{"id":205,"date":"2018-12-11T08:29:43","date_gmt":"2018-12-11T08:29:43","guid":{"rendered":"http:\/\/msp13.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=205"},"modified":"2018-12-11T09:17:49","modified_gmt":"2018-12-11T09:17:49","slug":"dielectric-properties-lecture-6","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp13\/chapter\/dielectric-properties-lecture-6\/","title":{"rendered":"Dielectric Properties Lecture 6"},"content":{"raw":"
\r\n
\"epgp<\/a>\r\n<\/span><\/div>\r\n\u00a0 \u00a0<\/em><\/strong>\r\n\r\n \r\n\r\n \r\n\r\nLearning Outcomes:<\/em><\/strong>\r\n\r\n \r\n\r\nFrom this module students may get to know about the following:<\/em><\/strong>\r\n\r\n1.\u00a0\u00a0\u00a0\u00a0 <\/em>Detailed study of complex permittivity<\/em>\r\n\r\n2.\u00a0\u00a0\u00a0\u00a0 <\/em>The fundamental Dielectric relaxation.<\/em>\r\n\r\n3.\u00a0\u00a0\u00a0\u00a0 <\/em>You will learn about Debye equations and its complex plane diagram.<\/em>\r\n\r\n \r\n

The dielectric constant and loss are important properties of interest to physics people because these two parameters, among others, decide the suitability of a material for a given application. The relationships between the dielectric constant and the polarizability under dc fields have been discussed in sufficient detail in the previous lectures. In this <\/span>chapter we examine the behavior of a polar material in an alternating field, and the discussion begins with the definition of complex permittivity and dielectric loss which are of particular importance in polar materials.<\/span><\/p>\r\n \r\n

6.1 COMPLEX PERMITTIVITY:<\/strong><\/p>\r\n \r\n

Consider a capacitor that consists of two plane parallel electrodes in a vacuum having an applied alternating voltage represented by the equation<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n

Here \u00a0is the instantaneous voltage,the maximum value of \u00a0and \u00a0is the angular frequency in radian per second. The current through the capacitor, is given by<\/p>\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\nIn this is the vacuum capacitance , sometimes also called the geometrical capacitance.\r\n

In an ideal dielectric the current leads the voltage by 90\u00b0 and there is no component of the current in phase with the voltage. If a material of dielectric constant \u00a0is now placed between the plates the capacitance increases to \u00a0and the current is given by<\/p>\r\n\"\"\r\n

It is noted that the usual symbol for the dielectric constant <\/span>is , but we omit the subscript for the sake of clarity, noting that <\/span>&<\/em>is dimensionless. The current phase or will not now be in phase with the voltage but by an angle (90\u00b0-) <\/span>where \u00a0is called the <\/span>loss angle.<\/strong> The dielectric constant is a complex quantity represented by<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

The current can be resolved into two components; the component in phase with the applied voltage is and the component leading the applied voltage by 90\u00b0 is(fig.1). This component is the charging current of the ideal capacitor.<\/p>\r\n \r\n

The component in phase with the applied voltage gives rise to dielectric loss. is the loss angle and is given by<\/p>\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\nis usually referred to as the loss factor and \u00a0the dissipation factor. To complete the definitions we note that\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n

Figure1. Real () and imaginary () parts of the complex dielectric constant () in an alternating electric field.<\/p>\r\n \r\n\r\nThe alternating current conductivity is given by\r\n\r\n\"\"\r\n\r\n \r\n\r\n6.2 Dielectric relaxation:<\/strong>\r\n\r\n \r\n

When a direct voltage applied to a dielectric for a sufficiently long duration is suddenly removed the decay of polarization to zero value is not instantaneous but takes a finite time. This is the time required for the dipoles to revert to a random distribution, in equilibrium with the temperature of the medium, from a field oriented alignment. Similarly the buildup of polarization following the sudden application of a direct voltage takes a finite time interval before the polarization attains its maximum value. This phenomenon is described by the general term dielectric relaxation.<\/strong><\/p>\r\n \r\n

When a dc voltage is applied to a polar dielectric let us assume that the polarization builds up from zero to a final value (fig. 2) according to an exponential law<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

