*Robert E. Newnham*

- Published in print:
- 2004
- Published Online:
- November 2020
- ISBN:
- 9780198520757
- eISBN:
- 9780191916601
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198520757.003.0017
- Subject:
- Earth Sciences and Geography, Geochemistry

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 ...
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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.
Less

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.

*Robert E. Newnham*

- Published in print:
- 2004
- Published Online:
- November 2020
- ISBN:
- 9780198520757
- eISBN:
- 9780191916601
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198520757.003.0021
- Subject:
- Earth Sciences and Geography, Geochemistry

The phenomenon of atomic and ionic migration in crystals is called solidstate diffusion, and its study has shed light on many problems of technological and scientific importance. Diffusion is ...
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The phenomenon of atomic and ionic migration in crystals is called solidstate diffusion, and its study has shed light on many problems of technological and scientific importance. Diffusion is intimately connected to the strength of metals at high temperature, to metallurgical processes used to control alloy properties, and to many of the effects of radiation on nuclear reactor materials. Diffusion studies are important in understanding the ionic conductivity of the materials used in fuel cells, the fabrication of semiconductor integrated circuits, the corrosion of metals, and the sintering of ceramics. When two miscible materials are in contact across an interface, the quantity of diffusing material which passes through the interface is proportional to the concentration gradient. The atomic flux J is given by where J is measured per unit time and per unit area, c is the concentration of the diffusing material per unit volume, and Z is the gradient direction. The proportionality factor D, the diffusion coefficient, is measured in units of m2/s. This equation is sometimes referred to as Fick’s First Law. It describes atomic transport in a form that is analogous to electrical resistivity (Ohm’s Law) or thermal conductivity. There are several objections to Fick’s Law, as discussed in Section 19.5. Strictly speaking, it is valid only for self-diffusion coefficients measured in small concentration gradients. Since J and Z are both vectors, the diffusion coefficient D is a second rank tensor. As with other symmetric second rank tensors, between one and six measurements are required to specify Dij, depending on symmetry. The relationship between structure and anisotropy is more apparent in PbI2. Lead iodide is isostructural with CdI2 in trigonal point group.m. The self-diffusion of Pb is much easier parallel to the layers where the Pb atoms are in close proximity to one another. Diffusion is more difficult along Z3 = [001] because Pb atoms have a very long jump distance in this direction. The mineral olivine, (Mg, Fe)2SiO4, is an important constituent of the deeper parts of the earth’s crust.
Less

The phenomenon of atomic and ionic migration in crystals is called solidstate diffusion, and its study has shed light on many problems of technological and scientific importance. Diffusion is intimately connected to the strength of metals at high temperature, to metallurgical processes used to control alloy properties, and to many of the effects of radiation on nuclear reactor materials. Diffusion studies are important in understanding the ionic conductivity of the materials used in fuel cells, the fabrication of semiconductor integrated circuits, the corrosion of metals, and the sintering of ceramics. When two miscible materials are in contact across an interface, the quantity of diffusing material which passes through the interface is proportional to the concentration gradient. The atomic flux J is given by where J is measured per unit time and per unit area, c is the concentration of the diffusing material per unit volume, and Z is the gradient direction. The proportionality factor D, the diffusion coefficient, is measured in units of m2/s. This equation is sometimes referred to as Fick’s First Law. It describes atomic transport in a form that is analogous to electrical resistivity (Ohm’s Law) or thermal conductivity. There are several objections to Fick’s Law, as discussed in Section 19.5. Strictly speaking, it is valid only for self-diffusion coefficients measured in small concentration gradients. Since J and Z are both vectors, the diffusion coefficient D is a second rank tensor. As with other symmetric second rank tensors, between one and six measurements are required to specify Dij, depending on symmetry. The relationship between structure and anisotropy is more apparent in PbI2. Lead iodide is isostructural with CdI2 in trigonal point group.m. The self-diffusion of Pb is much easier parallel to the layers where the Pb atoms are in close proximity to one another. Diffusion is more difficult along Z3 = [001] because Pb atoms have a very long jump distance in this direction. The mineral olivine, (Mg, Fe)2SiO4, is an important constituent of the deeper parts of the earth’s crust.