Tensor transformation

Since a is an arbitrary vector, from the second relation it follows that an indifferent second-order tensor transforms as... 

Thus, the Tsai-Wu tensor failure criterion is obviously of more general character than the Tsai-Hill or Hoffman failure criteria. Specific advantages of the Tsai-Wu failure criterion include (1) invariance under rotation or redefinition of coordinates (2) transformation via known tensor-transformation laws (so data interpretation is eased) and (3) symmetry properties similar to those of the stiffnesses and compliances. Accordingly, the mathematical operations with this tensor failure criterion are well-known and relatively straightforward. 

This type of transformation will be used to assist in the definition of various orders of tensors. Each tensor will be defined on the basis of the type of transformation it satisfies. Tensors transform according to the relations... 

Second-rank tensors transform according to the expression... 

X, ..., X. Such a transformation induces a trivial tensor transformation for the instantaneous force Tip(t). We show in the Appendix, Section H, by evaluating the time and ensemble average of the instantaneous force over a short time interval, that, in the case of a nonsingular mobility matrix, such a transformation creates a transformed force bias... 

Note that the term involving a derivative of In / in Eq. (2.331) is identical to the velocity arising from the second term on the RHS of Eq. (2.286) for the transformed force bias in the traditional interpretation of the Langevin equation. The traditional interpretation of the Langevin equation yields a simple tensor transformation rule for the drift coefficient A , but also yields a contribution to Eq. (2.282) for the drift velocity that is driven by the force bias. The kinetic interpretation yields an expression for the drift velocity from which the term involving the force bias is absent, but, correspondingly, yields a nontrivial transformation mle for the overall drift coefficient. 

The only common factor is that the charge-current 4-tensor transforms in the same way. The vector representation develops a time-like component under Lorentz transformation, while the tensor representation does not. However, the underlying equations in both cases are the Maxwell-Heaviside equations, which transform covariantly in both cases and obviously in the same way for both vector and tensor representations. 

For a vibration to be Raman active, there must be a change in the pofonzability tensor. We need not go into the details of this bere,24 but merely note that the components of the polarizability tensor transform as the quadratic (unctions of x, y, and s. Therefore, in the character tables we are looking fbr x2, y2, r2, xy, x=, yz, or their combinations such as x2 — >>2. Because the irreducible representation fbr x2 is Al and that for yz is B2, all three vibrations of the water molecule are Raman active as well. 

A rank-two property tensor is diagonal in the coordinate system defined by its eigenvectors. Rank-two tensors transform like 3x3 square matrices. The general rule for transformation of a square matrix into its diagonal form is... 

By contrast, when the active space is restricted to the spin-only kets, the influence of all attainable excited states manifests itself in the filling of the MPs (tensors). In such a case the g-tensor deviates considerably from the free-electron value, the TIP appears substantial, and the spin-spin interaction tensor transforms to high values of the ZFS parameters (D and E). 

Guha and Chase reported that a Raman spectrum could be observed only in experiments that select a component of the polarizability tensor transforming as Ee or Ee [1]. The electronic matrix elements in these polarization geometries are [1]... 

C.2 Tensor Transformation Laws 1167 Then, if A is a second order tensor, or a dyad, the divergence of A is ... 

The values expected for pi in a few cases of special interest will now be discussed. First, in cubic symmetries, inspection of character tables shows that the trace of the scattering tensor transforms (like +y + z ) alone under the totally symmetric irreducible representation. Thus a totally symmetric vibrational mode will display pure isotropic scattering in this case with p = 0. As mentioned previously, no dispersion of Pi is expected. Explicitly, this is because the allowed electric-dipole transition moments are triply degenerate and thus a resonance effect does not alter the equivalence of the trace elements. Any vibronic coupling contributions are also equivalent for the three different polarisation directions when a totally symmetric mode is involved. 

The determination of polymer structure at the atomic level is possible by analyzing orientation-dependent NMR interactions such as dipole-dipole, quadrupole and chemical shielding anisotropy as mentioned above. The outline of the atomic coordinate determination for oriented protein fibers used here is described more fully in Ref. [30]. The chemical shielding anisotropy (CSA) interaction for N nucleus in an amide (peptide) plane can be interpreted with the chemical shielding tensor transformation as shown in Fig. 8.3. [31, 32]. 

When using Eq. (11.11), care must be taken to rotate the constituent atomic susceptibility tensors, with their natural principal axis system oriented at the bond axes, into the principal inertia axis system of the molecule. With the components of the susceptibility tensor transforming like the corresponding coordinate products [compare Eq. (1.4)], the appropriate transformation is given by ... 

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