Where P(t)<\/em> is the polarization at time t and \u00a0is called the relaxation time, i is a function of temperature and it is independent of the time.<\/p>\r\n \r\n\r\nThe rate of buildup of polarization may be obtained, by differentiating equation (8) as\r\n\r\n\"\"\r\n\r\nSubstituting equation (8) in (8a) and assuming that the total polarization is due to the dipoles, we get\r\n\r\n \r\n\r\n\"\"\r\n

Neglecting\u00a0 atomic\u00a0 polarization\u00a0 the\u00a0 total\u00a0 polarization \u00a0PT(t)\u00a0 can\u00a0 be\u00a0 expressed\u00a0 as\u00a0 the\u00a0 sum\u00a0 of\u00a0 the orientational polarization at that instant, , and electronic polarization, \u00a0which is assumed to attain its final value instantaneously because the time required for it to attain saturation value is in the optical frequency range. Further, we assume that the instantaneous polarization of the material in an alternating voltage is equal to that under dc voltage that has the same magnitude as the peak of the alternating voltage at that instant.<\/p>\r\n \r\n\r\nWe can write total polarization as\r\n\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n

<\/div>\r\n
\r\n\r\n\"\"\r\n

where C is a constant. At time t, sufficiently large when compared with , the first term on the right side of equation (17) becomes so small that it can be neglected and we get the solution for P(t)<\/em> as<\/p>\r\n\"\"\r\n

Equation (20) shows that \u00a0is a sinusoidal function with the same frequency as the applied voltage. The instantaneous value of flux density D is given by<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

Equations (25) and (26) are known as Debye equations and they describe the behavior of polar dielectrics at various frequencies. The temperature enters the discussion by way of the parameter \u00a0as will be described in the following section. The plot of is known as the relaxation curve and it is characterized by a peak at = 0.5. It is easy to show = 3.46 for this ratio and one can use this as a guide to determine whether Debye relaxation is a possible mechanism.<\/p>\r\n \r\n\r\n6.3 DEBYE EQUATIONS:<\/strong>\r\n\r\n \r\n\r\nOne another way to express the Debye equations are\r\n\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n

<\/div>\r\n
\r\n\r\n\u00a0 \u00a0 Equations (25)-(27) are shown in fig 2. The conclusion of these equations shows are the following\r\n\r\n(1)\u00a0 \u00a0For small values of, the real part because of the squared term in the denominator of equation (28) and \u00a0is also small for the same reason. Of course, at = 0, we get = 0 as expected because this is dc voltage.\r\n\r\n(2)\u00a0 For very large values of, and is small.\r\n\r\n(3)\u00a0 For intermediate values of frequencies is a maximum at some particular value of.\r\n\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n

figure2. Schematic representation of Debye equations plotted as a function of <\/span>log.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\nThe values of \u00a0and at this value of \u00a0are\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n

The dissipation factor tan also increases with frequency, reaches a maximum, and for further increase in frequency, it decreases. The frequency at which the loss angle is a maximum can also be found by differentiating tanwith respect to \u00a0and equating the differential to zero. This leads to<\/p>\r\n \r\n\r\n\"\"\r\n\r\nThese equations shows that\r\n\r\n\"\"\r\n

Fig<\/span>.(2) shows that, at the relaxation frequency defined by <\/span>equation , decreases sharply over a relatively small <\/span>band width. This fact may be used to determine whether relaxation occurs in a material at a specified frequency. If we <\/span>measure \u00a0as a function of temperature at <\/span>constant frequency it will decrease rapidly with <\/span>temperature at relaxation frequency. Normally in the absence of relaxation s' should increase with decreasing temperature.<\/span><\/p>\r\n \r\n

Variation <\/span>of \u00a0as a function of frequency is referred to as dispersion in the literature on dielectrics. Variation of as a function of frequency is called absorption though the two terms are often used interchangeably, possibly because dispersion and absorption are associated phenomena.<\/span><\/p>\r\n \r\n

