Second-Rank Tensors

Diamagnetic shielding tensor Second rank tensor -1 -2... 

The electrostriction coefficient is a fourth-rank tensor because it relates a strain tensor (second rank) to the various cross-products of the components of E or D in the. v, y and z directions. 

Second derivatives of the energy with respect to the elements of a uniform electric field, 14,14, and 14, make up a tensor (second rank matrix) called the dipole polarizability, a. 

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. 

In contrast, the second term in (4.6) comprises the full orientation dependence of the nuclear charge distribution in 2nd power. Interestingly, the expression has the appearance of an irreducible (3 x 3) second-rank tensor. Such tensors are particularly convenient for rotational transformations (as will be used later when nuclear spin operators are considered). The term here is called the nuclear quadrupole moment Q. Because of its inherent symmetry and the specific cylindrical charge distribution of nuclei, the quadrupole moment can be represented by a single scalar, Q (vide infra). 

Raman spectroscopy is an inelastic light scattering experiment for which the intensity depends on the amplitude of the polarizability variation associated with the molecular vibration under consideration. The polarizability variation is represented by a second-rank tensor, oiXyZ, the Raman tensor. Information about orientation arises because the intensity of the scattered light depends on the orientation of the Raman tensor with respect to the polarization directions of the electric fields of the incident and scattered light. Like IR spectroscopy, Raman... 

This matrix describes the transformation from x y z to xyz as a rotation about the z axis over angle a, followed by a rotation about the new y" axis over angle /), followed by a final rotation over the new z " axis over angle y (Watanabe 1966 148). Formally, the low-symmetry situation is even a bit more complicated because the nondiagonal g-matrix in Equation 8.11 is not necessarily skew symmetric (gt] -g. Only the square g x g is symmetric and can be transformed into diagonal form by rotation. In mathematical terms, g x g is a second-rank tensor, and g is not. 

Second rank (bilinear after pseudospin) zero-field splitting tensor Dap (i.e. the conventional D tensor), its main values and the main anisotropy axes (Xa,Ya,Za). 

The products of second-rank tensor components, such as A1-, j ( P/ JA1-, j (0Fi), can be expressed in coupled form using the Clebsch-Gordan coefficients, yielding tensor terms, of spatial rank / = 0, 2, and 4 ... 

The EFG is a second-rank tensor with the principal components, Vxx, Vyy, Vzz. The quadrupole coupling constant is given by ... 

These ideas may be extended to define tensors of any rank. There are three varieties of second rank tensors, defined by the transformations... 

They are called contravariant, covariant and mixed tensors, respectively. A useful mixed tensor of the second rank is the Kronecker delta... 

Hooke s law is the relationship between strain and stress second rank tensors ... 

In Equation 6, the dlffuslvlty and mobility are second rank tensors whose positional dependence is a consequence of the hydrodynamic wall effect and F represents the probabllllty that the Brownian particle, initially at some fixed point, will be at some position in space R at a later time t. At low concentrations, P is replaced by the number concentration, C (25). Conceptually the approach followed is similar to that developed by Brenner and Gaydos (25), however, one needs to include an expression for the flux of particles at the wall due to exchange with the pores. Upon averaging over the interstitial tube cross section of Figure 2, one arrives at the following expression (29) for the area averaged rate equation for the mobile phase transport. 

The tensor quantities given in this chapter are all second rank, and are sometimes referred to as matrices, according to common usage, so that the two terms, tensor and matrix, are used interchangeably. In many cases, the components (or coefficients) of second-rank tensors are represented by 3 x 3 matrices. Symbols for tensors (matriees) are printed in bold italic type, while symbols for the components are printed in italic type. In general, the base tensors are those for a rectangular Cartesian coordinate system. 

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