u00a0 \u0444 1\u00a0 , \u0444 2\u00a0 and\u00a0 \u0444 3 along the x,y and z axes.<\/p>\n<p>As in the one-dimensional case above, here also:<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-373\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-24.png\" alt=\"\" width=\"442\" height=\"284\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-24.png 442w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-24-300x193.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-24-65x42.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-24-225x145.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-24-350x225.png 350w\" sizes=\"auto, (max-width: 442px) 100vw, 442px\" \/><\/p>\n<p>Which is in agreement with equation no. 23.1<\/p>\n<p>&nbsp;<\/p>\n<p><strong>23.3\u00a0\u00a0\u00a0 Perio dic Boun dary Con di tions &amp; Wave\u00a0 Vector .<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In the So mmerfeld\u2019s model, the free electrons are assumed\u00a0 to travel in a potential wh ich is constant everywhere inside the metal. The electron is considered bound inside a cubical crystal of side say \u2018l\u2019.The potential inside the crystal is assumed to be zero and the potential outside the crystal is infinity. The motion of the electrons is calculated by solving the Schrodinger equation :<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">2\u03a8+( 8\u03c02 m\/h2 )E \u03a8 = 0<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Instead of assuming that the electron wave function vanishes at x=0 and at x= l, it is considered more realistic to assume that the electron wave function is periodic with a period l. That means there must be integral number o f wavelengths of the electron wave (de Broglie wave) in the distance \u2018l\u2019 of the crystal. It can be exp ressed mathematically as:<\/p>\n<p>&nbsp;<\/p>\n<p>\u03a8\u00a0 [ ( x+l), y , z ] = \u03a8 (x,y,z) ,<\/p>\n<p>&nbsp;<\/p>\n<p>with similar exp ressions for y and z coo rd inates.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">This is popularlyknown as periodic boundarycondition, also known as cyclic boundary condition .It is an important concept in solid state physics.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Considering the fact that atoms in a crystal are arranged in an orderly manner consequently leading to periodicity of the lattice, one can express the periodic boundary condition for the electron wave function as:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">\u03a8 ( x + a ) =\u00a0 \u03a8 (x ) ,<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>where a is the lattice constant \u03a8 ( x + a ) = \u03a8 (x ) ,<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The relation between the wave vector k and the energy E is given by : K = 2\u03c0 (2mE) 1\/2 h The De Broglie waves, also known as Sommerfeld waves or Bloch waves in so far as motion of electrons in a crystalline solids is concerned, are expressed in terms of the wave vector. We know<\/p>\n<p>&nbsp;<\/p>\n<p>.E =p 2 \/2m, Since p = h\/\u03bb, E = h 2 \/2\u03bb 2m<\/p>\n<p>&nbsp;<\/p>\n<p>Substituting this expression for E in the above equation, we get<\/p>\n<p>&nbsp;<\/p>\n<p>K = 2\u03c0\/\u03bb<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Therefore, k is a vector whose magnitude is 2\u03c0\/\u03bband whosedirection is the same as the direction of propagation of electron wave or the De Broglie wave (So mmerfeld waves or Bloch waves). It is also called wave vector or wave number. The kinetic energy of the electron is proportional to the square of the wave number and the relationship between E and k for a free electron is parabolic as shown in the figure 23.1<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-374\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-25.png\" alt=\"\" width=\"362\" height=\"285\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-25.png 362w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-25-300x236.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-25-65x51.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-25-225x177.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-25-350x276.png 350w\" sizes=\"auto, (max-width: 362px) 100vw, 362px\" \/><\/p>\n<p style=\"text-align: center\">Figure 23.1: Plot of E Vs. wave vector k fo r a free electron<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">23.4 Bloch Theorem<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">What has been discussed above is regarding general functions with the periodicity of the lattice. It is general case of any such function. We shall now consider here a specific function of periodic potential. We know that the atoms orions in a crystalline solid are positioned in regular periodic arrays and assuch theelectrostatic potential exhib ited by them is also periodic which may be exp ressed as:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">V ( r + R ) =\u00a0 V ( r ) \u2026\u2026\u2026\u2026\u2026\u202623.11<\/span><\/p>\n<div>\n<p>&nbsp;<\/p>\n<p>for all Bravais lattice\u00a0 vectors r .<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">This potential assumes significance when we deal with an understanding of motion ofelectrons in a crystalline solid. The periodicity of the electrostatic potential V is of the order of 1\u01fawhich is almost\u00a0<span style=\"text-align: initial;text-indent: 1em;font-size: 1em\">crystal is free fro m any imperfection wh ich is rather an ideal concept. The real crystalline solid is not free fro m any imperfection because of which the periodicity breaks down in the region around imperfection. The imperfections influence the physical properties of crystalline solids. However, it is necessary to assume that we are dealing with an ideal crystalline solid in order to be able to develop the theory regarding physical properties of solids in the background of symmetries and periodicity in crystals. It is in this ideal crystal that we shall consider the motion of electrons as represented by one-electron potential V(r) which fo llo ws the expression no. 23.11 for an ideal crystal, i.e .,<\/span><\/p>\n<p>&nbsp;<\/p>\n<p>V (r + R) = V (r)<\/p>\n<p>&nbsp;<\/p>\n<p>The motion of electrons is described by Schrodinger equation:<\/p>\n<p>&nbsp;<\/p>\n<p>H\u03a8 = (- \u01272\/2m 2\u00a0 +V (r)\u00a0 ) \u03a8 = E \u03a8 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202623.12<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Where H stands for Hamiltonian, m is the effective mass of the electron;\u0127= h\/2\u03c0 (where h represents Planck\u2019s constant , \u03a8 is the wave function , 2is the operator \u22022\/\u2202r2 and E is the total energy). In order to describe the motion of the electrons in crystalline solid , equation 23.12 is required to be solved with the periodic potential of an ideal crystal lattice V (r + R ) = V ( r ). The solution of this equation is given by :<\/p>\n<p>&nbsp;<\/p>\n<p>\u03a8k( r )\u00a0 = Vk (r) eik. r ,\u2026\u2026\u2026\u2026\u2026\u2026.23.13<\/p>\n<p>&nbsp;<\/p>\n<p>Where\u00a0\u00a0\u00a0\u00a0 Vk( r + R ) = Vk(r) \u2026\u2026\u2026\u2026.23.14<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Functions of the type of equation23.13 are called Bloch functions,k the wave vector (= \u03bb\/2\u03c0). It was Bloch who in 1928 showed that the solutions are of the form as given by equation 23.13.