6.4 COMPLEX PLANE DIAGRAM:<\/strong><\/p>\r\n \r\n

Cole and Cole showed <\/span>that, in a material exhibiting Debye relaxation a plot of <\/span>against , each point corresponding to a particular frequency yields a semi-circle. This can easily be demonstrated by rearranging equations (25) and (26) to give<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n\r\nThe right side of equation (35) may be simplified using equation (25) resulting in\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
\r\n\r\npractice very few materials completely agree with Debye equations, the discrepancy <\/span>being attributed to what is generally referred to as <\/span>distribution of relaxation times.<\/strong>\r\n\r\n<\/div>\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\nSummary:<\/em><\/strong>\r\n\r\n \r\n

In this present chapter Debye equation was derived and its complex plane diagram was discussed. Complex permittivity was defined and derived. Dielectric relaxation were discussed and described the behavior of polar dielectrics at various frequencies.<\/p>\r\n\r\n\r\n\r\n\r\n
you can view video on Dielectric Properties Lecture 6<\/strong><\/td>\r\n\"\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
\r\n\r\n\u00a0 \u00a0References:<\/em><\/strong>\r\n
    \r\n \t
  1. Peter Y. Yu, Manuel Cardona (2001). <\/em>Fundamentals of Semiconductors: Physics and<\/em> Materials Properties. <\/em>Berlin: Springer. p. 261. <\/em>ISBN <\/em>3-540-25470-6.<\/em><\/li>\r\n \t
  2. Jos\u00e9 Garc\u00eda Sol\u00e9, Jose Sol\u00e9, Luisa Bausa, (2001). <\/em>An introduction to the optical spectroscopy<\/em> of inorganic solids. <\/em>Wiley. Appendix A1, pp, 263. <\/em>ISBN <\/em>0-470-86885-6.<\/em><\/li>\r\n \t
  3. John H. Moore, Nicholas D. Spencer (2001). <\/em>Encyclopedia of chemical physics and physical<\/em> chemistry. <\/em>Taylor and Francis. p. 105. <\/em>ISBN <\/em>0-7503-0798-6.<\/em><\/li>\r\n \t
  4. Chiang, Y. et al.: <\/em>Physical Ceramics, <\/em>John Wiley & Sons <\/em>1997, New York<\/em><\/li>\r\n \t
  5. Mussig & Hans-Joachim, Semiconductor capacitor with praseodymium oxide as dielectric, <\/em>U.S. Patent <\/em>7,113,388published 2003-11-06, issued 2004-10-18, assigned to IHP GmbH- Innovations for High Performance Microelectronics\/Institute Fur Innovative Mikroelektronik<\/em><\/li>\r\n<\/ol>\r\n\u00a0 \u00a0\u00a0<\/em>References and Suggestive Readings<\/em><\/strong>\r\n
      \r\n \t
    1. C.J.F. Bottcher, O.C. von Belle & Paul Bordewijk (1973) Theory of Electric Polarization: Dielectric Polarization, volume 1, (1978) volume 2, Elsevier <\/em>ISBN 0-444-41579-3<\/em>.<\/em><\/li>\r\n \t
    2. Arthur R. von Hippel <\/em>(1954) Dielectrics and Waves <\/em>ISBN 0-89006-803-8<\/em><\/li>\r\n \t
    3. Arthur von Hippel editor (1966) Dielectric Materials and Applications: papers by 22 contributors <\/em>ISBN 0-89006-805-4.<\/em><\/li>\r\n \t
    4. Jackson, John David <\/em>(August 10, 1998). <\/em>Classical Electrodynamics <\/em>(3 rd ed.). <\/em>John Wiley &<\/em> Sons. <\/em>ISBN <\/em>978-0-471-30932-1. <\/em>808 or 832 pages.<\/em><\/li>\r\n \t
    5. Scaife, Brendan <\/em>(September 3, 1998). <\/em>Principles of Dielectrics (Monographs on the Physics &<\/em> Chemistry of Materials) <\/em>(2 nd ed.)<\/em>.Oxford University Press. <\/em>ISBN <\/em>978-0198565574.<\/em><\/li>\r\n<\/ol>\r\n\u00a0 \u00a0\u00a0<\/em>Web Links<\/strong><\/em>\r\n\r\n \r\n\r\n1.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>http:\/\/www.lightandmatter.com\/html_books\/0sn\/ch11\/ch11.html<\/em>\r\n\r\n2.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>http:\/\/wiki.4hv.org\/index.php\/Dielectric_Sphere_in_Electric_Field<\/em>\r\n\r\n3.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>http:\/\/www.doitpoms.ac.uk\/tlplib\/dielectrics\/index.php<\/em>\r\n\r\n4.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>http:\/\/falsecolour.com\/aw\/eps_plots\/eps_plots.pdf<\/em>\r\n\r\n<\/div>\r\n
      <\/div>\r\n
      \r\n\r\n\u00a0 \u00a0 Additional Topics to be studied<\/em><\/strong>\r\n\r\n \r\n\r\n1)\u00a0\u00a0 Dielectric relaxation time\r\n\r\n2)\u00a0 Complex permittivity of metals\r\n\r\n \r\n\r\nDebye relaxation<\/em><\/strong>\r\n\r\n \r\n

