Tensor quantity

The shear viscosity is a tensor quantity, with components T] y, t],cz, T)yx> Vyz> Vzx> Vzy If property of the whole sample rather than of individual atoms and so cannot be calculat< with the same accuracy as the self-diffusion coefficient. For a homogeneous fluid the cor ponents of the shear viscosity should all be equal and so the statistical error can be reducf by averaging over the six components. An estimate of the precision of the calculation c then be determined by evaluating the standard deviation of these components from tl average. Unfortunately, Equation (7.89) cannot be directly used in periodic systems, evi if the positions have been unfolded, because the unfolded distance between two particl may not correspond to the distance of the minimum image that is used to calculate the fore For this reason alternative approaches are required. 

The susceptibility tensors give the correct relationship for the macroscopic material. For individual molecules, the polarizability a, hyperpolarizability P, and second hyperpolarizability y, can be defined they are also tensor quantities. The susceptibility tensors are weighted averages of the molecular values, where the weight accounts for molecular orientation. The obvious correspondence is correct, meaning that is a linear combination of a values, is a linear combination of P values, and so on. 

Motivated by the qualitative observations made above, a set of internal state variables deseribing the internal strueture of the material will be intro-dueed ab initio, denoted eolleetively by k. Their physieal meaning or preeise properties need not be established at this point, and they may inelude sealar, veetor, or tensor quantities. The following eonstitutive assumptions are now made ... 

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. 

The summations are over all nuclei in radicals A and B for which aj and are finite. The a-values are tensor quantities but are treated as isotropic here. 

In principle, the electronic quadrupole can also be extracted from the calculated valence charge distribution. This is a tensor quantity and its components are defined by the matrix... 

The flow velocity, pressure and dynamic viscosity are denoted u, p and fj and the symbol (...) represents an average over the fluid phase. Kim et al. used an extended Darcy equation to model the flow distribution in a micro channel cooling device [118]. In general, the permeability K has to be regarded as a tensor quantity accounting for the anisotropy of the medium. Furthermore, the description can be generalized to include heat transfer effects in porous media. More details on transport processes in porous media will be presented in Section 2.9. 

The fundamental equation (1) describes the change in dipole moment between the ground state and an excited state jte expressed as a power series of the electric field E which occurs upon interaction of such a field, as in the electric component of electromagnetic radiation, with a single molecule. The coefficient a is the familiar linear polarizability, ft and y are the quadratic and cubic hyperpolarizabilities, respectively. The coefficients for these hyperpolarizabilities are tensor quantities and therefore highly symmetry dependent odd order coefficients are nonvanishing for all molecules but even order coefficients such as J3 (responsible for SHG) are zero for centrosymmetric molecules. Equation (2) is identical with (1) except that it describes a macroscopic polarization, such as that arising from an array of molecules in a crystal (10). 

The concept of magic-angle spinning arises from the understanding of the shielding constant, cr (Eq. 4). This constant is a tensor quantity and, thus, can be related to three principal axes ... 

We should recognise that stress and strain are tensor quantities and not scalars. This will not present any difficulties in this text but we should bear it in mind because the consequences can be both dramatic and useful. To illustrate the mathematical problem, we can think about what happens when we apply a strain to an element of our material. The strain is made up of three orthogonal components which can be further subdivided into three elements, each of which is lined up with one of our axes. This is shown in Figure 1.3. 

In a crystal, the electronic and ionic conductivities are generally tensor quantities relating the current density Iq to the applied electric field E in accordance with Ohm s law. The scalar expression for the mobile-ion current density in the different principal crystallographic directions has the form... 

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. 

In this equation ut should be interpreted as the volumetric flux density (directional flow rate per unit total area). The indexes range from 1 to 3, and repetition of an index indicates summation over that index according to the conventional summation convention for Cartesian tensors. The term superficial velocity is often used, but it is in our opinion that it is misleading because n, is neither equal to the average velocity of the flow front nor to the local velocity in the pores. The permeability Kg is a positive definite tensor quantity and it can be determined both from unidirectional and radial flow experiments [20], Darcy s law has to be supplemented by a continuity equation to form a complete set of equations. In terms of the flux density this becomes ... 

While many chemical investigations of solids seek to emulate solution investigations by averaging all orientation-dependent properties, this approach may be shortsighted in that it necessarily reduces the information conveyed by the full anisotropic tensor quantities. 2D MAS experimental techniques originally developed by Bax, Szeverenyi, and Maciel (40) and refined by Grant and co-workers (41) can provide isotropic chemical shifts... 

Many materials properties are anisotropic they vary with direction in the material. When anisotropic materials properties are characterized, the values used to represent the properties must be specified with respect to particular coordinate axes. If the material remains fixed and the properties are specified with respect to some new set of coordinate axes, the properties themselves must remain invariant. The way in which the properties are described will change, but the properties themselves (i.e., the material behavior) will not. The components of tensor quantities transform in specified ways with changes in coordinate axes such transformation laws distinguish tensors from matrices [6]. 