<\/p>\n<p>&nbsp;<\/p>\n<p>Fro m equations 23.13, 23.14, we have:<\/p>\n<p>&nbsp;<\/p>\n<p>\u03a8k( r + R ) = e ik.R . \u03a8k\u00a0 (r)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.23.15<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Bloch\u00a0 function\u00a0 can\u00a0 also\u00a0 be exp ressed\u00a0 by the following\u00a0 equation by selecting<\/p>\n<p>&nbsp;<\/p>\n<p>Eigen states of H in such a way\u00a0 that corresponding to each \u03a8 there is a wave vector k so that:<\/p>\n<p>&nbsp;<\/p>\n<p>\u03a8\u00a0 ( r + R ) = e ik.R \u03a8 (r) \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..23.16 for all possible Bravais lattice vectors R .<\/p>\n<p>&nbsp;<\/p>\n<p>Referring\u00a0 to Schrodinger equation no. 23.12, its solution is given by:<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-size: 1em;text-align: initial\">\u03a8( r )\u00a0 = f ( r ) u ( r )\u2026\u2026\u2026\u2026\u2026\u2026\u2026.23.17<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">where u ( r ) is a periodic function . Since the potential u(r) is periodic, all the quantities associated with the electron are also periodic. The quantity \u2502 \u03a8 (r)\u25022has also to be periodic provided f(r) satisfies the condition :<\/span><\/p>\n<\/div>\n<div>\n<p>\u2502\u00a0 f ( r + R ) \u25022\u00a0 = \u2502 f ( r )\u25022 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202623.18<\/p>\n<p>&nbsp;<\/p>\n<p>The solution of this equation is expressed to be of this form:<\/p>\n<p>&nbsp;<\/p>\n<p>\u03a8\u00a0\u00a0 (r) = u (r) e ikr \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..23.19 This equation represents Bloch theorem.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">If one considers the periodicity of atoms in the crystalline solid, it is obvious that the electron moves in the field of periodic potential. For example, figure 23.2 indicates the qualitative form of the electrostatic potential energy of conduction electrons in the field of positive ion cores of monatomic linear lattice. The ion cores left out by conduction electrons are positively charged. The potential energy of an electron in the field of these positive ion cores is negative.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-375\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-26.png\" alt=\"\" width=\"403\" height=\"233\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-26.png 403w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-26-300x173.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-26-65x38.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-26-225x130.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-26-350x202.png 350w\" sizes=\"auto, (max-width: 403px) 100vw, 403px\" \/><\/p>\n<p style=\"text-align: center\">Figure 23.2: Variat ion\u00a0 of electrostatic potential energy of electrons\u00a0 in the field of ions mo no ato mic linear\u00a0 lattice<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>23.5<\/strong>\u00a0<strong>Kronig -Penney Mo del<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Here, we consider the motion of electrons in a simple one-dimensional periodic potential and use Bloch theorem to understand about the allowed and forbidden electronic energy bands in a onedimensional lattice. For this purpose we shall consider Kronig-Penney model in which the potential energy variation is assumed to be of the form as shown in figure 23.3.It shows an ideal periodic square well potential used in Kronig-Penney Model to illustrate the general characteristics of the quantum behaviour of the electrons in periodic lattices.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The essential features of the behaviour of electrons in a periodic potential may be studied by considering the model first discussed by Kronig and Penney (1931) and so is named after them. In this model, it is assumed that the potential energy of an electron moving in onedimensional perfect crystal lattice is represented in the form of a periodic array of rectangular wells as indicated in the figure; the period of potential being (a+b).In regions ranging 0 &lt; x &lt; a potential energy is assumed to be zero whereas in the regions ranging \u2500 b &lt; x &lt; 0 , the potential energy is V0 . Each of the potential energy may be treated as a rough approximation for the potential in the vicinity of an atom. With this infinite one-dimensional rectangular well potential, it is possible to obtain an exact solution of the Schrodinger equation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-376\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-27.png\" alt=\"\" width=\"421\" height=\"247\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-27.png 421w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-27-300x176.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-27-65x38.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-27-225x132.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-27-350x205.png 350w\" sizes=\"auto, (max-width: 421px) 100vw, 421px\" \/><\/p>\n<p style=\"text-align: center\">Figure 23.3: A one-dimensional periodic potential of periodicity (a+b) for the Kronig-Penney<br \/>\nmodel<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Although it is somewhat an idealized periodic potential wh ich in reality is just a crude approximation towhat in reality exists in the real crystal, it is nevertheless very useful because it serves the purpose of illustrating the most explicit way with the help of which several important characteristic features of the quantum behaviour of electrons in a periodic lattice can be described.<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-377\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-28.png\" alt=\"\" width=\"389\" height=\"65\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-28.png 389w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-28-300x50.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-28-65x11.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-28-225x38.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-28-350x58.png 350w\" sizes=\"auto, (max-width: 389px) 100vw, 389px\" \/><\/p>\n<p style=\"text-align: center\">And\u00a0\u00a0\u00a0\u00a0 V (x + a + b) = V (x)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202623.21<\/p>\n<p>&nbsp;<\/p>\n<p>So, the periodicity of the potential as represented\u00a0 by the figure is (a + b). The Schrodinger\u00a0 equations\u00a0for the two regions are given by:<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-378\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-29.png\" alt=\"\" width=\"475\" height=\"69\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-29.png 475w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-29-300x44.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-29-65x9.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-29-225x33.