      Debye relaxation is the dielectric relaxation response of an ideal, <\/span>non interacting population of dipoles to an alternating external electric field. It is usually expressed in the complex permittivity \u03b5 of a medium as a function of the field's <\/span>frequency \u03c9:<\/span><\/p>\r\n\r\n<\/div>\r\n

      \r\n\r\n\"\"\r\n\r\n \r\n\r\nwhere \u03b5\u221e is the permittivity at the high frequency limit, \u0394\u03b5 = \u03b5s \u2212 \u03b5\u221e where \u03b5s is the static, low frequency permittivity, and \u03c4 is the characteristic relaxation time of the medium.\r\n\r\n \r\n\r\nThis relaxation model was introduced by and named after the physicist Peter Debye (1913).\r\n\r\n \r\n\r\nVariants of the Debye equation<\/em><\/strong>\r\n\r\n \r\n\r\n1)\u00a0 Cole\u2013Cole equation:\r\n\r\n \r\n\r\nThis equation is used when the dielectric loss peak shows symmetric broadening The Cole\u2013Cole equation is a relaxation model that is often used to describe dielectric relaxation in polymers.\r\n\r\n \r\n\r\nIt is given by the equation\r\n\r\n\"\"\r\n\r\n \r\n\r\nwhere \u00a0is the complex dielectric constant, \u00a0and \u00a0are the \"static\" and \"infinite frequency\" dielectric constants, \u00a0is the angular frequency and \u00a0is a time constant.\r\n\r\n \r\n

      The exponent parameter , which takes a value between 0 and 1, allows to describe different spectral shapes. When , the Cole-Cole model reduces to the Debye model. When , the relaxation is stretched<\/em>, i.e. it extends over a wider range on a logarithmic scale than Debye relaxation.<\/p>\r\n \r\n

      Cole-Cole relaxation constitutes a special case of Havriliak-Negami relaxation when the symmetry parameter (\u03b2) is equal to 1 - that is, when the relaxation peaks are symmetric. Another special case of Havriliak-Negami relaxation (\u03b2<1, \u03b1=0) is known as Cole-Davidson relaxation, for an abridged and updated review of anomalous dielectric relaxation in dissored systems see Kalmykov.<\/p>\r\n \r\n\r\n2)\u00a0\u00a0 Cole\u2013Davidson equation\r\n\r\n \r\n\r\nThis equation is used when the dielectric loss peak shows asymmetric broadening.<\/span>\r\n\r\n \r\n\r\n3)<\/span>Havriliak\u2013Negami relaxation\r\n\r\n \r\n\r\nThis equation considers both symmetric and asymmetric broadening.<\/span>\r\n\r\n \r\n

      Havriliak\u2013Negami relaxation is an empirical modification of the <\/span>Debye relaxation model, accounting for the <\/span>asymmetry and broadness of the <\/span>dielectric dispersion curve. The model was first used to describe the dielectric relaxation of some <\/span>polymers, by adding two <\/span>exponential parameters to the Debye equation:<\/span><\/p>\r\n\r\n<\/div>\r\n

      \r\n\r\n\"\"\r\n\r\n \r\n\r\n<\/div>\r\n
      \"\"<\/div>\r\n
      <\/div>\r\n\"\"\r\n
      \r\n\r\n\"\"\r\n\r\n<\/div>\r\n ","rendered":"
      \n
      \"epgp<\/a>
      \n<\/span><\/div>\n

