{"id":58,"date":"2018-12-10T06:58:57","date_gmt":"2018-12-10T06:58:57","guid":{"rendered":"http:\/\/msp13.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=58"},"modified":"2018-12-10T09:07:55","modified_gmt":"2018-12-10T09:07:55","slug":"dielectric-properties-lecture-2","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp13\/chapter\/dielectric-properties-lecture-2\/","title":{"rendered":"Dielectric Properties Lecture 2"},"content":{"raw":"
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\"epgp<\/a>\r\n<\/span><\/div>\r\n\u00a0 \u00a0 <\/em><\/strong>\r\n\r\n \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\n \r\n\r\nThe mechanism and different type of polarization.<\/em>\r\n\r\n\u25cf\u00a0\u00a0 electronic polarization<\/em>\r\n\r\n\u25cf\u00a0\u00a0 ionic polarization<\/em>\u00a0<\/em>\r\n\r\n\u25cf\u00a0\u00a0 orientation polarization <\/em>\r\n\r\nThe static dielectric of gases<\/em>\r\n\r\n \r\n\r\n2.\u00a0\u00a0 <\/span>Type of polarizability<\/strong>\r\n\r\n \r\n\r\n2.1 Electronic polarizability :<\/em><\/strong>\r\n\r\n \r\n

In a free atom, the charge distribution is such that the dipole moment in the absence of an external field vanishes; the center of gravity of the electron distribution coincides with the nucleus. Consider now an atom in a static homogeneous external field <\/span>E.<\/em> The force exerted on the positive nucleus will then be oppositely directed to the forces exerted on the electrons. As a result, the external field tends to draw the center of gravity of the electrons away from the nucleus. On the other hand, the attractive forces between the electrons and the nucleus tend to preserve a vanishing dipole moment in the atom. Consequently, an equilibrium situation is reached in which the atom bears a finite dipole moment. This has been represented schematically in Fig<\/span>.(1). The resulting dipole moment is thus induced by the field as a result of an elastic displacement of the electronic charge distribution relative to the nucleus. The induced moment<\/span><\/p>\r\n \r\n

may be represented by<\/span><\/p>\r\n \r\n

?<\/span>???<\/sub>=?<\/span>?<\/sub>?\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 .........(1)<\/span><\/p>\r\n \r\n

Where is called the electronic polarizability of the <\/span>atom. It should be noted that (1) is actually only the first term of a power series in the field strength. For the usual fields employed in dielectric measurements, however, (1) is a very good approximation.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

Figure (a) Schematic illustration of the displacement of the electron orbit relative to the nucleus for a hydrogen atom under influence of an external field E.<\/em><\/p>\r\n

(b)Simplified model for estimating the magnitude of the electronic polarizability of an atom, as described in the text.<\/p>\r\n \r\n

To obtain an idea of the magnitude of ,<\/em> consider the following simplified model; Suppose the atom is represented by a nucleus of charge Ze<\/em> and a homogeneous negative charge distribution inside a sphere\u00a0of radius <\/span>r.<\/em> If the nucleus is displaced over a distance <\/span>d,<\/em> under the influence of the field E as shown in Fig<\/span>.(2). So it can be shown through the laws of electrostatics that when a field E is applied to this atom, the nucleus is displaced from the <\/span>centre of <\/span>sphere by a distance<\/span><\/p>\r\n\r\n<\/div>\r\n

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

Where r is the radius of the sphere (the atomic radius), and Ze is the nuclear charge. the atom is thus polarized, and the dipole moment<\/p>\r\n \r\n\r\n?=?? ?\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 .........(3)\r\n\r\n \r\n\r\nPut the value of d from equation (2) we gets\r\n\r\n\"\"\r\n\r\n \r\n

Table 1. Electronic Polarizabilities for some inert gases and closed-Shell Alkali and Halogenic Ions (in units of 10\u221240 farad. meter2).<\/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\n \r\n

It will be evident that in general atoms with many electrons tend to have a larger polarizability than those with few electrons. Electrons in the outer electronic shells will contribute more to than do electrons in the inner shells, because the former are not so strongly bound to the nucleus as the latter. Positive ions therefore will have relatively small polarizabilities compared with the corresponding neutral atoms; for negative ions the reverse is true.<\/p>\r\n \r\n\r\n2.2 Ionic Polarizability:<\/em><\/strong>\r\n\r\n \r\n

So far, we have considered only simple atoms and ions. For molecules one is faced with two more possible influences of an external field:<\/p>\r\nI. Molecules may have permanent dipole moments which may be aligned in an external field.\r\n\r\n2. The distances between ions or atoms may be influenced by an external field.\r\n\r\n \r\n