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-29-350x51.png 350w\" sizes=\"auto, (max-width: 475px) 100vw, 475px\" \/><\/p>\n<\/div>\n<div>\n<p><span style=\"text-align: initial;font-size: 1em\">Here, it may be assumed that the energy E of the electrons is smaller than V0 i.e., E &lt;V0 . We also put two real quantities \u03b1 and \u03b2 by:<\/span><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-379\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-30.png\" alt=\"\" width=\"477\" height=\"58\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-30.png 477w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-30-300x36.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-30-65x8.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-30-225x27.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-30-350x43.png 350w\" sizes=\"auto, (max-width: 477px) 100vw, 477px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>E &lt; V0, because \u03b1 and \u03b2 are real quant ities.<\/p>\n<p>If E &gt;V0 then \u03b2 will be imaginary.<\/p>\n<p>&nbsp;<\/p>\n<p>The Schrodinger equation 23.22 and 23.23 may be written as :<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-380\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-31.png\" alt=\"\" width=\"467\" height=\"55\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-31.png 467w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-31-300x35.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-31-65x8.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-31-225x26.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-31-350x41.png 350w\" sizes=\"auto, (max-width: 467px) 100vw, 467px\" \/><\/p>\n<\/div>\n<div><\/div>\n<div style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">From Bloch theorem we know that the solution of wave equation for a period ic potential will be of the form o f a plane wave modulated with the periodicity of the lattice o f the form:<\/span><\/div>\n<div style=\"text-align: justify\">\n<p><span style=\"text-align: initial;font-size: 1em\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-381\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-32.png\" alt=\"\" width=\"413\" height=\"41\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-32.png 413w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-32-300x30.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-32-65x6.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-32-225x22.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-32-350x35.png 350w\" sizes=\"auto, (max-width: 413px) 100vw, 413px\" \/><\/span>Where uk(x) is the periodic function in x with the period (a+b).<\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-382\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-33.png\" alt=\"\" width=\"433\" height=\"54\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-33.png 433w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-33-300x37.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-33-65x8.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-33-225x28.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-33-350x44.png 350w\" sizes=\"auto, (max-width: 433px) 100vw, 433px\" \/><\/p>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-383\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-34.png\" alt=\"\" width=\"662\" height=\"97\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-34.png 662w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-34-300x44.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-34-65x10.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-34-225x33.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-34-350x51.png 350w\" sizes=\"auto, (max-width: 662px) 100vw, 662px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Substituting fro m equations 23.27 and 23.29 in equation no. 23.25 we get :<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-384\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-35.png\" alt=\"\" width=\"528\" height=\"36\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-35.png 528w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-35-300x20.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-35-65x4.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-35-225x15.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-35-350x24.png 350w\" sizes=\"auto, (max-width: 528px) 100vw, 528px\" \/><\/p>\n<p>For the sake of convenience let us drop the subscripts and write the above expression as:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-385\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-36.png\" alt=\"\" width=\"486\" height=\"47\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-36.png 486w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-36-300x29.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-36-65x6.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-36-225x22.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-36-350x34.png 350w\" sizes=\"auto, (max-width: 486px) 100vw, 486px\" \/><\/p>\n<p>Similarly, substituting equations 23.27 and 23.29 in equation 23.26, we have:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-386\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-37.png\" alt=\"\" width=\"649\" height=\"85\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-37.png 649w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-37-300x39.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-37-65x9.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-37-225x29.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-37-350x46.png 350w\" sizes=\"auto, (max-width: 649px) 100vw, 649px\" \/><\/p>\n<p>Representing the value of u(x) in the interval 0&lt; x &lt; a by u1(x) and in the interval -b &lt; x &lt; 0 by u2(x), equation 23.30 and 23.31 may be written as :<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-387\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-38.png\" alt=\"\" width=\"536\" height=\"60\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-38.png 536w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-38-300x34.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-38-65x7.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-38-225x25.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-38-350x39.png 350w\" sizes=\"auto, (max-width: 536px) 100vw, 536px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>The differential equations 23.32 and 23.33 are easily solved by the standard procedure as follo ws:<\/p>\n<p>In solving these differential equations, it is assumed that the solution is of the follo wing form\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 :<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-388\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-39.png\" alt=\"\" width=\"748\" height=\"561\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-39.png 748w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-39-300x225.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-39-65x49.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-39-225x169.