      \u00a0 \u00a0<\/em><\/strong><\/p>\n

       <\/p>\n

       <\/p>\n

      Learning Outcomes:<\/em><\/strong><\/p>\n

       <\/p>\n

      From this module students may get to know about the following:<\/em><\/strong><\/p>\n

      1.\u00a0\u00a0\u00a0\u00a0 <\/em>Detailed study of complex permittivity<\/em><\/p>\n

      2.\u00a0\u00a0\u00a0\u00a0 <\/em>The fundamental Dielectric relaxation.<\/em><\/p>\n

      3.\u00a0\u00a0\u00a0\u00a0 <\/em>You will learn about Debye equations and its complex plane diagram.<\/em><\/p>\n

       <\/p>\n

      The dielectric constant and loss are important properties of interest to physics people because these two parameters, among others, decide the suitability of a material for a given application. The relationships between the dielectric constant and the polarizability under dc fields have been discussed in sufficient detail in the previous lectures. In this <\/span>chapter we examine the behavior of a polar material in an alternating field, and the discussion begins with the definition of complex permittivity and dielectric loss which are of particular importance in polar materials.<\/span><\/p>\n

       <\/p>\n

      6.1 COMPLEX PERMITTIVITY:<\/strong><\/p>\n

       <\/p>\n

      Consider a capacitor that consists of two plane parallel electrodes in a vacuum having an applied alternating voltage represented by the equation<\/span><\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

      Here \u00a0is the instantaneous voltage,the maximum value of \u00a0and \u00a0is the angular frequency in radian per second. The current through the capacitor, is given by<\/p>\n

       <\/p>\n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      In this is the vacuum capacitance , sometimes also called the geometrical capacitance.<\/p>\n

      In an ideal dielectric the current leads the voltage by 90\u00b0 and there is no component of the current in phase with the voltage. If a material of dielectric constant \u00a0is now placed between the plates the capacitance increases to \u00a0and the current is given by<\/p>\n

      \"\"<\/p>\n

      It is noted that the usual symbol for the dielectric constant <\/span>is , but we omit the subscript for the sake of clarity, noting that <\/span>&<\/em>is dimensionless. The current phase or will not now be in phase with the voltage but by an angle (90\u00b0-) <\/span>where \u00a0is called the <\/span>loss angle.<\/strong> The dielectric constant is a complex quantity represented by<\/span><\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      The current can be resolved into two components; the component in phase with the applied voltage is and the component leading the applied voltage by 90\u00b0 is(fig.1). This component is the charging current of the ideal capacitor.<\/p>\n

       <\/p>\n

      The component in phase with the applied voltage gives rise to dielectric loss. is the loss angle and is given by<\/p>\n

       <\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      is usually referred to as the loss factor and \u00a0the dissipation factor. To complete the definitions we note that<\/p>\n

      \"\"<\/p>\n

      \"\"<\/p>\n<\/div>\n

      \n

      Figure1. Real () and imaginary () parts of the complex dielectric constant () in an alternating electric field.<\/p>\n

       <\/p>\n

      The alternating current conductivity is given by<\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      6.2 Dielectric relaxation:<\/strong><\/p>\n

       <\/p>\n

      When a direct voltage applied to a dielectric for a sufficiently long duration is suddenly removed the decay of polarization to zero value is not instantaneous but takes a finite time. This is the time required for the dipoles to revert to a random distribution, in equilibrium with the temperature of the medium, from a field oriented alignment. Similarly the buildup of polarization following the sudden application of a direct voltage takes a finite time interval before the polarization attains its maximum value. This phenomenon is described by the general term dielectric relaxation.<\/strong><\/p>\n

       <\/p>\n

      When a dc voltage is applied to a polar dielectric let us assume that the polarization builds up from zero to a final value (fig. 2) according to an exponential law<\/span><\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

      Where P(t)<\/em> is the polarization at time t and \u00a0is called the relaxation time, i is a function of temperature and it is independent of the time.<\/p>\n

