Some Tensor Properties

For cubic crystals, which iaclude sUicon, properties described by other than a zero- or a second-rank tensor are anisotropic (17). Thus, ia principle, whether or not a particular property is anisotropic can be predicted. There are some properties, however, for which the tensor rank is not known. In addition, ia very thin crystal sections, the crystal may have two-dimensional characteristics and exhibit a different symmetry from the bulk, three-dimensional crystal (18). Table 4 is a listing of various isotropic and anisotropic sUicon properties. Table 5 gives values for the more common physical properties and for some of the thermodynamic properties. Figure 5 shows some thermal properties. 

Characterization of Molecular Hyperpolarizabilities Using Third Harmonic Generation. Third harmonic generation (THG) is the generation of light at frequency 3co by the nonlinear interaction of a material and a fundamental laser field at frequency co. The process involves the third-order susceptibility x 3K-3 , , ) where —3 represents an output photon at 3 and the three s stand for the three input photons at . Since x(3) is a fourth (even) rank tensor property it can be nonzero for all material symmetry classes including isotropic media. This is easy to see since the components of x(3) transform like products of four spatial coordinates, e.g. x4 or x2y2. There are 21 components that are even under an inversion operation and thus can be nonzero in an isotropic medium. Since some of the terms are interrelated there are only four independent terms for the isotropic case. 

Similarly as the trace, the anisotropy of the polarizability tensor of diatomic colli-sional systems can also be related to some macroscopic properties, namely to the refractive properties of atomic gases. The so-called Kerr constant, the anisotropy of the refractive index in the parallel and perpendicular directions to the external static electric field is given by,... 

Over the past few years, a number of approaches have emerged for the interpretation of RDCs in terms of carbohydrate structure. However, a key feature of all these methods is that they require the alignment tensor to be determined. Order matrix analysis uses experimental RDCs, while some molecular properties, such as molecular shape or mass distribution provide the alignment tensor a priori without the need for the experimental RDCs. 

Let us now derive phenomenological equations of the kind (5.193) corresponding to the expression (5.205). As has been mentioned before, each flux is a linear function of all thermodynamic forces. However the fluxes and thermodynamic forces that are included in the expression (5.205) for the dissipative function, have different tensor properties. Some fluxes are scalars, others are vectors, and the third one represents a second rank tensor. This means that their components transform in different ways under the coordinate transformations. As a result, it can be proven that if a given material possesses some symmetry, the flux components cannot depend on all components of thermodynamic forces. This fact is known as Curie s symmetry principle. The most widespread and simple medium is isotropic medium, that is, a medium, whose properties in the equilibrium conditions are identical for all directions. For such a medium the fluxes and thermodynamic forces represented by tensors of different ranks, cannot be linearly related to each other. Rather, a vector flux should be linearly expressed only through vectors of thermodynamic forces, a tensor flux can be a liner function only of tensor forces, and a scalar flux - only a scalar function of thermodynamic forces. The said allows us to write phenomenological equations in general form... 

Since the expressions (8.5) and (8.15) are identical, they can be combined, by introducing so-called global tensors of friction f and mobility V, including translational and rotational components. In the Stokes flow, these tensors have some universal properties [2], of which the most important are dependence on instant configuration and independence of velocity, as well as symmetry and positive definiteness of matrixes fj and V. ... 

The second-order tensors are characterized by three invariants, that is, it is possible to combine the nine components in three ways to get quantities that are independent of the coordinate systems and express some fundamental properties. For the siuface component there are only two such invariants. The first of these can be written as tr (xj, where tr is short for trace and the operation that sums the diagonal elements of the tensor. The second is l/2 [tr (xj] - XjiXj, where a double dot product has been introduced. Since the trace itself is an invariant, some authors drop this term from the second invariant. In addition, the second invariant of this symmetric siuface tensor is the same as the third invariant in three dimensions, which is the determinant of x (see the remark after Equation 7.E1.8). There is a very important second-order surface tensor in the form of... 

The multipole tensors (2.1.1) have some important properties. One of them... 

Many properties of interest for liquid crystals are second rank tensors, and these have some special properties, since they... 

Taking into account certain restrictions originating from the symmetry properties of isotropic liquids at equilibrium (Curie s theorem), after some tensor algebra we obtain the Navier-Stokes equations for single-component atomic fluids. 

It is assumed that the reader is familiar with some basic properties of Cartesian tensors, such as those that may be found in the book by Spencer [256]. In this book we shall define the divergence of a second order tensor Tij to be the tensor... 

Some Crystal Properties Represented by Polar Tensors... 

In order to consider the inelastic stress rate relation (5.111), some assumptions must be made about the properties of the set of internal state variables k. With the back stress discussed in Section 5.3 in mind, it will be assumed that k represents a single second-order tensor which is indifferent, i.e., it transforms under (A.50) like the Cauchy stress or the Almansi strain. Like the stress, k is not indifferent, but the Jaumann rate of k, defined in a manner analogous to (A.69), is. With these assumptions, precisely the same arguments... 

It is possible to assume other transformation properties for k. For example, for some purposes it may be more desirable to attribute strainlike properties obeying a transformation law like (A. 19), in which case the equations of this section will take a somewhat different form. Of course, k may be taken to be comprised of a number of such tensors, and it is not difficult to extend the theory to include a number of indifferent scalars and vectors, if desired. 

The mathematical operations in the study of mechanics of composite materials are strongly dependent on use of matrix theory. Tensor theory is often a convenient tool, although such formal notation can be avoided without great loss. However, some of the properties of composite materials are more readily apparent and appreciated if the reader is conversant with tensor theory. 

We begin by describing the HPP model, which satisfies all of the above requirements except for the isotropy of the momentum flux density tensor. As we shall, however, this early model nonetheless has some very interesting and suggestive properties, despite not being able to reproduce Navier-Stokes-like behavior exactly. 

It is evident that methods analogous to the ones developed here could be applied to molecular properties which, instead of being pseudoscalar, belong to some other representation of the skeleton point group (vector, tensor, etc. properties). To treat such properties, one needs only to induce from a different representation of than the chiral one. 

In order to extract some more information from the csa contribution to relaxation times, the next step is to switch to a molecular frame (x,y,z) where the shielding tensor is diagonal (x, y, z is called the Principal Axis System i.e., PAS). Owing to the properties reported in (44), the relevant calculations include the transformation of gzz into g x, yy, and g z involving, for the calculation of spectral densities, the correlation function of squares of trigonometric functions such as cos20(t)cos20(O) (see the previous section and more importantly Eq. (29) for the definition of the normalized spectral density J((d)). They yield for an isotropic reorientation (the molecule is supposed to behave as a sphere)... 

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