Polarisability relative permittivity

The written set of equations has a simple solution for the components of the polarisation vector. We use them to write, in accordance to equation (10.1), the relative permittivity tensor... 

The value of the refractive index n of light in the anisotropic medium depends on the direction of propagation s and on the direction of the polarisation of the light. For the given relative permittivity tensor ji, the refractive index can be determined from the relation (Born and Wolf 1970 Landau et al. 1987)... 

This is the fundamental electric field equation that applies at any point in an isotropic medium. In this context the quantity e0e is the absolute permittivity of the material, and the ratio e, which we have called the dielectric constant of the material, is more properly termed the relative permittivity (with respect to the absolute permittivity of free space e0) and we shall use this term. The flux of dielectric displacement begins and ends on free charge and otherwise is continuous, even at an interface between two media. Electric field, on the other hand, is discontinuous at an interface between two different materials as a result of the different degrees of polarisation. 

Figure 2.4 depicts the characteristic stepwise fall in polarisation of a material as the measurement frequency is raised, rendering it impossible for preceding components of molecular polarisation to make their contribution. The relative permittivity follows a similar pattern. 

In contrast to molar polarisation calculated from optical refractivities, that calculated from relative permittivities observed at lower frequencies is by no means always independent of temperature. Actually, materials tend to fall into one of two classes. Those in one class show a relatively constant molar polarisation in accord with the simple Clausius-Mosotti relation, whilst the members of the other class, which contains materials with high relative permittivities, show a molar polarisation that decreases with increase in temperature. Debye recognised that permanent molecular dipole moments were responsible for the anomalous behaviour. From theories of chemical bonding we know that certain molecules which combine atoms of different electronegativity are partially ionic and consequently have a permanent dipole moment. Thus chlorine is highly electronegative and the carbon-chlorine... 

The situation is much more complicated in solids because the intermolecular effects can no longer be ignored, i.e. the approximation EM = 0 inherent in the simple formula for the local field (2.29) is not generally true. Consequently, although we can predict the molecular dipole moment from known group moments, it is not possible to calculate the molar polarisation and thereby the relative permittivity, without further elaboration of the dielectric model. In the case of a polymer there are further complications which arise from the flexibility of the long chains. 

Conjugated polymers, which contain multiple carbon-carbon bonds, can be expected to have somewhat higher relative permittivities, since the bond polarisabilities of multiple bonds are higher than those of single C-C bonds, see Table 2.2. Most measurements of relative permittivities have been made on the conductive forms of these polymers, but these are outside the scope of this chapter and will be discussed later. The intrinsic energy gap in the... 

An increase in fractional free volume will reduce the number of polarisable groups per unit volume, and thereby reduce the relative permittivity of the polymer. Quantitatively, the effect may be estimated by means of the Clausius-Mossotti/Lorenz-Lorentz model for dielectric mixing (Bottcher, 1978) ... 

Generally, we must conclude that control of free volume is just as important as selection of groups with low polarisability, in order to achieve polymer molecular structures with low relative permittivities. 

Polarisation effects at electrodes become most prominent when the material of a specimen shows some appreciable bulk conductivity. Characteristically, there is an apparent increase in the relative permittivity at low frequencies. The anomaly originates in a high-impedance layer on the electrode surface. This may be caused by imperfect contact between the metal electrode and the specimen, aggravated by the accumulation of the products of electrolysis, etc. At low frequencies there is sufficient time for any slight conduction through the specimen to transfer the entire applied field across the very thin electrode layers, and the result is an enormous increase in the measured capacitance. For a purely capacitive impedance Ce at the electrodes, in.series with the specimen proper (geometrical capacitance C0), Johnson and Cole (1951) showed that the apparent relative permittivity takes the approximate form ... 

To relate polarisability, a, to the relative permittivity, it is necessary to remember that each... 

Using Equations (11.3)-(11.5) it is possible to derive the most widely used relationship between relative permittivity and polarisability, the Clausius-Mossotti relation. Equation (11.6), usually written ... 

As the total polarisability of a material, a, is made up several contributions, the relative permittivity, e, can also be thought of as made up from the same contributions. In a static electric field, all the various contributions wiU be important, and both a and r wiU arise from electrons, ions, dipoles, defects and surfaces. However, if a variable, especially alternating, electric field acts on the solid the situation changes. 

This relationship as such is not well obeyed for most compounds if the static or low-frequency relative permittivity is used, as can be judged from Table 11.1. The relationship can be correctly interpreted by using the relative permittivity due to electronic polarisation in the equation. With this in mind, substitution of the relationship given in Equation (11.10) into the Clausius-Mossotti equation yields the Lorentz-Lorenz equation ... 

The relative permittivity (as well as the refractive index) for such crystals is quoted as three values corresponding to the polarisations projected onto the axes. Some representative values are given in Table 11.1. 

It may sometimes be necessary to estimate the polarisability of a solid in the absence of experimental data. Polarisability is not particularly easy to measure, but the relative permittivity is. The Clau-sius-Mossotti equation. Equation (11.6), is generally used to obtain polarisability from relative permittivity. The equation gives reasonable values for isotropic solids showing only ionic and electronic polarisation. If the refractive index is known, the... 

Vaiies with crystal direction. Uj r and lower values are given when these differ substantially. Note 7c, Curie temperature P, spontaneous polarisation r, relative permittivity. 

The fact that an applied field can cause the polarisation to alter its direction implies that the atoms involved make only small movements and that the energy barrier between the different states is low. With increasing temperature the thermal motion of the atoms will increase, and eventually they can overcome the energy barrier separating the various orientations. Thus at high temperatures the distribution of atoms becomes statistical and the crystal behaves as a normal dielectric and no longer as a polar material. This is referred to as the paraelectric state. The temperature at which this occurs is known as the Curie temperature, Tc, or the transition temperature. The relative permittivity often rises to a sharp peak in the neighbourhood of Tb. 

Some crystals, however, exhibit relative permittivity values many orders of magnitude higher than found in normal dielectrics. By analogy with magnetic behaviour, this behaviour is called ferroelec-tricity, and the materials are called ferroelectrics. Ferroelectrics also possess a spontaneous polarisation, Ps, in the absence of an electric field and a mechanical distortion. They are, therefore, a subset of pyroelectrics and, as such, all ferroelectrics are also pyroelectrics and piezoelectrics. The feature that distinguishes ferroelectrics from pyroelectrics is that the direction of the spontaneous polarisation, Ps, can be switched (changed) in an applied electric field. 

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