       <\/p>\n

      The rate of buildup of polarization may be obtained, by differentiating equation (8) as<\/p>\n

      \"\"<\/p>\n

      Substituting equation (8) in (8a) and assuming that the total polarization is due to the dipoles, we get<\/p>\n

       <\/p>\n

      \"\"<\/p>\n

      Neglecting\u00a0 atomic\u00a0 polarization\u00a0 the\u00a0 total\u00a0 polarization \u00a0PT(t)\u00a0 can\u00a0 be\u00a0 expressed\u00a0 as\u00a0 the\u00a0 sum\u00a0 of\u00a0 the orientational polarization at that instant, , and electronic polarization, \u00a0which is assumed to attain its final value instantaneously because the time required for it to attain saturation value is in the optical frequency range. Further, we assume that the instantaneous polarization of the material in an alternating voltage is equal to that under dc voltage that has the same magnitude as the peak of the alternating voltage at that instant.<\/p>\n

       <\/p>\n

      We can write total polarization as<\/p>\n

       <\/p>\n

      \"\"<\/p>\n<\/div>\n

      <\/div>\n
      \n

      \"\"<\/p>\n

      where C is a constant. At time t, sufficiently large when compared with , the first term on the right side of equation (17) becomes so small that it can be neglected and we get the solution for P(t)<\/em> as<\/p>\n

      \"\"<\/p>\n

      Equation (20) shows that \u00a0is a sinusoidal function with the same frequency as the applied voltage. The instantaneous value of flux density D is given by<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      Equations (25) and (26) are known as Debye equations and they describe the behavior of polar dielectrics at various frequencies. The temperature enters the discussion by way of the parameter \u00a0as will be described in the following section. The plot of is known as the relaxation curve and it is characterized by a peak at = 0.5. It is easy to show = 3.46 for this ratio and one can use this as a guide to determine whether Debye relaxation is a possible mechanism.<\/p>\n

       <\/p>\n

      6.3 DEBYE EQUATIONS:<\/strong><\/p>\n

       <\/p>\n

      One another way to express the Debye equations are<\/p>\n

       <\/p>\n

      \"\"<\/p>\n<\/div>\n

      <\/div>\n
      \n

      \u00a0 \u00a0 Equations (25)-(27) are shown in fig 2. The conclusion of these equations shows are the following<\/p>\n

      (1)\u00a0 \u00a0For small values of, the real part because of the squared term in the denominator of equation (28) and \u00a0is also small for the same reason. Of course, at = 0, we get = 0 as expected because this is dc voltage.<\/p>\n

      (2)\u00a0 For very large values of, and is small.<\/p>\n

      (3)\u00a0 For intermediate values of frequencies is a maximum at some particular value of.<\/p>\n

       <\/p>\n

      \"\"<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      figure2. Schematic representation of Debye equations plotted as a function of <\/span>log.<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      The values of \u00a0and at this value of \u00a0are<\/p>\n

       <\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      The dissipation factor tan also increases with frequency, reaches a maximum, and for further increase in frequency, it decreases. The frequency at which the loss angle is a maximum can also be found by differentiating tanwith respect to \u00a0and equating the differential to zero. This leads to<\/p>\n

       <\/p>\n

      \"\"<\/p>\n

      These equations shows that<\/p>\n

      \"\"<\/p>\n

      Fig<\/span>.(2) shows that, at the relaxation frequency defined by <\/span>equation , decreases sharply over a relatively small <\/span>band width. This fact may be used to determine whether relaxation occurs in a material at a specified frequency. If we <\/span>measure \u00a0as a function of temperature at <\/span>constant frequency it will decrease rapidly with <\/span>temperature at relaxation frequency. Normally in the absence of relaxation s’ should increase with decreasing temperature.<\/span><\/p>\n

       <\/p>\n

      Variation <\/span>of \u00a0as a function of frequency is referred to as dispersion in the literature on dielectrics. Variation of as a function of frequency is called absorption though the two terms are often used interchangeably, possibly because dispersion and absorption are associated phenomena.<\/span><\/p>\n