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-39-350x263.png 350w\" sizes=\"auto, (max-width: 748px) 100vw, 748px\" \/><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-389\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-40.png\" alt=\"\" width=\"721\" height=\"377\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-40.png 721w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-40-300x157.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-40-65x34.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-40-225x118.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-40-350x183.png 350w\" sizes=\"auto, (max-width: 721px) 100vw, 721px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The first two conditions under equations 23.37(a) and 23.37(b) are imposed because of the requirements of continuity of the wave functions. Because of continuity at x=0, the two wave functions and their derivatives must have the same value at x=0. The other two conditions viz., 23.38(a) &amp; 23.38(b) are required because of the periodicity of uk(x). Thus, the function uk(x) should have the same value at x=a and x= \u2500 b.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Imposition of the first two conditions 23.37(a) &amp; 23.37 (b) in equations 23.34 and 23.36 leads to:<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-390\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-41.png\" alt=\"\" width=\"585\" height=\"67\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-41.png 585w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-41-300x34.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-41-65x7.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-41-225x26.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-41-350x40.png 350w\" sizes=\"auto, (max-width: 585px) 100vw, 585px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>The derivation of equation no.23.40 is given as under:<\/p>\n<p>Fro m equation 23.35, we have:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-391\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-42.png\" alt=\"\" width=\"407\" height=\"44\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-42.png 407w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-42-300x32.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-42-65x7.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-42-225x24.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-42-350x38.png 350w\" sizes=\"auto, (max-width: 407px) 100vw, 407px\" \/><\/p>\n<p>Similarily, from equation 23.36, we have:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-392\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-43.png\" alt=\"\" width=\"433\" height=\"81\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-43.png 433w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-43-300x56.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-43-65x12.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-43-225x42.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-43-350x65.png 350w\" sizes=\"auto, (max-width: 433px) 100vw, 433px\" \/><\/p>\n<p>Equation 23.40 is thus derived here. Moving forward we have:<\/p>\n<p>Imposition of the next two conditions 23.38 (a &amp; b ) in equations 23.34 and 23.36, we have:<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-393\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-44.png\" alt=\"\" width=\"559\" height=\"86\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-44.png 559w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-44-300x46.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-44-65x10.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-44-225x35.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-44-350x54.png 350w\" sizes=\"auto, (max-width: 559px) 100vw, 559px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-394\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-45.png\" alt=\"\" width=\"675\" height=\"23\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-45.png 675w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-45-300x10.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-45-65x2.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-45-225x8.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-45-350x12.png 350w\" sizes=\"auto, (max-width: 675px) 100vw, 675px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Thus application of 23.37(a ,b) on equation no. 23.34 and 23.36 leads to four linear homogeneous equations in the constants A, B , C and D. The coefficients A, B, C and D can thus be determined as the solution of a set of these four simu ltaneous linear homogeneous equations in these quantities. There is no solution other than A =B =C =D =0, unless the determinant of the coefficients vanishes. That means these four equations 23.39 to 23.42 have a solution only if the determinant of the coefficients of A,B,C, and D van ishes.<\/p>\n<\/div>\n<div><\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-395\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-46.png\" alt=\"\" width=\"673\" height=\"119\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-46.png 673w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-46-300x53.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-46-65x11.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-46-225x40.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-46-350x62.png 350w\" sizes=\"auto, (max-width: 673px) 100vw, 673px\" \/><\/p>\n<p>On expansion of this determinant, one can show after rigorous and straight forward algebra that equation 23.43 can be expressed as:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-396\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-47.png\" alt=\"\" width=\"571\" height=\"43\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-47.png 571w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-47-300x23.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-47-65x5.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-47-225x17.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-47-350x26.png 350w\" sizes=\"auto, (max-width: 571px) 100vw, 571px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In order to obtain a more convenient equation, Kronig and Penney considers the case for which the potential barriers become delta functions i.e., V0 \u2192 \u221e &amp; b \u2192 0 but the product<\/p>\n<p>&nbsp;<\/p>\n<p>V<sub>0b<\/sub> or \u03b2<sub>2b<\/sub> remains finite. Such a function is known as delta function. Under these circu mstances equation 23.44 reduces to:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-397\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-48.png\" alt=\"\" width=\"403\" height=\"49\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-48.png 403w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-48-300x36.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-48-65x8.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-48-225x27.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-48-350x43.png 350w\" sizes=\"auto, (max-width: 403px) 100vw, 403px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">It is because when b \u2192 0 sinh \u03b2b \u2192 \u03b2b and cosh \u03b2b \u2192 1 and because E &lt; V0 and V0 \u2192 \u221e so E is very small, \u03b12 may be neglected but \u03b2 cannot be neglected on account of involvement of the term (V0 \u2500 E).