Tensor local

The polarization P is given in tenns of E by the constitutive relation of the material. For the present discussion, we assume that the polarization P r) depends only on the field E evaluated at the same position r. This is the so-called dipole approximation. In later discussions, however, we will consider, in some specific cases, the contribution of a polarization that has a non-local spatial dependence on the optical field. Once we have augmented the system of equation B 1.5.16. equation B 1.5.17. equation B 1.5.18. equation B 1.5.19 and equation B 1.5.20 with the constitutive relation for the dependence of Pon E, we may solve for the radiation fields. This relation is generally characterized tlirough the use of linear and nonlinear susceptibility tensors, the subject to which we now turn. 

The structure of the section is as follows. In Section 2.8.2 we give necessary definitions and construct a Borel measure n which describes the work of the interaction forces, i.e. for a set A c F dr, the value /a(A) characterizes the forces at the set A. The next step is a proof of smoothness of the solution provided the exterior data are regular. In particular, we prove that horizontal displacements W belong to in a neighbourhood of the crack faces. Consequently, the components of the strain and stress tensors belong to the space In this case the measure n is absolutely continuous with respect to the Lebesgue measure. This confirms the existence of a locally integrable function q called a density of the measure n such that... 

The axial resonance is assigned to ruthenium A with its D4h local symmetry (compare gj = 2.51, gy = 1.64 in traw.s-RuCl4(PEt3)J) while the rhombic signal is assigned to ruthenium B , where the local symmetry is D2h and three different components of the g-tensor are expected. 

A value of 0 = 0° corresponds to a pure ground state, and 6 = 90° to a pure 3,2 ground state. Since the d orbital rotation matrix elements are different for the d and d -y orbitals, this will lead to a variation of the local g tensor of the Fe" site with the mixing angle d ... 

According to Eqs. (8) and (9), the g tensor of [4Fe-4S] clusters depends on four ferrous gi, 2, gi, gi) and two ferric gs,gs) local g tensors. This holds true even if the mixed-valence pair is fully delocalized Ca = Cb = i Owing to the low-lying excitation energies of the... 

The range of g values predicted by the standard model can be roughly estimated by assuming that all the local g tensors are isotopic and take only two different g values g(Fe(III)) = 2.02 andg(Fe(II)) = 2.00 + Ag, with Ag > 0. One obtauns... 

For the example in Figure 2.14 it would be possible to perform the coordinate transformation analytically by introducing cylindrical coordinates. However, in general, geometries are too complex to be described by a simple analytical transformation. There are a variety of methods related to numerical curvilinear coordinate transformations relying on ideas of tensor calculus and differential geometry [94]. The fimdamental idea is to establish a numerical relationship between the physical space coordinates and the computational space curvilinear coordinates The local basis vectors of the curvilinear system are then given as... 

We restrict ourselves to the local valence part of the EFG tensor to illustrate the principle. Since the EFG operator is spin-free, there are no off-diagonal elements M M and an inspection of Table 5.6 reveals that there are also no off-diagonal components between different configurations I A J- Hence ... 

La2 xSrxCu04 EFG tensor, disagrees with point charge model, charge state, holes from substitution of La by Sr are localized mainly at the oxygen sites in the CU-O2 plane... 

RBa2Cu40g (R = Sm, Y, Er) Nuclear-quadrupole coupling parameters at the rare-earth metal and copper sites from Cu ( Zn) and Ga( Zn) Mossbauer emission spectroscopy, EEG tensor in comparison with point charge model, shows that holes in lattices are localized primarily at chain-oxygen sites... 

HgBa2Ca iCu 02 +2 (n = 1, 2, 3) EEG tensor at the copper, barium, and mercury sites, by Cu( Zn), Ba( Cs), and Hg ( Au) Mossbauer emission spectroscopy. Comparison with point-charge approximation and Cu NMR data showed that the holes originating from defects are localized primarily in the sublattice of the oxygen lying in the copper plane (for HgBa2Ca2Cu30g, in the plane of the Cu(2) atoms)... 

Y2Ba4Cu7025 Nuclear quadrupole interaction at copper sites, EFG tensor at all sites is calculated using the point charge model, conclusion that holes in the Y2Ba4Cu70i5 lattice are localized predominantly at positions of chain oxygen... 

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