       <\/p>\n

      6.4 COMPLEX PLANE DIAGRAM:<\/strong><\/p>\n

       <\/p>\n

      Cole and Cole showed <\/span>that, in a material exhibiting Debye relaxation a plot of <\/span>against , each point corresponding to a particular frequency yields a semi-circle. This can easily be demonstrated by rearranging equations (25) and (26) to give<\/span><\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

      The right side of equation (35) may be simplified using equation (25) resulting in<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      \"\"<\/p>\n<\/div>\n

      \n

      practice very few materials completely agree with Debye equations, the discrepancy <\/span>being attributed to what is generally referred to as <\/span>distribution of relaxation times.<\/strong><\/p>\n<\/div>\n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      Summary:<\/em><\/strong><\/p>\n

       <\/p>\n

      In this present chapter Debye equation was derived and its complex plane diagram was discussed. Complex permittivity was defined and derived. Dielectric relaxation were discussed and described the behavior of polar dielectrics at various frequencies.<\/p>\n\n\n\n
      you can view video on Dielectric Properties Lecture 6<\/strong><\/td>\n\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
      \n

      \u00a0 \u00a0References:<\/em><\/strong><\/p>\n

        \n
      1. Peter Y. Yu, Manuel Cardona (2001). <\/em>Fundamentals of Semiconductors: Physics and<\/em> Materials Properties. <\/em>Berlin: Springer. p. 261. <\/em>ISBN <\/em>3-540-25470-6.<\/em><\/li>\n
      2. Jos\u00e9 Garc\u00eda Sol\u00e9, Jose Sol\u00e9, Luisa Bausa, (2001). <\/em>An introduction to the optical spectroscopy<\/em> of inorganic solids. <\/em>Wiley. Appendix A1, pp, 263. <\/em>ISBN <\/em>0-470-86885-6.<\/em><\/li>\n
      3. John H. Moore, Nicholas D. Spencer (2001). <\/em>Encyclopedia of chemical physics and physical<\/em> chemistry. <\/em>Taylor and Francis. p. 105. <\/em>ISBN <\/em>0-7503-0798-6.<\/em><\/li>\n
      4. Chiang, Y. et al.: <\/em>Physical Ceramics, <\/em>John Wiley & Sons <\/em>1997, New York<\/em><\/li>\n
      5. Mussig & Hans-Joachim, Semiconductor capacitor with praseodymium oxide as dielectric, <\/em>U.S. Patent <\/em>7,113,388published 2003-11-06, issued 2004-10-18, assigned to IHP GmbH- Innovations for High Performance Microelectronics\/Institute Fur Innovative Mikroelektronik<\/em><\/li>\n<\/ol>\n

        \u00a0 \u00a0\u00a0<\/em>References and Suggestive Readings<\/em><\/strong><\/p>\n

          \n
        1. C.J.F. Bottcher, O.C. von Belle & Paul Bordewijk (1973) Theory of Electric Polarization: Dielectric Polarization, volume 1, (1978) volume 2, Elsevier <\/em>ISBN 0-444-41579-3<\/em>.<\/em><\/li>\n
        2. Arthur R. von Hippel <\/em>(1954) Dielectrics and Waves <\/em>ISBN 0-89006-803-8<\/em><\/li>\n
        3. Arthur von Hippel editor (1966) Dielectric Materials and Applications: papers by 22 contributors <\/em>ISBN 0-89006-805-4.<\/em><\/li>\n
        4. Jackson, John David <\/em>(August 10, 1998). <\/em>Classical Electrodynamics <\/em>(3 rd ed.). <\/em>John Wiley &<\/em> Sons. <\/em>ISBN <\/em>978-0-471-30932-1. <\/em>808 or 832 pages.<\/em><\/li>\n
        5. Scaife, Brendan <\/em>(September 3, 1998). <\/em>Principles of Dielectrics (Monographs on the Physics &<\/em> Chemistry of Materials) <\/em>(2 nd ed.)<\/em>.Oxford University Press. <\/em>ISBN <\/em>978-0198565574.<\/em><\/li>\n<\/ol>\n

          \u00a0 \u00a0\u00a0<\/em>Web Links<\/strong><\/em><\/p>\n

           <\/p>\n

          1.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>http:\/\/www.lightandmatter.com\/html_books\/0sn\/ch11\/ch11.html<\/em><\/p>\n