For example, a molecule such as HCI may in first approximation be considered to consist of two ions; the permanent dipole moment is thus equal to the effective charge per ion times the separation of the ions. Symmetric molecules like <\/span>H2 ,CO2, CCl4, etc. evidently have no permanent dipole moment. An external electric field will tend to orient permanent dipoles along the field direction, and one speaks of orientational polarization.<\/span><\/p>\r\n \r\n

In molecules as well as in atoms an external field will displace the electrons with respect to the corresponding nuclei. Over and above this, however, a displacement of atoms or ions within the molecule may be caused by an external field. For example, in an HCI <\/span>molecule an external field will change the inter-ionic distance to some extent, leading to a change in the dipole moment. Similarly, in a molecule like CCl4 (which has no permanent dipole moment) a change in the bond angles between the CCl groups will produce a dipole moment because each of these groups by itself does have a dipole moment. This kind of induced polarization is called atomic or ionic polarization because it is a consequence of the displacement of atoms within the molecule. The induced electric dipole moment resulting from elastic displacements of ions within the molecule may again be represented by an expression of the type equation (5<\/span>) , by replacing by the atomic <\/span>polarizability .It should be noted that refers to an average over all possible orientations of the molecule with respect to the field. In <\/span>lecture (1) it will be shown that may be considered a constant up to frequencies in the infrared spectrum. For most molecules, is of the order of 10 <\/span>per cent of .<\/p>\r\n\r\n<\/div>\r\n

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

Figure 2: (a) <\/strong>A NaCl chain in the NaCl crystal without an applied field. Average or net dipole moment per ion is zero.(b)<\/strong> In the presence of an applied field, the ions become slightly displaced, which leads to a net average dipole moment per ion.<\/p>\r\n \r\n\r\n3.3 Orientational (Dipolar) Polarization:<\/em><\/strong>\r\n\r\n \r\n

Certain molecules possess permanent dipole moments. For example, the HC1 molecule shown in Figure 3 (a). a has a permanent dipole moment 0from the Cl- ion to the H+ ion. In the liquid or gas phases, these molecules, in the absence of an electric field, are randomly oriented as a result of thermal\u00a0agitation, as shown in Figure 3(b). When an electric field E is applied, E <\/span>ries to align the dipoles parallel to itself, as depicted in Figure 3(c). The Cl- and H+ charges experience forces in opposite directions. But the nearly rigid bond between Cl- and H+ holds them together, which means that the molecule experiences a torque r about its center of mass. This torque acts to rotate the molecule to align 0 with E.<\/span><\/p>\r\n \r\n

However, due to their thermal energy, the molecules move around randomly and collide with each other and with the walls of the container. These collisions destroy the dipole alignments. Thus the thermal energy tries to randomize the orientations of the dipole moments. A snapshot of the dipoles in the material in the presence of a field can be pictured as in Figure 3(d) in which the dipoles have different orientations. There is, nonetheless, a net average dipole moment per molecule that is finite and directed along the field. Thus the material exhibits net polarization, which leads to a dielectric constant that is determined by this orientational polarization.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

Figure 3. dipoles in the material in the presence of a field<\/p>\r\n \r\n

To find the induced average dipole moment along E, we need to know the average potential energy Edip of a dipole placed in a field E and how this compares with the average thermal energy 52 per molecule as in the present case of five degrees of freedom. Edip represents the average external work done by the field in aligning the dipoles with the field. If 52 is much greater than Edipthen the average thermal energy of collisions will prevent any dipole alignment with the field. If, however, Edip is much greater than 52 , then the thermal energy is insufficient to destroy the dipole alignments.<\/p>\r\n \r\n

A dipole at an angle ? to the field experiences a torque ? that tries to rotate it, as shown in Figure 3(c). Work done dW by the field in rotating the dipole by ?? is ? ??.<\/p>\r\n \r\n

(as in F dx). This work dW represents a small change dE in the potential energy of the dipole. No work is done if the dipole is already aligned with E, when ?=0, which\u00a0corresponds to the minimum in PE, On the other hand, maximum work is done when the torque has to rotate the dipole from ?=180\u00b0 to ?=0\u00b0 (either clockwise or counterclockwise, it doesn't matter).<\/p>\r\n \r\n

Let us define the potential energy of a dipole making a 90 angle with the external field as zero. The potential energy corresponding to an angle? between ? and E is then equal to<\/p>\r\n \r\n

? ?cos?=?.?\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0.........(6)<\/span><\/p>\r\n \r\n

According to statistical mechanics, the probability- for a dipole to make an angle between and with the electric field is then proportional to<\/span><\/p>\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n

Where is the solid angle between?and ?+??. Hence the average component of the dipole moment along the field direction is equal to<\/p>\r\n