<\/p>\n<p>&nbsp;<\/p>\n<p>Equation 23.45 can be further exp ressed as:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-398\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-49.png\" alt=\"\" width=\"209\" height=\"54\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-49.png 209w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-49-65x17.png 65w\" sizes=\"auto, (max-width: 209px) 100vw, 209px\" \/><\/p>\n<p>Let us put \u03b22ab\/2 = P in the above equation, we get:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-399\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-50.png\" alt=\"\" width=\"221\" height=\"57\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-50.png 221w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-50-65x17.png 65w\" sizes=\"auto, (max-width: 221px) 100vw, 221px\" \/><\/p>\n<p>The quantity P is defined by the exp ression:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-400\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-51.png\" alt=\"\" width=\"221\" height=\"54\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-51.png 221w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-51-65x16.png 65w\" sizes=\"auto, (max-width: 221px) 100vw, 221px\" \/><\/p>\n<p>potential barrier. The physical significance of this quantity is that if P is increased, the area of the potential barrier is increased and so a given electron is bound more strongly to a particular potential well. When P \u2192 0, the potential barrier beco mes very weak which in other words means that electrons are free electrons<\/p>\n<p>&nbsp;<\/p>\n<p>When P \u2192 0 i.e., for free electrons, expression at 23.46 can be written as:<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-401\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-52.png\" alt=\"\" width=\"660\" height=\"151\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-52.png 660w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-52-300x69.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-52-65x15.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-52-225x51.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-52-350x80.png 350w\" sizes=\"auto, (max-width: 660px) 100vw, 660px\" \/><\/p>\n<p>The expression at 23.47 resembles the result as was obtained by considering the So mmerfeld model of<\/p>\n<p>metals.<\/p>\n<p>A\u00a0\u00a0 plot of the function (Psin\u03b1a\/\u03b1a + cos\u03b1a) when drawn against \u03b1a for P = 3\u03c0\/2 appears as shown in figure 23.4.<\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-402\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-53.png\" alt=\"\" width=\"736\" height=\"301\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-53.png 736w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-53-300x123.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-53-65x27.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-53-225x92.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-53-350x143.png 350w\" sizes=\"auto, (max-width: 736px) 100vw, 736px\" \/><\/p>\n<div>\n<p style=\"text-align: center\">Figure 23.4: Variation of function (P s in \u03b1a\/ \u03b1a + cos\u03b1a) with \u03b1a for P = 3\u03c0\/2.The allowed values of energy are given by those ranges of \u03b1 =( 2mE\/\u01272)1\/2 for wh ich the function lies between \u2500 1 and\u00a0 + 1<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">We know that \u03b1 = (2mE\/\u01272)1\/2\u00a0 which suggests\u00a0 that \u03b12\u00a0 is proportional to energy E. The abscissa is a\u00a0measure of energy and in order to calculate the energy represented by the function at a point, the value of \u03b1a corresponding to that point is determined. The values of \u03b1a which satisfy equation 23.46are found out by drawing a line parallel to \u03b1a axis at a distance coska from it. If ka is continuously varied from 0 to \u03c0, i.e., coska from +1 to \u2500 1, one is able to find all possible values of \u03b1a and hence that of energy. It is important to realize that the right hand side of equation 23.46 can accept only values between \u2500 1 and + 1 (i.e., ka continuously varying from 0 to \u03c0 and coska from +1 to \u2500 1) as indicated by the horizontal lines in the figure. Therefore, the condition as imposed by equation 23.46 can be met only for values of \u03b1a for which the L.H.S. lies between \u00b11.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;text-indent: 1em;font-size: 1em\">In figure 23.4, the function: f(\u03b1a) = Psin \u03b1a\/\u03b1a + cos \u03b1a is plotted against \u03b1a. Obviously, for k to be real \u2502f (\u03b1a) \u2502should be less than 1. Those values of E for which \u2502 (\u03b1a) \u2502&gt; 1 will be forbidden.Those values of E for which f(\u03b1a) \u2264 1 will correspond to allowed values of energy. It may also be noted that if k is replaced by k+2n\u03c0\/a (where n= \u00b11, \u00b12, \u00b13\u2026\u2026), the right hand side of equation 23.46 will remain the same. That means one cannot uniquely determine the value of k. It is usually restricted to the domain:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">which is known as the first Brillouin zone.<\/p>\n<p style=\"text-align: justify\">\u2500 \u03c0\/a\u2264 k\u2264 + \u03c0\/a<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The E \u2013 k diagram showing the allowed and forbidden energy bands are shown in figures 23.5 &amp; 23.6. At the zone boundaries ( k=\u00b1 \u03c0\/a ) the Bloch wave function satisfies the Bragg condition and the group velocity [=( 1\/\u0127)dE\/dk] is zero which corresponds to a standing wave.<\/p>\n<\/div>\n<div><\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-403\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-54.png\" alt=\"\" width=\"714\" height=\"449\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-54.png 714w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-54-300x189.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-54-65x41.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-54-225x141.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-54-350x220.png 350w\" sizes=\"auto, (max-width: 714px) 100vw, 714px\" \/><\/p>\n<p style=\"text-align: center\">Figure 23.5: E Vs. wave vector K for electron in a periodic potential<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-404\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-55.png\" alt=\"\" width=\"709\" height=\"438\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-55.png 709w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-55-300x185.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-55-65x40.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-55-225x139.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-55-350x216.png 350w\" sizes=\"auto, (max-width: 709px) 100vw, 709px\" \/><\/p>\n<p style=\"text-align: center\">Figure 23.6: Full curve of E Vs. wave vector k showing d iscontin uit ies at k=\u00b1n \u03c0\/ a (where n=1, 2, 3\u2026)<\/p>\n<p>&nbsp;<\/p>\n<p>Let us analyse the details of what may be concluded fro m the figure drawn on the basis of equation 23.46.