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Navbar Search Filter Mobile Enter search term Search Navbar Search Filter Enter search term SearchThe physical and chemical properties of crystals and textured materials often depend on direction. An understanding of anisotropy requires a mathematical description together with atomistic arguments to quantify the property coefficients in various directions. Tensors and matrices are the mathematics of choice and the atomistic arguments are partly based on symmetry and partly on the basic physics and chemistry of materials. These are subjects of this book: tensors, matrices, symmetry, and structure–property relationships. We begin with transformations and tensors and then apply the ideas to the various symmetry elements found in crystals and textured polycrystalline materials. This brings in the 32 crystal classes and the 7 Curie groups. After working out the tensor and matrix operations used to describe symmetry elements, we then apply Neumann’s Law and the Curie Principle of Symmetry Superposition to various classes of physical properties. The first group of properties is the standard topics of classical crystal physics: pyroelectricity, permittivity, piezoelectricity, elasticity, specific heat, and thermal expansion. These are the linear relationships between mechanical, electrical, and thermal variables as laid out in the Heckmann Diagram (Fig. 1.1). These standard properties are all polar tensors ranging in rank from zero to four. Axial tensor properties appear when magnetic phenomena are introduced. Magnetic susceptibility, the relationship between magnetization and magnetic field, is a polar second rank tensor, but the linear relationships between magnetization and thermal, electrical, and mechanical variables are all axial tensors. As shown in Fig. 1.2, magnetization can be added to the Heckmann Diagram converting it into a tetrahedron of linear relationships. Pyromagnetism, magnetoelectricity, and piezomagnetism are the linear relationships between magnetization and temperature change, electric field, and mechanical stress. Examples of tensors of rank zero through four are given in Table 1.1. In this book we will also treat many of the nonlinear relationships such as magnetostriction, electrostriction, and higher order elastic constants. The third group of properties is transport properties that relate flow to a gradient. Three common types of transport properties relate to the movement of charge, heat, and matter. Electrical conductivity, thermal conductivity, and diffusion are all polar second rank tensor properties.
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