Tensorial quantities

D is the zero-field splitting tensor, a traceless, rank-two tensorial quantity. The ZFS tensor is a property of a molecule or a paramagnetic complex, with its origin in the mixing of the electrostatic and spin-orbit interactions (80). In addition, the dipole dipole interaction between individual electron spins can contribute to the ZFS (81), but this contribution is believed to be unimportant... 

It should be stressed that in the literature one can come across a wide variety of notations for creation and annihilation operators. In this book we follow the authors [14, 95] who attach the sign of Hermitian conjugation to the electron annihilation operator, but not to the electron creation operator. Although the opposite notation is currently in wide use, it is inconvenient in the theory of the atom, since it is at variance with the common definitions of irreducible tensorial quantities. 

Equations of the kind (14.23) or (14.25) are inconsistent in the way that tensors in their left and right sides are defined in different (pseudostandard and standard) phase systems. This is done to underline that historically tensors composed of unit tensors were defined in pseudostandard phase systems whereas in this book main tensorial quantities obey standard phase systems. 

Actually, the nucleus senses a different field in different directions around it. Therefore, the shift is a tensorial quantity. Upon rotation in solution an average value a is obtained. The proton of CHCI3 will experience a smaller shielding constant because chlorine, being quite electronegative, will attract the electrons... 

Generally, e and are tensorial quantities. They reduce to scalars in the case of isotropic media, and then describe the longitudinal polarization effects. Our presentation is devoted to this simple transparent case. Complications introduced by anisotropic phenomena are not considered they do not change the main idea of nonlocal theory only making the notation cumbersome. 

The statistical average of any configuration-dependent tensorial quantity A(xw, t) is defined as... 

In our description of spin reorientational relaxation processes, tensorial quantities are used for which it is necessary to know the transformation properties concerning rotation. A clear and compact formulation is obtained by replacing the cartesian components with a representation in terms of irreducible spherical components. It is known that any representation of the group of rotations can be developed into a sum of irreducible rqpre-sentations D of dimension 2/ +1. If for the description of general rotation R(U) we use the Euler angles Q = (a, p, y), this rotation will be defined by... 

Magnetic saturation in an electric field. The inverse of equation (14) takes place when the magnetic permeability is analysed with a magnetic field Ha in the presence of a strong electric field Ep. The variation in magnetic permeability, a tensorial quantity in general, is now of the form ... 

The anisotropy of molecules and matter imposes that the proportionality factors (Eqs. (2) and (3)) are tensorial quantities (vide supra). As such, the relations between the dipole moment vector and the electric field vector can be defined by Eq. (6),... 

Both x and x are complex tensorial quantities and in general, the different experimental techniques complement each other in order to fully characterize and In the case of molecular materials, the response of the bulk can be related to the individual molecular response. One has to always keep in mind that the experimental setup has to meet the appropriate conditions imposed by the particular model applied to derive or x from the experimental measurements. 

Another output of Landau theory is that any other physical quantities (tensors) that are coupled to the primary order parameter p contain components that may exhibit also some anomaly. Typically, if has the same symmetry as p, then oc p. Otherwise, ocpm with exponent m = 2 although other values are possible. As a consequence, the phase transitions can be detected in an indirect manner by the measurement of any physical tensorial quantity that is coupled to the order parameter depending on the symmetry of the coupling, some components may become nonzero in the low-symmetry phase, or otherwise exhibit an anomalous behavior near the transition. Schematic evolution of different physical parameters at second-and first-order phase transitions is summarized in Fig. 3. These considerations are highly relevant to NMR because all interactions are second-rank tensors that may couple with the order parameter. 

If averages are to be taken of vectorial or tensorial quantities, Eq. 1.7 should be applied to each of their components. 

Viscoelasticity deals with the dynamic or time-dependent mechanical properties of materials such as polymer solutions. The viscoelasticity of a material in general is described by stresses corresponding to all possible time-dependent strains. Stress and strain are tensorial quantities the problem is of a three dimensional nature (8), but we shall be concerned only with deformations in simple shear. Then the relation between the shear strain y and the stress a is simple for isotropic materials if y is very small so that a may be expressed as a linear function of y,... 

