Subject tensor component

The interaction of nitric oxide (NO) with metal ions in zeolites has been one of the major subjects in catalysis and environmental science and the first topic was concerned with NO adsorbed on zeolites. NO is an odd-electron molecule with one unpaired electron and can be used here as a paramagnetic probe to characterize the catalytic activity. In the first topic focus was on a mono NO-Na" complex formed in a Na -LTA type zeolite. The experimental ESR spectrum was characterized by a large -tensor anisotropy. By means of multi-frequency ESR spectroscopies the g tensor components could be well resolved. The N and Na hyperfine tensor components were accurately evaluated by ENDOR spectroscopy. Based on these experimentally obtained ESR parameters the electronic and geometrical structures of the NO-Na complex were discussed. In addition to the mono NO-Na complex the triplet state (NO)2 bi-radical is formed in the zeolite and dominates the ESR spectrum at higher NO concentration. The structure of the bi-radicai was discussed based on the ESR parameters derived from the X- and Q-band spectra. Furthermore the dynamical ESR studies on nitrogen dioxides (NO2) on various zeolites were briefly presented. 

Transitioning from the stress state of a particle to the stress field of the continuum, the interaction of the Cauchy stress tensor components of neighboring points needs to be investigated. They have to satisfy the conditions of local equilibrium to be established with the aid of an arbitrary infinitesimal volume element. Such an element with faces in parallel to the planes of the Cartesian coordinate system is subjected to the volume force and on the faces to the components of the Cauchy stress tensor with additional increments in the form of the first element of Taylor expansions on one of the respective opposing faces. The balance of moments proves the symmetry of the stress tensor,... 

We now provide a preliminary link to thermodynamics consider a cubic crystal in unstrained form that is subjected to a small homogeneous stress the resulting strain is specified by the symmetric tensor components e,- (i = 1,2,..., 6) of Eq. (5.10.13a). The corresponding element of work is then represented as usual by the generalized force component multiplying the corresponding differential of the displacement response, as in Eq. (5.10.13d) ... 

A method for determining the macroscopic stress tensor components as integral functionals of the elemental deformations is now developed. The procedure is illustrated in Figure 3 for the xx component of the stress tensor, and consists basically of summing the force contributions of those elements that are "cut" by the plane under consideration. For convenience let the macroscopic body be subjected to a homogeneous stress and deformation field, and let the reference axes x, y, z be both the symmetry axes for the distribution of elements [8] as well as the principal axes of deformation (this restriction is discussed in more detail below). 

A further term that can contribute to E(1)yAa is the ZFS (59,60). As implied by its name, ZFS splits the components of a state in the absence of a magnetic field. For states that are only spin degenerate, ZFS occurs when the spin S>l/2. Like the g-tensor, ZFS causes the axis of spin quantization to deviate from the direction of the magnetic field. The consequences with respect to spin integration and orientational averaging are similar to those caused by the use of a non-isotropic g-tensor. ZFS is made up of two terms, one second-order in spin-orbit coupling and the other from spin-spin coupling (59). The calculation of ZFS within DFT has been the subject of several recent publications (61-65). 

For dipolar chromophores that are the subject of this chapter, only one component of the molecular hyperpolarizability tensor, Pzzz, is important. Thus, the summation in Eq. (8) disappears. Electric field poling induces Cv cylindrical polar symmetry. Assuming Kleinman [12] symmetry, only two independent components of the macroscopic second-order nonlinear optical susceptibility tensor... 

We will present the effective Hamiltonian terms which describe the interactions considered, sometimes using cartesian methods but mainly using spherical tensor methods for describing the components. These subjects are discussed extensively in chapters 5 and 7, and at this stage we merely quote important results without justification. We will use the symbol T to denote a spherical tensor, with the particular operator involved shown in brackets. The rank of the tensor is indicated as a post-superscript, and the component as a post-subscript. For example, the electron spin vector A is a first-rank tensor, T1 (A), and its three spherical components are related to cartesian components in the following way ... 

Before leaving this aspect of the subject we must see how the term used by Ramsey [52] to describe the magnetic susceptibility in the effective Hamiltonian (8.15) arises. In what follows, the direction of the magnetic field defines the p = 0 space-fixed direction, but the components of the magnetic susceptibility tensor are defined in the molecule-fixed axis system (< / ). Note that B and iU)H are equivalent in a vacuum. 

Freund and Miller [24] determined the values of a number of parameters, some composite, for the N = 1 level of para-H2,3nu, in its v = 0 to 3 levels. In particular they determined the value of a parameter A containing contributions from both the spin-orbit and spin-rotation terms, and components of the spin-spin interaction tensor. They were also able to identify the effects of breakdown in the Bom Oppenheimer approximation, a subject explored theoretically in more depth by Miller [29],... 