          2.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>http:\/\/wiki.4hv.org\/index.php\/Dielectric_Sphere_in_Electric_Field<\/em><\/p>\n

          3.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>http:\/\/www.doitpoms.ac.uk\/tlplib\/dielectrics\/index.php<\/em><\/p>\n

          4.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>http:\/\/falsecolour.com\/aw\/eps_plots\/eps_plots.pdf<\/em><\/p>\n<\/div>\n

          <\/div>\n
          \n

          \u00a0 \u00a0 Additional Topics to be studied<\/em><\/strong><\/p>\n

           <\/p>\n

          1)\u00a0\u00a0 Dielectric relaxation time<\/p>\n

          2)\u00a0 Complex permittivity of metals<\/p>\n

           <\/p>\n

          Debye relaxation<\/em><\/strong><\/p>\n

           <\/p>\n

          Debye relaxation is the dielectric relaxation response of an ideal, <\/span>non interacting population of dipoles to an alternating external electric field. It is usually expressed in the complex permittivity \u03b5 of a medium as a function of the field’s <\/span>frequency \u03c9:<\/span><\/p>\n<\/div>\n

          \n

          \"\"<\/p>\n

           <\/p>\n

          where \u03b5\u221e is the permittivity at the high frequency limit, \u0394\u03b5 = \u03b5s \u2212 \u03b5\u221e where \u03b5s is the static, low frequency permittivity, and \u03c4 is the characteristic relaxation time of the medium.<\/p>\n

           <\/p>\n

          This relaxation model was introduced by and named after the physicist Peter Debye (1913).<\/p>\n

           <\/p>\n

          Variants of the Debye equation<\/em><\/strong><\/p>\n

           <\/p>\n

          1)\u00a0 Cole\u2013Cole equation:<\/p>\n

           <\/p>\n

          This equation is used when the dielectric loss peak shows symmetric broadening The Cole\u2013Cole equation is a relaxation model that is often used to describe dielectric relaxation in polymers.<\/p>\n

           <\/p>\n

          It is given by the equation<\/p>\n

          \"\"<\/p>\n

           <\/p>\n

          where \u00a0is the complex dielectric constant, \u00a0and \u00a0are the “static” and “infinite frequency” dielectric constants, \u00a0is the angular frequency and \u00a0is a time constant.<\/p>\n

           <\/p>\n

          The exponent parameter , which takes a value between 0 and 1, allows to describe different spectral shapes. When , the Cole-Cole model reduces to the Debye model. When , the relaxation is stretched<\/em>, i.e. it extends over a wider range on a logarithmic scale than Debye relaxation.<\/p>\n

           <\/p>\n

          Cole-Cole relaxation constitutes a special case of Havriliak-Negami relaxation when the symmetry parameter (\u03b2) is equal to 1 – that is, when the relaxation peaks are symmetric. Another special case of Havriliak-Negami relaxation (\u03b2<1, \u03b1=0) is known as Cole-Davidson relaxation, for an abridged and updated review of anomalous dielectric relaxation in dissored systems see Kalmykov.<\/p>\n

           <\/p>\n

          2)\u00a0\u00a0 Cole\u2013Davidson equation<\/p>\n

           <\/p>\n

          This equation is used when the dielectric loss peak shows asymmetric broadening.<\/span><\/p>\n

           <\/p>\n

          3)<\/span>Havriliak\u2013Negami relaxation<\/p>\n

           <\/p>\n

          This equation considers both symmetric and asymmetric broadening.<\/span><\/p>\n

           <\/p>\n

          Havriliak\u2013Negami relaxation is an empirical modification of the <\/span>Debye relaxation model, accounting for the <\/span>asymmetry and broadness of the <\/span>dielectric dispersion curve. The model was first used to describe the dielectric relaxation of some <\/span>polymers, by adding two <\/span>exponential parameters to the Debye equation:<\/span><\/p>\n<\/div>\n

          \n

          \"\"<\/p>\n

           <\/p>\n<\/div>\n

          \"\"<\/div>\n
          <\/div>\n

          \"\"<\/p>\n

          \n

          \"\"<\/p>\n<\/div>\n

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