Basic equations of the homogenization theory are applicable not only for scalar values of the constiments dielectric constants e, but also for the tensorial quantities. Indeed, evaluation of these equations does not demand any special requirements regarding the character of the electric displacement D and electric intensity E relation D = eeoE in which dielectric constant e can be tensorial. This fact enables us to calculate the effective dielectric constant Ceff for the composites with anisotropic constituents. It can be, for example, polymer composites with magnetic granular, the permittivity of which is tensorial quantity. The explicit form of the e tensor for magnetic media will be given in the next section. 

The description of the mechanical deformation of the membrane is cast in terms of principal force restiltants and principal extension ratios of the surface. The force resultants, like conventional three-dimensional strains, are generally expressed in terms of a tensorial quantity, the components of which depend on coordinate rotation. For the purposes of describing the constitutive behavior of the surface, it is convenient to express the surface resultants in terms of rotationally invariant quantities. These can be either the principal force resultants Ni and Nj, or the isotropic resultant N and the maximum shear resultant Ns- The surface strain is also a tensorial quantity, but maybe expressed in terms of the principal extension ratios of the surface. >.1 and Xj- The rate of surface shear deformation is given by (Evans and Skalak, 1979] ... 

Note Owing to its single crystalline structure, the coefficient of expansion and magnetic susceptibility of sapphire are tensorial quantities (the left side, values parallel, right side perpendicular to the principal axis of the system). (After ref. 36.)... 

Shear Modulus. The shear modulus determined from torsion measurements exhibits some dependence on temperature. For the Kevlar composite, the increase was 1.5, and for carbon fiber composite it was 1.2, from 293 to 4.2 K. Of course, both the damping and storage shear moduli represent tensorial quantities, and this must be included in the analysis. For anisotropic fibers, both the tensorial quantities of the fibers and those of the composite are involved. Here, only one tensor element, which was expected to be sensitive to temperature, was considered. 

The chemical shift of a nuclear spin is a tensorial quantity. Its value depends on the orientation of the electronic distribution about the nucleus with respect to the external magnetic field. In a liquid, due to the rapid molecular motions, this interaction is averaged to zero and the observed chemical shift is the trace of the tensor. In contrast, in a powder, in the absence of motions, all the orientations have the same probability and the signal obtained for each carbon is the sum of the elementary chemical shifts corresponding to the different orientations. When local motions occur in the bulk below 7g, they usually induce a partial averaging of the chemical shift anisotropy. 

Solid state NMR spectroscopy is especially suited for the determination of local order and segmental dynamics, providing that local and whole aggregate motions take place on different time scales. Thus, tensorial quantities such as quadrupolar or dipolar couplings and anisotropic Zeeman interactions can be directly observed. These interactions are averaged in small aggregate struetures such as sonicated vesicles or small mixed micelles. The desired information can be obtained, however, in mesomorphic... 

The experimental investigation of the chiral induction of the amino-anthraquinones and binaphthyls showed, on the one hand, that the induction by a dopant molecule depends strongly on the orientation of the orientation axis with respect to its skeleton and, on the other hand, on the orientation of a chiral group with respect to the principal axes of the order tensor of the molecule. It became clear that the quantitative contribution to the HTP of a chiral ligand of a molecule of class A depends on the orientation of this chiral ligand with respect to the skeleton. These properties are typical for the behavior of tensorial quantities and thus a tensorial description has to be developed. 

We propose a way to obtain averaged macroscopic quantities like density, momentum flux, stress, and strain from "microscopic" numerical simulations of particles in a two-dimensional ring-shear-cell. In the steady-state, a shear zone is found, about six particle diameters wide, in the vicinity of the inner, moving wall. The velocity decays exponentially in the shear zone, particle rotations are observed, and the stress and strain-tensors are highly anisotropic and, even worse, not co-linear. From combinations of the tensorial quantities, one can obtain, for example, the bulk-stifftiess of the granulate and its shear modulus. 

The chemical shift is a tensorial quantity, i.e. it is anisotropic and depends on the orientation of the molecule with respect to the external magnetic field. 

We first recap the equations of electromagnetism, considering them to be deduced from the balance of various tensorial quantities in timespace [EIN 05, LAN 82]. Thus, the formulation is relativistic, but we believe this simplifies the reasoning process. The drawback is that the conventional balance equations in Aerothermochemistry are not relativistic. Hence, at first glance, this presentation seems non-homogeneous. In reality, though, it is not so at aU here, the homogeneity stems from a unique presentation... 

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