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The physical properties discussed thus far are linear relationships between two measured quantities. This is only an approximation to the truth, and often not a very good approximation, especially for materials near a phase transformation. A more accurate description can be obtained by introducing higher order coefficients. To illustrate nonlinearity we discuss electrostriction, magnetostriction, and higher order elastic, and dielectric effects. These phenomena are described in terms of fourth and sixth rank tensors. Many of the recent innovations in the field of electroceramics have exploited the nonlinearities of material properties with factors such as electric field, mechanical stress, temperature, or frequency. The nonlinear dielectric behavior of ferroelectric ceramics (Fig. 15.1), for example, has opened up new markets in electronics and communications. In these materials the electric polarization saturates under high fields. Electric displacement Di varies with applied electric field Ej as … Di = εijEj + εijkEjEk + εijklEjEkEl +· · ·, … where εij is the dielectric permittivity and εijk and εijkl are higher order terms. The data in Fig. 15.1 were collected for a relaxor ferroelectric in its paraelectric state above Tc where the symmetry is centrosymmetric. Therefore the third rank tensor εijk is zero, and the shape of the curve is largely controlled by the first and third terms. For cubic crystals, the fourth rank tensor εijkl is similar in form to the elastic constants discussed in Chapter 13. Tunable microwave devices utilize nonlinear dielectrics in which the polarization saturates as in Fig. 15.1. By applying a DC bias the dielectric constant can be adjusted over a wide range.

The Lorentz force that a magnetic field exerts on a moving charge carrier is perpendicular to the direction of motion and to the magnetic field. Since both electric and thermal currents are carried by mobile electrons and ions, a wide range of galvanomagnetic and thermomagnetic effects result. The effects that occur in an isotropic polycrystalline metal are illustrated in Fig. 20.1. As to be expected, many more cross-coupled effects occur in less symmetric solids. The galvanomagnetic experiments involve electric field, electric current, and magnetic field as variables. The Hall Effect, transverse magnetoresistance, and longitudinal magnetoresistance all describe the effects of magnetic fields on electrical resistance. Analogous experiments on thermal conductivity are referred to as thermomagnetic effects. In this case the variables are heat flow, temperature gradient, and magnetic field. The Righi–Leduc Effect is the thermal Hall Effect in which magnetic fields deflect heat flow rather than electric current. The transverse thermal magnetoresistance (the Maggi–Righi–Leduc Effect) and the longitudinal thermal magnetoresistance are analogous to the two galvanomagnetic magnetoresistance effects. Additional interaction phenomena related to the thermoelectric and piezoresistance effects will be discussed in the next two chapters. In tensor form Ohm’s Law is …Ei = ρijJj, … where Ei is electrical field, Jj electric current density, and ρij the electrical resistivity in Ωm. In describing the effect of magnetic field on electrical resistance, we expand the resistivity in a power series in magnetic flux density B. B is used rather than the magnetic field H because the Lorentz force acting on the charge carriers depends on B not H.