Mean tensor

All NMR interactions are represented by tensors of rank 2, and enter the static NMR Hamiltonian after averaging over the fast degree of freedoms. The lineposition and lineshape are determined by this mean tensor, which should reflect the local symmetry. How the NMR interaction tensors are transformed by the symmetry operations is consequently highly relevant to structural phase transition that are characterized by a loss of some symmetry operations. 

In the parent phase = 0, the lineposition and shape (see Eq. (2)) depend on the variance of the fluctuations (din)o and should be approximatively linear in temperature if the classical equipartition theorem can be applied. In the phase characterized by 0, the total mean tensor (T ) should obey the new site symmetry of the new phase. Since the linear term is preponderant, the isotropic part of the tensor is thus a linear function of and its behavior as a function of temperature should reflect whether the phase transition is continuous or discontinuous. The exact behavior of the lineshape cannot be... 

We consider the problem of splitting for 2-D tensor field by using its representation by means of the tensor u... 

The dipole polarizability tensor characterizes the lowest-order dipole moment induced by a unifonu field. The a tensor is syimnetric and has no more than six independent components, less if tire molecule has some synnnetry. The scalar or mean dipole polarizability... 

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. 

Intensities of Raman transitions are proportional to R and therefore, from Equation (6.13), to (da/dx)g. Since a is a tensor property we cannot illustrate easily its variation with x instead we use the mean polarizability a, where... 

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 ... 

Dimensionless groups for a proeess model ean be easily obtained by inspeetion from Table 13-2. Eaeh of the three transport balanees is shown (in veetor/tensor notation) term-by-term under the deseription of the physieal meanings of the respeetive terms. The table shows how various well-known dimensionless groups are derived and gives the physieal interpretation of the various groups. Table 13-3 gives the symbols of the dimensions of the terms in Table 13-2. 

Cartesian tensors, i.e., tensors in a Cartesian coordinate system, will be discussed. Three Independent quantities are required to describe the position of a point in Cartesian coordinates. This set of quantities is X where X is (x, X2, X3). The index i in X has values 1,2, and 3 because of the three coordinates in three-dimensional space. The indices i and j in a j mean, therefore, that a j has nine components. Similarly, byi has 27 components, Cp has 81 components, etc. The indices are part of what is called index notation. The number of subscripts on the symboi denotes the order of the tensor. For example, a is a zero-order tensor... 

Gelbart (1974) has reviewed a number of theories of the origins of the depolarized spectrum. One of the simplest models is the isolated binary collision (IBC) model of McTague and Bimbaum (1968). All effects due to the interaction of three or more particles are ignored, and the scattering is due only to diatomic collision processes. In the case that the interacting particles A and B are atoms or highly symmetrical molecules then there are only two unique components of the pair polarizability tensor, and attention focuses on the anisotropy and the incremental mean pair polarizability... 

Here g and go are a set of angular variables, which define a molecular orientation at instants of time 0 and t, respectively and ft is the orientation at instant t which was g0 at t = 0. By difference of arguments we mean the difference of turns. In the molecular frame (MS), where the axes are oriented along the main axes of the inertia tensor, [Pg.86]

The form of the effective mobility tensor remains unchanged as in Eq. (125), which imphes that the fluid flow does not affect the mobility terms. This is reasonable for an uncharged medium, where there is no interaction between the electric field and the convective flow field. However, the hydrodynamic term, Eq. (128), is affected by the electric field, since electroconvective flux at the boundary between the two phases causes solute to transport from one phase to the other, which can change the mean effective velocity through the system. One can also note that even if no electric field is applied, the mean velocity is affected by the diffusive transport into the stationary phase. Paine et al. [285] developed expressions to show that reversible adsorption and heterogeneous reaction affected the effective dispersion terms for flow in a capillary tube the present problem shows how partitioning, driven both by electrophoresis and diffusion, into the second phase will affect the overall dispersion and mean velocity terms. 

Trans-polyenes H-(-HC=CH-),, -H, trans-polyenynes H-(HC=CH-C=C) -H, cumulenes H2C=(C=C) =CH2 and polyynes H-(C=C) -H have been studied (M=N-1). For eentrosymmetrie molecules, the first order hyperpolarizability p is equal to zero so that non linear effects are of second order nature. Furthermore, (the x axis goes through the middle of the C-C bonds of the polyenes, or is the intemuclear axis in the case of linear molecules) is the most important component of the second order y hyperpolarizability tensor, the other components being negligible. Both y and the mean hyperpolarizability... 

The major contribution to the components of the D tensor as well as the deviations of the g values from 2.0023 arises from the mixing of ligand field states by SOC other contributions to D result from direct spin-spin coupling, which mixes states of the same spin S. The D tensor and the g matrix both carry chemical information as they are related to the strength and symmetry of the LF, which is competing and counteracting to the effects of SOC. Details on the chemical interpretation of the parameters by quantum chemical means is found in Chap. 5. 

Subsequently, Mitchell s group in Vancouver, by means of a tensor-LEED study17 of the Cu (110)-(2 x 3)N surface structure, supported a reconstruction model in which the topmost layer is described as a pseudo-(100)-c(2 x 2)N overlayer with metal corrugation of about 0.52 A in the reconstructed layer. Each nitrogen adatom is almost coplanar with the local plane formed by the four neighbouring copper atoms. Of the four N atoms present in the unit mesh, three are also bonded to Cu atoms in the layer below and therefore are five coordinate. 

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