Tensorial quantities tensor

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... 

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. 

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. 

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. 

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. 

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. 

Geometrically speaking, the material medium which we are about to describe is a deformable svuface, which we shall now define in mathematical terms. The physical quantities used to describe the state of the material medium will then be introduced and, when we are dealing with tensorial quantities (pressure tensor, electrical and magnetic fields, momentum), compatibility conditions wiU need to be satisfied. 

If we want to apply equation (54) to a molecule M in solution, the external electric field E should be replaced by the local field El, defined as the field actually felt by M as a result of the presence of solvent molecules. The relation between local and external fields, and consequently between (hyper)polarizability and susceptibility tensors can be expressed in terms of some additional functions historically called local field factors (they should be tensorial quantities but are often reduced to scalars) ... 

It maps the discrete arrangement of N spins (y on the superlattice sites i to the real number F(totally equivalent to E(physical quantities - such as the bulk modulus, Zener s ratio, or other scalar and tensorial quantities - depend on the configuration and can be treated with the CE method. 

While finding the numerical values of any physical quantity one has to express the operator under consideration in terms of irreducible tensors. In the case of Racah algebra this means that we have to express any physical operator in terms of tensors which transform themselves like spherical functions Y. On the other hand, the wave functions (to be more exact, their spin-angular parts) may be considered as irreducible tensorial operators, as well. Having this in mind, we can apply to them all operations we carry out with tensors. As was already mentioned in the Introduction (formula (4)), spherical functions (harmonics) are defined in the standard phase system. 

Let us present the main definitions of tensorial products and their matrix or reduced matrix (submatrix) elements, necessary to find the expressions for matrix elements of the operators, corresponding to physical quantities. The tensorial product of two irreducible tensors and is defined as follows ... 

When constructing many-electron wave functions it is necessary to ensure their antisymmetry under permutation of any pair of coordinates. Having introduced the concepts of the CFP and unit tensors, Racah [22, 23] laid the foundations of the tensorial approach to the problem of constructing antisymmetric wave functions and finding matrix elements of operators corresponding to physical quantities. 

Apart from irreducible tensors (14.30) we can also introduce other operators that are expressed in terms of irreducible tensorial products of second-quantization operators, and establish commutation relations for them. As was shown in [12, 102, 103], using relations of this kind, we can relate standard quantities of the theory, which at first sight seem totally different. Consider the operator... 

In Chapter 19 a number of operators corresponding to physical quantities will be expressed in coordinate representation in terms of irreducible tensors. In the most general form, the tensorial structure of one-electron operator / can be written as follows ... 

Tensors (15.52) in the quasispin method are the main operators. Their irreducible tensorial products are used to expand operators corresponding to physical quantities. That is why we shall take a closer look at their properties. Examining the internal structure of tensor Wyields... 

Equations of this kind can also be derived for the special cases of reduced matrix elements of operators composed of irreducible tensors. Then, using the relation between CFP and the submatrix element of irreducible tensorial operators established in [105], we can obtain several algebraic expressions for two-electron CFP [92]. Unfortunately such algebraic expressions for CFP do not embrace all the required values even for the pN shell, which imposes constraints on their practical uses by preventing analytic summation of the matrix elements of operators of physical quantities. It has turned out, however, that there exist more general and effective methods to establish algebraic expressions for CFP, which do not feature the above-mentioned disadvantages. 

In Chapter 14 we have shown how an expansion in terms of irreducible tensors in the spaces of orbital and spin angular momenta for one shell can be obtained for the operators corresponding to physical quantities. The tensors introduced above enable the terms of a similar expansion to be also defined in the space of a two-shell configuration. So, for the one-particle operator of the most general tensorial structure (14.51) we find, instead of (14.52),... 

In this method tensors of such a type are the basic operators, and the operators for the pertinent physical quantities will be given in terms of irreducible tensorial products of these tensors. Specifically, unit tensor Tp (7.4) will be... 

Thus, the double tensor, defined by (23.52), is a convenient standard quantity for studies of mixed configurations. Let us turn now to the properties of the tensorial products of two-shell operators (23.52). Proceeding in the same way as in derivation of (23.11), we arrive at... 

The above relationships describe the behaviour of tensors (23.52) and their tensorial products. A further task is to express the operators corresponding to physical quantities in terms of these tensors. Specifically, two-particle operator (23.27) is expanded by going over to the second-quantization representation and coupling the quasispin ranks... 

During the last two decades a number of new versions of the Racah algebra or its improvements have been suggested [27]. So, the exploitation of the community of transformation properties of irreducible tensors and wave functions allows one to adopt the notion of irreducible tensorial sets, to deduce new relationships between the quantities considered, to simplify further on the operators already expressed in terms of irreducible tensors, or, in general, to offer a new method of calculating the matrix elements, as an alternative to the standard Racah way. It is based on the utilization of tensorial products of the irreducible operators and wave functions, also considered as irreducible tensors. 

This principle as originally stated by Curie in 1908, is quantities whose tensorial characters differ by an odd number of ranks cannot interact (couple) in an isotropic medium. Consider a flow J, with tensorial rank m. The value of m is zero for a scalar, it is unity for a vector, and it is two for a dyadic. If a conjugate force A) also has a tensorial rank m, than the coefficient Ltj is a scalar, and is consistent with the isotropic character of the system. The coefficients Lij are determined by the isotropic medium they need not vanish, and hence the flow J, and the force A) can interact or couple. If a force A) has a tensorial rank different from m by an even integer k, then Ltj has a tensor at rank k. In this case, Lfj Xj is a tensor product. Since a tensor coefficient Lt] of even rank is also consistent with the isotropic character of the... 

It is possible for more than two forces to couple. There exists a criterion which allows one to deduce a priori the number of effective couplings. This is Curie s principle of symmetry. The principle states that a macroscopic phenomenon in the system never has more elements of symmetry than the cause that produces it. For example, the chemical affinity (which is a scalar quantity) can never cause a vectorial heat flux and the corresponding coupling coefficient disappears. A coupling is possible only between phenomenon which have the same tensor symmetry. Thus Onsager reciprocity relation is not valid for a situation when the fluxes have different tensorial character. 

Some operators, such as the interelectronic electrostatic interaction e2/ nj, are obviously scalar quantities. Others are scalar products of two tensorial operators. A tensor of rank zero is a scalar. A tensor of rank one is a vector. There are several ways of combining two vector operators the scalar product... 

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