\"\"<\/p>\r\n

The function <\/span>L(a)<\/em> is called the Langevin <\/span>function, since this formula was first derived by Langevin in 1905 in connection with the theory of paramagnetism. In Fig. 4<\/span>,L(a)<\/em> has been<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n

plotted as a function of\u00a0.<\/em>Note that for very large values of a, <\/em>i.e., for high field strengths, the function approaches the saturation value unity. This situation would correspond to complete alignment of the dipoles in the field direction, because then\u00a0 ?\u2329cos?\u232a=?.<\/p>\r\n \r\n

As long as the field strength is not too high and the temperature is Fig. 6-5. The Langevin function L(a).<\/em>not too low, the situation may be For a <\/em>< 1, the slope is 1\/3.strongly simplified by making the approximation a<\/em>1 or .EkT. <\/em>Under these circumstances the Langevin function L(a) <\/em>= a\/3, <\/em>so that then<\/em><\/p>\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n

<\/div>\r\n
\u00a0 \u00a0As an example of the condition implied in (10), consider a field of 3000 volts per cm. The dipole moment p of a molecule is of the order of 10-10to <\/span>esu of charge times 10-8 cm, i.e., about 10-18cgs units, so that .<\/span>E<\/em> 10-17 in cgs units. On the other hand, <\/span>kT<\/em> at room temperature is of the order of 10-14 erg\u00a0<\/span>and for this <\/span>example the condition is certainly satisfied. In this <\/span>example saturation would be approached only in the vicinity of 1\u00b0K. It may be noted that the quantum mechanical treatment of this problem leads essentially to the same results as obtained here.<\/span><\/div>\r\n
\r\n\r\n\"\"\r\n\r\n \r\n

Figure 4. The Langavin curve L(a) , for a\u226a 1 the slope is 1\/3.<\/p>\r\n \r\n\r\n2.4 The static dielectric constant of gases:<\/em><\/strong>\r\n\r\n \r\n

We are now in a position to give an atomic interpretation of the static dielectric constant of a gas. It will be assumed that the number of molecules per unit volume is small enough so that the interaction between them may be neglected. In that case, the field acting at the location of a\u00a0particular molecule is to a good approximation equal to the applied field E. Suppose the gas contains N molecules per unit volume; the properties of the molecules will be characterized by an electronic polarizability an ionic polarizability and a permanent dipole moment .<\/em> From the discussions in the preceding two sections it follows. that, as a result of the external field E,<\/em> there will exist a resulting dipole moment per unit volume or polarization is:<\/p>\r\n\r\n<\/div>\r\n

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

Note that only the permanent dipole moment gives a temperature dependent contribution, because and are essentially independent of T.<\/em> If the gas fills the space between two capacitor plates of area A<\/em> and separation d,<\/em> the total dipole moment between the plates will be equal to<\/p>\r\n \r\n\r\n??????<\/sub>=? ? ?\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 .........(12)\r\n\r\n \r\n

This simple relation shows immediately that the same total dipole moment would be obtained by assuming that the dielectric acquires an induced surface charge density <\/span>P<\/em> at the boundaries facing the capacitor plates. It represented the induced surface charge density at the dielectric-plate interface.\u00a0<\/span>Therefore, combination of (\u00a0\u00a0 \u2212 1) = 0 = and (1l) leads immediately to the Debye formula for the static dielectric constant of a gas<\/span><\/p>\r\n \r\n\r\n<\/div>\r\n

\"\"<\/div>\r\n
\r\n\r\n\u00a0 As an example of an application of this formula, we show in Fig.(5) the temperature dependence of some organic substances in the gaseous state. Note that (??\u22121) has been plotted versus the reciprocal of the absolute temperature, leading to straight lines, in agreement with formula (13). From the slope of the lines and knowledge of the number of moleculesper unit volume, the dipole moment ?may be obtained. Also, from the extrapolated intercept of the lines with the ordinate, one can calculate(??+??). The determination of dipole moments has contributed a great deal to our knowledge of molecular structure. For example, CCl4 and CH4, according to Fig.(5), do not possess permanent dipole moments, in agreement with the symmetric structure of these molecules. Similarly, the fact that H2O has a dipole moment of 1.84 Debye units, whereas CO2 has no dipole moment, indicates that the CO2 molecule has a linear structure, whereas in H2O the two OH bonds must make an angle different from 180o with each other.<\/span>\"\"\r\n\r\n \r\n

Figure5. Temperature variation of the static dielectric constant of some vapors<\/p>\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n

<\/div>\r\n
\u00a0 \u00a0 Summary:<\/em><\/strong><\/div>\r\n
    \r\n \t
  1. We considered mechanisms of polarization and determined polarizability<\/li>\r\n<\/ol>\r\n