The x/jj may be scalar, vector, or, generally, tensor quantities however, each product in Eq. 2.5 must be a scalar. 

On the theoretical side, Dmitriev and Bonchkovskaya (D8) have shown that in principle turbulence should spread from waves. Kapitsa (K9) has calculated a general tensor quantity, termed the coefficient of wavy transfer, which is applicable to any flow with periodic disturbances, such as pulsations or surface waves. This treatment predicts an appreciable increase in the rates of heat and mass transfer in wavy films, though this increase does not appear to be as large as that observed experimentally under certain conditions. 

Since P = %E (Equation (10)), 4tuxE is the internal electric field created by the induced displacement (polarization) of charges. Usually, the induced polarization causes the spatial orientation of the internal electric field to differ from the applied electric field and, like ay (co), xij( ) is a tensor quantity that reflects the anisotropy of the internal electric field. 

Now the magnitude of the polarization depends on the direction of displacement (Figure 14). For the covalent (e.g. titanyl or vanadyl) M=0 bond, in general, one expects that the electron cloud would be more easily polarized towards the oxygen atom. This direction dependency means that the polarization coefficients must be described using tensor quantities. 

Most of the ab initio studies of NMR chemical shifts in peptides and proteins have focused on the average or isotropic value of the NMR chemical shift. The chemical shift is a tensor quantity and is, therefore, capable of providing six independent pieces of information, namely, the magnitude and direction of each of the three principal components. In general, the shielding tensor can be antisymmetric, leading to nine independent components. However, only the symmetric part of the shielding tensor is relevant to the experiments... 

Multipole Moments.—For any system of charges (1 < n) a set of electric multipoles can be defined. They are tensor quantities, the 2n-th pole being an n-rank tensor. For a set of charges a with position vectors n relative to some origin, the first few are ... 

The selection rules for the Raman effect are quite different from those for IR spectroscopy. The mechanism involves interaction between the incident radiation and the fluctuating polarisability of the molecule, in contrast to the fluctuating dipole moment in IR absorption. The dipole moment is a vector quantity, and can be resolved into components along three Cartesian axes. The polarisability is a tensor quantity, whose components can be written as products of Cartesian axes. For a molecule having no symmetry at all, or having only a plane of symmetry, all... 

The most comprehensive information obtained from a Mossbauer spectrum is contained in Bint that depends on the magnetic hyperfine tensor A and, through (S), on the ZFS, the electronic g tensor (and exchange couplings when we consider polynuclear systems). For samples containing randomly oriented molecules, such as poly crystalline powders or molecules in frozen solution, the Mossbauer spectrum depends on the orientation of the molecule relative to the direction of the applied field,4 6 which is fixed in the laboratory and is generally either parallel or perpendicular to the direction of Mossbauer radiation. As a consequence, the spectrum is a powder average from which we have to extract the various tensor quantities of... 

To determine what stresses are generated in the torsional disk flow of a CEF fluid, we assume that its flow field is that of a pure viscous fluid then we calculate the tensor quantities Vv, y, co, y y, co y, and v Vy that appear in the CEF equation. Obtaining these quantities, we substitute them in the constitutive equation to find out which are the nonzero stress components. 

Chemical Shift Anisotropy. Details of chemical shift theory are dealt with by several authors, 3 4,6-8 Of relevance to solid-state studies, however, is the fact that the chemical shift is a tensor quantity with three components, an, a22,... 

The aim of this decomposition is that, as we shall see, the sets of scalar quantities A, B, C, E, vector quantities >,. Ei and tensor quantities Eij evolve independently from each others. Note that we are left with four scalars A, B, C, E), two vectors (Bi, Ei) which both have three components but which obey obey divergenceless constraint so that we have four independent components, and one tensor (Eij) which is a 3 x 3 symmetric matrix with one traceless constraint and three divergenceless constraints, which therefore has only two independent components. As expected, we are still left with ten independent components for the metric perturbations. As we said, four of these perturbations are in fact unphysical. Let us decompose the infinitesimal coordinate change 4 / into... 

Multiplets in NMR. The second effect yields a spin-spin coupling constant / (usually quoted in hertz), which it generates a multiplet structure that is due to nuclear spin-nuclear spin interactions between equivalent or inequivalent protons (in H1 NMR). The spin interaction is actually a tensor quantity due to... 

C. J. Jameson, Reply to "conventions for tensor quantities used in nuclear magnetic resonance, nuclear quadrupole resonance and electron spin resonance spectroscopy". Solid State Magn. Reson., 1998,11,265-268. 

R. F. Schneider, Asymmetry in magnetic second-rank tensor quantities, f. Chem. Phys., 1968, 48 (11), 4905-4909. 

Tensor quantities may be printed in bold face italic sans-serif type. 

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