Examples of Tensor Properties

In this section we will present a number of equilibrium properties of crystals such as the electric polarization, elasticity and piezoelectricity. Transport properties will not be covered in this book. 

In a vacuum, the electric displacement D is proportional to the electric field E, 

O- D is the charge density induced into a metal plate, or the charge density necessary to create the field = E in a condenser. 

P is the electric dipole moment per unit volume, P = P being the polarization charge per unit area normal to the vector P. D = 1 D 1 also expresses the charge density of a condenser required to maintain the field E at the interior of the dielectric (Fig. 4.11). 

COMMENT. The field E created in the dielectric by a homogeneous field E, generally depends on the shape of the dielectric. The dielectric creates a depolarization field Ej such that E = E, H- E. Clearly, for the plate represented in Fig. 4.11, these values are given by E = E, / , hence E = — ( — 1)E = — %E and P = — oE. Fora long bar of isotropic material whose axis is parallel to E, E = E, and Ej = 0 because the field is continuous across the vacuum/solid interface. In general, E is not homogeneous, even if E, is. 

---

The electric moments are examples of tensor properties the charge is a rank 0 tensor (which i the same as a scalar quantity) the dipole is a rank 1 tensor (which is the same as a vectoi with three components along the x, y and z axes) the quadrupole is a rank 2 tensor witl nine components, which can be represented as a 3 x 3 matrix. In general, a tensor of ran] n has 3" components. 

The dipole polarizability, the field gradient and the quadrupole moment are all examples of tensor properties. A detailed treatment of tensors is outside the scope of the text, but you should be aware of the existence of such entities. 

Einstein coefficient b, in (5) for viscosity 2.5 by a value dependent on the ratio between the lengths of the axes of ellipsoids. However, for the flows of different geometry (for example, uniaxial extension) the situation is rather complicated. Due to different orientation of ellipsoids upon shear and other geometrical schemes of flow, the correspondence between the viscosity changed at shear and behavior of dispersions at stressed states of other types is completely lost. Indeed, due to anisotropy of dispersion properties of anisodiametrical particles, the viscosity ceases to be a scalar property of the material and must be treated as a tensor quantity. 

However, perpendicular to the pyridine planes, a small positive n interaction was unambiguously defined 50 cm 1 < e < 130 cm"1. Any value for e outside this range, produced hopelessly inadequate descriptions of the magnetic properties, and it is certainly this small amount of n interaction that is responsible for the low symmetry of the ligand field and the unusual orientation of the susceptibility tensor. This is an example of a sharply fitting parameter. 

In order to apply group theory to the physical properties of crystals, we need to study the transformation of tensor components under the symmetry operations of the crystal point group. These tensor components form bases for the irreducible repsensentations (IRs) of the point group, for example x, x2 x3 for 7(1) and the set of infinitesimal rotations Rx Ry Rz for 7(1 )ax. (It should be remarked that although there is no unique way of decomposing a finite rotation R( o n) into the product of three rotations about the coordinate axes, infinitesimal rotations do commute and the vector o n can be resolved uniquely... 

Since the intrinsic lineshape has finite width, the experimentally observed lineshape is the convolution of 1(f) with one of the lineshape functions g(f). A powder lineshape for an axially symmetric chemical shielding tensor is shown in Fig. 4 and a typical example of a general powder lineshape is shown in Fig. 5. Many systems yield lineshapes close to that of a powder pattern and the mathematical properties of these lineshapes are discussed in detail by Alexander et al.iA... 

Most engineering materials are isotropic in nature, and thus they have the same properties in all directions. For such materials we do not need to be concerned about the variation of properties with direction. But in anisotropic materials such as (he fibrous or composite materials, (he properties may change with direction. For example, some of the properties of wood along the grain are different than those in (he direction normal to the grain. In such cases the thermal conductivity may need to be expressed as a tensor quantity to account for the variation with direction. I he treatment of such advanced topics is beyond the scope of tlus text, and we will assume the thermal conductivity of a material to be independent of direction. 

As for all second-rank tensor properties, the orientational variation in chemical shift can be visualized as an ellipsoidal surface for which the symmetry and alignment of the principal axes must conform to the point symmetry at the atom position. For example. 

The data points are fitted in a least-square sense to a fourth degree polynomial, and the properties thereby obtained are presented in Table 3. Since the atom possesses spherical symmetry there is only a single independent component of the a-tensor as well as the y-tensor. The curvature of the energy, or the polarizability, at the SCF level differs by less than 5% compared to the FCI result, and the MP2 value captures slightly more than half of the correlation effect. Electron correlation plays a more important role in the determination of the fourth-order property y. Again the MP2 method captures slightly more than half of the total contribution, which amounts to 21% at the FCI level of theory. The trends we have seen here in the example of the helium atom are more or less representative for closed-shell molecules in general. 

Section 4.1.1 reviews second harmonic generation (SHG) for para-nitroaniline (PNA), Section 4.1.2 the polarizability and second hyperpolarizability of nitrogen and benzene, Section 4.1.3 the second hyperpolarizability of Cgo, Section 4.2 the excited state polarizability of pyrimidine and r-tetrazine. Section 4.3 three-photon absorption, and finally, in Section 4.5 the electronic g-tensor and the hyperfine coupling tensor are reviewed as examples of open shell DFT response properties. 

What this equation tells us is that a particular state of stress is nothing more than a linear combination (albeit perhaps a tedious one) of the entirety of components of the strain tensor. The tensor Cijn is known as the elastic modulus tensor or stiffness and for a linear elastic material provides nearly a complete description of the material properties related to deformation under mechanical loads. Eqn (2.52) is our first example of a constitutive equation and, as claimed earlier, provides an explicit statement of material response that allows for the emergence of material specificity in the equations of continuum dynamics as embodied in eqn (2.32). In particular, if we substitute the constitutive statement of eqn (2.52) into eqn (2.32) for the equilibrium case in which there are no accelerations, the resulting equilibrium equations for a linear elastic medium are given by... 

It is clear that a consequence of the tensor equation for the conductivity (4.2) is that E and J are not necessarily parallel, and that the absolute value J 1 is a function of the direction of E. The property is thus anisotropic. Isotropy is characterized by the equation Oij = S a. Figure 4.1 shows an example of a tensor relationship. 

---