<\/p>\n<p>&nbsp;<\/p>\n<p>(i) There are infinite nu mbers of allo wed energy bands separated by intervals in which there are no energy<\/p>\n<p>levels. Such regions are known as forbidden regions.<\/p>\n<p>&nbsp;<\/p>\n<p>The boundaries of the allowed ranges of \u03b1a correspond to the values for which coska=\u00b11<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-size: 1em;text-align: initial\">That happens if ka = n\u03c0, or k = n\u03c0\/a, where n = 1, 2, 3\u2026\u2026.<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">(ii) The first term in the expression for f (\u03b1a) i.e ., P.sin\u03b1a\/\u03b1a decreases on an average with increas ing \u03b1a. It suggests that the width of the allowed energy bands increases and the forbidden regions get narrower as \u03b1a increases or as energy increases.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">(iii) As P increases, the width of the allowed band decreases. Increase in the value of P means the increase in \u201cbinding energy\u201d of the electrons. In the extreme case, when P is infinite, the allowed energy bands get infinitely narrower, consequently the energy spectrum becomes line spectrum and are independent of k.<\/p>\n<p>If P\u2192 \u221e, the allowed energy ranges of \u03b1a reduce to points given by:<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-405\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-56.png\" alt=\"\" width=\"478\" height=\"131\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-56.png 478w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-56-300x82.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-56-65x18.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-56-225x62.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-56-350x96.png 350w\" sizes=\"auto, (max-width: 478px) 100vw, 478px\" \/><\/p>\n<p>It shows that E is independent of k.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The energy levels in this case are discrete and the electron is completely bound. It is trapped within the potential wells and moves only in one cell of width \u2018a \u2019.<\/p>\n<p>&nbsp;<\/p>\n<p>Fro m the discussion on analysis of equation 23.46, we co me to the conclusion that in the one-dimensional problem for the limit ing case, the spectrum of energy values that are permitted is found to be consisting of continuous regions separated by finite intervals. By vary ing the quantity<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">P (i.e., V<sub>0<\/sub> b) fro m zero to infinity, we move fro m the case of free electrons to that of bound electrons and so are able to study the changes in the allowed and forbidden ranges of energy and the wave function.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">We may also use equation 23.46 to study the variation of energy E with the wave number k.The variation is as shown in figures23.5 &amp; 23.6.Fro m these figures one arrives at the follo wing conclusions:<\/p>\n<p>&nbsp;<\/p>\n<p>(i) The energy spectrum of the electrons consists of a number of allowed energy bands separated by forbidden regions.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">(ii) The width of the allowed energy bands increases with increase of energy values.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">(iii) If V0 is made zero, the E versus k curve becomes a continuous parabola, the same as that of free electrons (see figure 23.1)<\/p>\n<p>&nbsp;<\/p>\n<p>(iv) There are d iscontinuities in the E versus k curve which occur for k = n\u03c0\/a, where n = \u00b11, \u00b12, \u00b13.The range of allowed values of k between \u2500 \u03c0\/a and + \u03c0\/a constitutes the first Brillouin zone. The<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">range of values of k between \u2500\u03c0\/a and \u2500 2\u03c0\/a and between + \u03c0\/a and + 2\u03c0\/a constitutes the second Brillouin zone.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The energy E is represented as function of k for P = 3\u03c0\/2.Analyzing the curve of E against k further regarding the discontinuities in the E versus k at k = n\u03c0\/a, where n= 1, 2, 3\u2026\u2026.The k-values define the boundaries of the first (I), second (II), third (III) and so on Brillouin zones. Figure 23.5 shown here provides only half of the E-k curve; the first zone in fact extends from k= \u2500 \u03c0\/a to + \u03c0\/a.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Similarly, the second Brillouin zone is composed of two parts; the one extending from +\u03c0\/a to + 2\u03c0\/a as shown in the figure and the other part extending fro m \u2500 \u03c0\/a to \u2500 2\u03c0\/a. Figure 23.6 represents the full E-k curve. We may call each position of the curve as a band. The curve of E-k has the follo wing characteristics:<\/p>\n<p>&nbsp;<\/p>\n<p>1.\u00a0 The curves are horizontal at the top and bottom.<\/p>\n<p>2.\u00a0 The curves are parabolic near the top and the bottom having curvatures in opposite direct ions.<\/p>\n<p>3.\u00a0 d2E\/dk2is positive in the lower portion of the band and negative in the upper port ion of the band.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Let us consider a linear lattice of lattice constant a. According to free electron theory E versus wave vector k is continuous as shown in figure 23.1. The low energy part of the band structure is shown in figure 23.7 for electrons which are nearly free but with an energy gap at wave vector k= \u00b1\u03c0\/a resulting into creation of forbidden gap as a result of our assumption that electron is moving in a periodic potential (unlike in the case of free electron theory where electron is assumed to be moving\u00a0<span style=\"font-size: 1em;text-align: initial\">in a constant potential field). The curve of E versus wave vector k for an electron in a monato mic linear lattice with lattice constant a appears as shown in figure 23.7. The energy gap Eg is associated with the first Bragg reflection at k = \u00b1\u03c0\/a.<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-406\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-57.png\" alt=\"\" width=\"618\" height=\"350\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-57.png 618w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-57-300x170.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-57-65x37.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-57-225x127.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-57-350x198.png 350w\" sizes=\"auto, (max-width: 618px) 100vw, 618px\" \/><\/p>\n<p style=\"text-align: center\">Figu re 23.7: E as a function\u00a0 of wave vector k fo r an electron\u00a0 in a monatomic\u00a0 linear\u00a0 lattice o f latt ice constant\u00a0 a. The forb idden\u00a0 band associated with the energy gap Egis shown The Band\u00a0 gap is associated\u00a0 with the first Bragg\u00a0 reflection at k=\u00b1\u03c0\/a .