Tensor trace

According to convention, if the three diagonal elements in Eq. (9) are arranged in a decreasing numerical order, these are called <5n, <522, and 533 (<5n > 822 > 833), respectively. The isotropic chemical shift <5jS0 is defined as the average of the tensor trace ... 

In addition, we used Bader s atoms-in-molecules (AIM) theory [56,57] to help analyze some of the results. For convenience, we give here a very brief overview of this approach. According to the AIM theory, every chemical bond has a bond critical point at which the first derivative of the charge density, p(r), is zero. The (> r) topology is described by a real, symmetric, second-rank Hessian-of-/3(r) tensor, and the tensor trace is related to the bond interaction energy by a local expression of the virial theorem ... 

On the contrary, in the case of laboratory investigations on rock specimens under uniaxial or triaxial load, volume changes in the source play an important role. Dilatancy can be explained as volume expansion caused by tensile opening. In contrast to the fault-plane solution method, the more complex moment tensor method is capable of describing sources with volumetric components like tensile cracks, deviatoric sources like shear cracks, or a mixture of both source types. The volumetric source components can be easily obtained using the isotropic part, or one-third of the moment tensor trace. With the moment tensor method, the source mechanisms are estimated in a least-squares inversion calculation from amplitudes of the first motion as well as from full waveforms of P and S waves. This method requires additional knowledge about the transfer function of the medium (the so-called Green s function) and sensor response. 

Fig. 4(a) FEG-SEM micrograph of the investigated polycrystalline alumina microstructure, (b) map from EBSD analysis (c) CL/PS residual stress (tensor trace) map. 

As implied by the trace expression for the macroscopic optical polarization, the macroscopic electrical susceptibility tensor at any order can be written in temis of an ensemble average over the microscopic nonlmear polarizability tensors of the individual constituents. 

Clearly, /, d, and w are spatial tensors with components relative to the current configuration. Since the trace of an antisymmetric tensor vanishes, from (A.9)... 

Such a matrix is not independent on the coordinate system, but the trace is. Cioslowski has proposed a definition of atomic charges as one-third of the trace over the APT, denoted Generalized Atomic Polar Tensor (GAPT) charges.The charge on atom A is defined as... 

Isotropy of the Momentum Flux Density Tensor If we trace back our derivation of the macroscopic LG Euler s and Navier-Stokes equations, we see that the only place where the geometry of the underlying lattice really enters is through the form for the momentum flux density tensor, fwhere cp = x ) + y ), k = 1,..., V... 

By this time Polya s Theorem had become a familiar combinatorial tool, and it was no longer necessary to explain it whenever it was used. Despite that, expositions of the theorem have continued to proliferate, to the extent that it would be futile to attempt to trace them any further. I take space, however, to mention the unusual exposition by Merris [MerRSl], who analyzes in detail the 4-bead 3-color necklace problem, and interprets it in terms of symmetry classes of tensors — an interpretation that he has used to good effect elsewhere (see [MerRSO, 80a]). 

By Eq. (1-55), we have px — m Jujaf dv. Since mut is the random momentum in the -direction (. e., the momentum associated with the -component of the random velocity), the (i,j) component of the pressure tensor is the average of the random flow in the -direction of the -directed momentum. From the definition of the temperature, Eq. (1-45), the hydrostatic pressure, defined as one-third of the trace of the pressure tensor, is... 

The trace vanishes because only p- and /-electrons contribute to the EFG, which have zero probability of presence at r = 0 (i.e. Laplace s equation applies as opposed to Poisson s equation, because the nucleus is external to the EFG-generating part of the electronic charge distribution). As the EFG tensor is symmetric, it can be diagonalized by rotation to a principal axes system (PAS) for which the off-diagonal elements vanish, = 0. By convention, the principal axes are chosen such that... 

Notice that the fine structure term found here has the same form (and the tensor is given the same symbol) as that obtained from the electron dipolar interaction. Unlike the dipolar D-tensor, however, the spin-orbit coupling D-tensor in general does not have zero trace. Nonetheless, we introduce analogous parameters ... 

The isotropic chemical shift, the trace of the chemical shift tensor, is one of the basic NMR parameters often measured for both spin-1/2 and quadrupolar nuclei. The CSA can also be measured in non-cubic environments, such as the n3Cd nuclei experience in the chalcopyrite structure of crystalline CdGeAs2 [141] or CdGeP2 [142], and the 31P nuclei in the latter compound [142], Although the isotropic chemical shift can be measured from the NMR spectrum of a static powder because the CSA is zero in many cases because of cubic symmetry of the lattice, improved resolution is obtained by using MAS to remove dipolar couplings. Two particular areas where the isotropic chemical shifts have proven very informative will now be discussed, semiconductor alloys and semiconductor polytypes. 

In the real world the stress tensor never vanishes and so requires a nonvanishing curvature tensor under all circumstances. Alternatively, the concept of mass is strictly undefined in flat Minkowski space-time. Any mass point in Minkowski space disperses spontaneously, which means that it has a space-like rather than a time-like world line. In perfect analogy a mass point can be viewed as a local distortion of space-time. In euclidean space it can be smoothed away without leaving any trace, but not on a curved manifold. Mass generation therefore resembles distortion of a euclidean cover when spread across a non-euclidean surface. A given degree of curvature then corresponds to creation of a constant quantity of matter, or a constant measure of misfit between cover and surface, that cannot be smoothed away. Associated with the misfit (mass) a strain field appears in the curved surface. 

Fig. 10.20. Theoretical spectral patterns for NMR of solid powders. The top trace shows the example of high symmetry, or cubic site symmetry. In this case, all three chemical shift tensor components are equal in value, a, and the tensor is best represented by a sphere. This gives rise to a single, narrow peak. In the middle trace, two of the three components are equal, so the tensor is said to have axial site symmetry. This tensor is best represented by an ellipsoid and gives rise to the assymetric lineshape shown. If all three chemical shift components are of different values, then the tensor is said to have low-site symmetry. This gives rise to the broad pattern shown in the bottom trace.

It has six independent components. It is convenient to separate the trace of this tensor from the rest and thus introduce spherical tensors... 

Freeman, B. E. (1977). Tensor difiusivity of a trace constituent in a stratified boundary layer. J. Atmos. Sci. 34, 124-136. 

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