Since the Raman techniques described are sensitive to (fl(t) - fi (0)>, we are interested in the properties of products of two elements of II, subject to the above constraints. Furthermore, it is necessarily true that any tensor elements of n that are related by reversed indices are identical, e.g., nxy = Flyx. This leaves us only two independent elements of R<3) to consider. It is conventional to express these independent elements in terms of rotationally invariant features of n One of these invariants (usually denoted a) is given by one third of the trace of FI (24). Since this invariant measures the average polarizability of the system, it is known as the isotropic component of n. The other invariant is generally denoted ft, and in the principal axis system of FI it is given by (24)... 

This was an introduction to rheology. If you want to go further, you will find that it is a difficult subject. For three-dimensional problems you need sophisticated mathematics to get anywhere. You will need tensor equations the three stresses Xxy, Xyx and Xyy that you have seen are three of the nine components of the stress tensor (Figure C4-17). And that is just... 

Here only noncrystalline symmetries, which are likely to play an important role in the linear viscoelastic behavior of materials, are considered. We follow Tschoegl s approach to this subject (5). Crystalline materials and their symmetries are described in many textbooks (6,7). In order to study how the symmetry of the system affects the number of independent components of Cijki, it is convenient to reduce the number of indices of both the stress and strain tensors. Following Voigt s formulation, the reduction is made by doing 11 -> 1, 22 2, 33 3, 12 -> 4, 23 -> 5, 13 6, so that... 

Case 3. Consider an infinite medium with a spherical cavity of radius R subjected to internal pressurization. If the boundary conditions P = p and = 0 re assumed, then the components of the stress tensor are... 

Similar to the ODF for texture, SODF can be subjected to a Fourier analysis by using generalized spherical harmonics. However, there are three important differences. The first is that in place of one distribution (ODF), six SODFs are analyzed simultaneously. The components of the strain, or the stress tensor can be used for analysis in the sample or in the crystal reference system. The second difference concerns the invariance to the crystal and the sample symmetry operations. The ODF is invariant to both crystal and sample symmetry operations. By contrast, the six SODFs in the sample reference system are invariant to the crystal symmetry operations but they transform similarly to Equation (65) if the sample reference system is replaced by an equivalent one. Inversely, the SODFs in the crystal reference system transform like Equation (65) if an equivalent one replaces this system and remain invariant to any rotation of the sample reference system. Consequently, for the spherical harmonics coefficients of the SODF one expects selection rules different from those of the ODF. As the third difference, the average over the crystallites in reflection (83) is structurally different from Equations (5)+ (11). In Equation (83) the products of the SODFs with the ODF are integrated, which, in comparison with Equation (5), entails a supplementary difficulty. 

On the basis of the above relationships one can conclude that the three principal components of the /1-tensor can be used to express the complete set of magnetic parameters that describes zero-field splitting in mononuclear complexes, i.e. gxx, gyy, gzz, D and E. In addition, they define the temperature-independent paramagnetic term x - Therefore it is possible to reconstruct the components of the /1-tensor having the set of magnetic parameters determined from an appropriate fit of experimental data. However, an opposite procedure is possible to consider the components of the /1-tensor as the principal quantities which determine the ZFS parameters. Then one can consider Axx, Ayy, Azz and X as a set of free parameters subjected to optimisation. In performing such a procedure, the following optimisation scheme can be followed. 

A nonspherical particle is generally anisotropic with respect to its hydro-dynamic resistance that is, its resistance depends upon its orientation relative to its direction of motion through the fluid. A complete investigation of particle resistance would therefore seem to require experimental data or theoretical analysis for each of the infinitely many relative orientations possible. It turns out, however, at least at small Reynolds numbers, that particle resistance has a tensorial character and, hence, that the resistance of a solid particle of any shape can be represented for all orientations by a few tensors. And the components of these tensors can be determined from either theoretical or experimental knowledge of the resistance of the particle for a finite number of relative orientations. The tensors themselves are intrinsic geometric properties of the particle alone, depending only on its size and shape. These observations and various generalizations thereof furnish most, but not all, of the subject matter of this section. 

Although smaller than the Hartree contribution, the remaining xc part of the electron-electron interaction should also be subjected to the DK transformation to obtain further improved two-component wave functions for calculating g values work in this direction is in progress in our group. We have good reason to assume that the difference of our g values from those calculated with the two-component KS method ZORA [112] can be rationalized by the fact that the xc potential remains untransformed in our present g tensor approach. 

For a three-dimensional deformation, a generalization of the strain as a measure of deformation when a material is subjected to deformation is the strain tensor. There are several ways to express the strain tensor, based on linear and nonlinear representations of the strain components. In Equation 22.3, the strain tensor is written by using a linear representation of the strain components ... 

Among these response properties the most known are the dipole polarizability tensors a, P, and y, that give flic first three components of flie expansion of a dipole moment t subjected to an external homogeneous field F ... 

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