<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>The Bragg condition in terms of the reciprocal lattice imp lies that:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-407\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-58.png\" alt=\"\" width=\"278\" height=\"39\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-58.png 278w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-58-65x9.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-58-225x32.png 225w\" sizes=\"auto, (max-width: 278px) 100vw, 278px\" \/><\/p>\n<p>for diffraction of a wave vector k, if taken in one dimension becomes on expanding the dot product and simp lify ing<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>where G is 2\u03c0 times a vector fro m the origin to a lattice point of the reciprocal lattice and k is a vector of magnitude 2\u03c0\/\u03bb along the direction of the incident x=ray beam. Equation 23.49 is the vector form of the Bragg equation.<\/p>\n<p>&nbsp;<\/p>\n<p>In one dimensions equation 23.49 beco mes:<\/p>\n<p>&nbsp;<\/p>\n<p>k = \u00b1 \u00bd G =\u00b1 n\u03c0\/a, \u2026\u2026\u2026\u2026\u2026\u2026\u202623.50 Where G = \u00b12n\u03c0\/a is the reciprocal lattice vector.<\/p>\n<p>&nbsp;<\/p>\n<p>The first reflection\u00a0\u00a0\u00a0 and\u00a0 the first energy gap\u00a0 results\u00a0 at k = \u00b1\u03c0\/a;\u00a0 other energy gaps\u00a0 result\u00a0 for other<\/p>\n<p>&nbsp;<\/p>\n<p>values of integer in equation 23.50.<\/p>\n<p>The reflection\u00a0 at the wave vector k =\u00b1\u03c0\/a takes\u00a0 place because the wave reflected\u00a0 fro m one atom in<\/p>\n<p>&nbsp;<\/p>\n<p>the linear lattice\u00a0 meets\u00a0 constructive\u00a0 interference\u00a0 with\u00a0 the wave fro m a nearest neighbour\u00a0 atom. The<\/p>\n<p>&nbsp;<\/p>\n<p>phase difference between the two reflected waves is \u00b12\u03c0 for these two values of k. The region in the k-space between \u2500\u03c0\/a and +\u03c0\/a is called the first Brillouin zone .<\/p>\n<p>&nbsp;<\/p>\n<p><strong>23.6<\/strong>\u00a0 \u00a0<strong>Brillouin Zo nes<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Kronig\u2013Penney model has shown that the energy discontinuities in the monatomic one dimensional lattice result when the wave vector satisfies the relation :<\/p>\n<p>&nbsp;<\/p>\n<p>k = n\u03c0\/a , where n is an integer which may be positive\u00a0 or negative.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">In the one-dimensional monatomic lattice a line representing the value of k is divided up by the energy discontinuities into segments of length \u03c0\/a as shown in figure 23.8<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-409\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-59.png\" alt=\"\" width=\"600\" height=\"152\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-59.png 600w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-59-300x76.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-59-65x16.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-59-225x57.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-59-350x89.png 350w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">Figure 23.8: Line rep resent ing the value of k for one -d imension al monatomic lattice is sho wn divided into segments of length \u00b1n\u03c0\/a (n=1, 2,3\u2026)<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">These line segments are known as Brillouin zones. The segment \u2500 \u03c0\/a &lt; k &lt; \u03c0\/a represents first Brillouin zone; the two segments \u2500 2\u03c0\/a &lt; k &lt; \u2500 \u03c0\/a and \u03c0\/a &lt; k &lt; 2\u03c0\/a form the second Brillouin zone and so on. Brillouin zones are characteristic of a particular crystal structure and as such each crystal structure form its own characteristic Brillouin zones.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Let us take the examp le of two dimensional simple square lattice as shown in figure 23.9. The first Brillouin zone for this lattice will be a square ABCD whose boundaries are defined by:<\/p>\n<p>&nbsp;<\/p>\n<p>kx = \u00b1\u03c0\/a;\u00a0\u00a0 ky\u00a0 =\u00b1 \u03c0\/a<\/p>\n<p>&nbsp;<\/p>\n<p>Similarly, the boundaries of the second Brillouin zone are defined by :<\/p>\n<p>\u00b1kx =\u00b1ky\u00a0 = \u03c0\/a as represented by the diagram EFGH.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-410\" src=\"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/106\/2018\/12\/2-60.png\" alt=\"\" width=\"576\" height=\"326\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-60.png 576w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-60-300x170.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-60-65x37.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-60-225x127.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-content\/uploads\/sites\/106\/2018\/12\/2-60-350x198.png 350w\" sizes=\"auto, (max-width: 576px) 100vw, 576px\" \/><\/p>\n<p>Figu re 23.9: Brillouin zones in two d imens ional simple square latt ice<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">One can apply the same principle for the three d imensional crystal structures. For a simp le cub ic lattice, the first Brillouin zone is a cube of edge 2\u03c0\/a.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">23.7<\/strong><span style=\"text-align: initial;font-size: 1em\">\u00a0<\/span><strong style=\"text-align: initial;font-size: 1em\">Significance of Brillouin Zones<\/strong><\/p>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">(i) Looking at the curve shown in figure 23.6 one finds that electron energy increases continuously from zero until the value of k reaches \u03c0\/a. Thereafter, it gets stopped as if it meets an obstacle or a wall at that instant and consequently gets reflected.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Treating\u00a0 electrons\u00a0 as waves and the propagation\u00a0 of electrons\u00a0 through a crystal as analogous\u00a0 to the propagation\u00a0 of electromagnetic\u00a0 waves, we arrive at a very important\u00a0 conclusion.\u00a0 We know that x-ray will\u00a0 suffer<\/p>\n<p>&nbsp;<\/p>\n<p>reflection\u00a0 if\u00a0 incident normal to a set of planes\u00a0 of interplanar\u00a0 spacing \u2018a\u2019 provided\u00a0 this equation is satisfied:<\/p>\n<p>&nbsp;<\/p>\n<p>n\u03bb = 2a sin90\u2070\u00a0 ( for x-ray is at right angles to the planes)<\/p>\n<p>&nbsp;<\/p>\n<p>n\u03bb = 2a<\/p>\n<p>&nbsp;<\/p>\n<p>For electron k = n\u03c0\/a and k = 2\u03c0\/\u03bb<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">So,\u00a0 2\u03c0\/\u03bb = n\u03c0\/a, or n\u03bb = 2a, wh ich is the same condition as for Bragg reflect ion. Thus, we may conclude\u00a0 that electron\u00a0\u00a0 suffers\u00a0\u00a0\u00a0 Bragg\u00a0\u00a0\u00a0 reflection\u00a0\u00a0\u00a0 when\u00a0\u00a0\u00a0 n\u00a0\u00a0 =\u00b11,\u00b12,\u00b13\u00a0\u00a0\u00a0 \u2026\u2026 corresponding to\u00a0 first, second, third\u00a0 orderreflections\u00a0 and\u00a0 so\u00a0 on. The\u00a0 zones\u00a0\u00a0 between\u00a0 the values\u00a0\u00a0 of k =\u00a0 \u2500 \u03c0\/a\u00a0 and\u00a0 + \u03c0\/a constitutes the first Brillouin zone and so on.<\/p>\n<p>&nbsp;<\/p>\n<p>(ii) The zone boundaries represent the maximu m energies that the electron can have without developing any discontinuity.<\/p>\n<p>&nbsp;<\/p>\n<p>(iii) The energy gap at the zero boundary is called the Forbidden zone or band; electrons cannot have those energies.<\/p>\n<p>&nbsp;<\/p>\n<p>(iv) Considering velocity of electrons in a periodic potential (given vg = 2\u03c0.d\u03bd\/dk; E = h\u03bd and so vg<\/p>\n<p>&nbsp;<\/p>\n<p>=2\u03c0\/h.d E\/dk)<\/p>\n<p>&nbsp;<\/p>\n<p>Slope d E\/dk = 0\u00a0\u00a0\u00a0\u00a0 when k = 0,<\/p>\n<p>&nbsp;<\/p>\n<p>&amp;\u00a0\u00a0 d E\/ dk = 0\u00a0\u00a0\u00a0\u00a0\u00a0 when k=\u00a0\u00a0\u00a0 \u03c0\/a.<\/p>\n<p>&nbsp;<\/p>\n<p>Therefore, the velocity of electron is zero both at the bottom and at the top of the first Brillou in zone or band<\/p>\n<p>At intermediate regions in the zone the electron velocity reaches the free electron velocity h k\/ 2\u03c0 m.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">(v) Let us now consider the motion\u00a0 of the electron\u00a0 in the first Brillouin zone,\u00a0 under the continued application\u00a0 of\u00a0a force F (either due to electric field or any other agency). Velocity of the electron at k = 0 will<br \/>\nincrease but when it approaches the value of k close to + \u03c0\/a its velocity begins to decrease and at k = + \u03c0\/a<br \/>\nthe velocity becomes zero, indicating that the electron wave packet suffers a Bragg reflection and begins to<br \/>\ntravel in a direction opposite to the applied force. The propagation in the negative direction continues until<br \/>\nit becomes equal to \u2500 \u03c0\/a. Once again the electron suffers Bragg reflection and the forward propagation<br \/>\nstarts until value of k reaches +\u03c0\/a. In other words, the electron is shunted back and forth in the first<br \/>\nBrillouin zone under the application of a constant unidirectional force. It means that a stationary electron<br \/>\nwave is set up in the first Brillouin zone instead of a travelling wave<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;text-indent: 1em;font-size: 1em\">vi) An electron remain ing in one zone or band cannot cross over to another zone by continuous application\u00a0 of force.Two consecutive Brillouin zones are separated by a forbidden energy gap. So, unless and until the electronin the first zone absorbs an energy equal to that of the forbidden gap in a single dose, the electron<br \/>\ncannot cross over to the next Brillouin zone irrespective of the duration of the applied force<\/span><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">vii) If the electrons continue to move back and forth in their respective Brillouin zones there is no change in the sum of the mo menta of all the electrons and, as a result, no net current is conducted by the crystal. It, obviously, means that the material is an insulator. Now suppose the forbidden gap is small enough or if impurity atoms provide some localized electronic states within the forb idden energy gap, one will find a semi conductor behaviour. The third situation is that there is no forbidden energy gap. In that case there is no restriction for increase in the value of k and so electric conduction can take place. The material in this case is a good conductor.<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>SUMMAR Y<\/strong><\/p>\n<ul>\n<li>We have discussed motion of electrons, both in a constant potential field (as assumed by Somerfield) as well\u00a0<span style=\"font-size: 1em;text-align: initial\">as in a period ic potential field as assumed in Kronig Penney model.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Significance of t ranslational period icity in a crystalline solid is explained.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">The concept of De Broglie wave or So merfield waves or Bloch waves are given.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">It is shown that Somerfield&#8217;s model of motion of a free electrons in a constant potential field leads to a relation between energy and wave vector for a free electron that is parabolic.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Bloch theorem is explained.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">It is shown that if electron is assumed to travel in a periodic potential as proposed by Kronig Penney model it leads to energy versus wave vector tha shows discontinuities at<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">The concept of brillioun zones and their significance in crystallography is exp lained<\/span><\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on Motionof electronsinperiodicpotential<\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/H_V2xP923iQ\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>References.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>1.Verma,A.R. &amp;Srivastava,O.N.\u201d Crystallography for Solid State Physics\u201d,Wiley Eastern Ltd., N.Delhi,1982.<\/p>\n<p>2. Seitz ,F.: \u201c Modern Theory of Solids\u201d, McGraw-Hill, N.Y.,1940.<\/p>\n<p>3. Omar,M.A. \u201c Elementary Solid State Physics\u201d, Addison-Wesley, Readin,1975.<\/p>\n<p>4. Kittel, C.:\u201dIntroduction to Solid State Physics\u201d,Wiley-Eastern Ltd.,N.Delhi,1985.<\/p>\n<p>5. Wannier, G.H.: \u201c Elements of Solid State Theory\u201d,Cambridge Univ. Press,1959.<\/p>\n<p>6. Ziman,J.M.: \u201c Principles of the Theory of Solids\u201d, Cambridge Univ. Press,1964.<\/p>\n<p>7. Clark,H.: \u201c Solid State Physics\u201d,Macmillan,London,1968.<\/p>\n<p>8. Pillai,S.O. : \u201cSolid State Physics\u201d, New Age Int.(P) Ltd.Publishers, N.Delhi,1997.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Suggested Reading.<\/strong><\/p>\n<ol>\n<li>Madelung,O :\u201d Introduction to Solid State Theory\u201d, Springer-Verlag,N.Y.,1978.<\/li>\n<li>Ghatak, A.K. &amp;Kothari,L.S.: \u201c Introduction to Lattice Dynamics\u201d,Addison Wesley,Reading,1971.<\/li>\n<li>Ashcroft,N.W. &amp;Mermin,N.D.: \u201cSolid State Physics\u201d,New York: Holt,Rinchart and Winston,1976.<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n","protected":false},"author":3,"menu_order":23,"template":"","meta":{"_acf_changed":false,"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["prof-p-n-kotru"],"pb_section_license":""},"chapter-type":[],"contributor":[58],"license":[],"class_list":["post-368","chapter","type-chapter","status-publish","hentry","contributor-prof-p-n-kotru"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-json\/pressbooks\/v2\/chapters\/368","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":6,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-json\/pressbooks\/v2\/chapters\/368\/revisions"}],"predecessor-version":[{"id":614,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-json\/pressbooks\/v2\/chapters\/368\/revisions\/614"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-json\/pressbooks\/v2\/chapters\/368\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-json\/wp\/v2\/media?parent=368"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-json\/pressbooks\/v2\/chapter-type?post=368"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-json\/wp\/v2\/contributor?post=368"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp06\/wp-json\/wp\/v2\/license?post=368"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}