Difference tensors

The different tensor elements of the third-order response can all be described in terms of the isotropic and anisotropic components of FF The... 

Multiple CT leading to three level contributions and the possibility of ordering through electrical poling is combined with dipolar 2D NLO-phores. The establishment of structure-property relationships for this type of molecule, reviewed recently (Wolff and Wortmann, 1998), is still in its infancy because application of a single analytical method is clearly inadequate to unravel the combination of different tensor elements. It is convenient to keep the number of numerically different tensor elements as low as possible and to study planar molecules of C2v symmetry. Out of the seven /3 components that are significant for this case, only hve are independent. In addition, the components in the x-direction, dehned as perpendicular to the molecular plane (y,z) are negligibly small, so only four components remain = y y, yy and A... 

Because not much is known experimentally about general interaction tensors, and especially not about their long range behaviour, we will not pursue this line of approach, but rather give a brief review of the existing work which has concentrated on two different tensors the pair dipole (order 1 tensor) and the pair polarizability (order 2 plus order 0 tensor). 

CODEX analysis based on reorientation angle and difference tensor... 

In Eq. (2), the argument of the cosine function, which depends on 8Ntr and the reorientation angles (aR, fiR, yR), can be considered as the phase acquired within a time of N/2tr under the action of the chemical-shift difference tensor coA = (a>2 - principal values of the difference tensor are WA22 = 0, and the full-width anisotropy coA33 - ouAl, of A, i.e. the range of possible frequency differences, is 20 56... 

Note that the orientational averaged hyperpolarizability is dependent on the symmetry of the molecule investigated and that there are different relations to the molecular tensor elements. It is also important to mention that the vectorial part of the hypcrpolarizability is constituted out of different tensor elements than the orientational averaged hypcrpolarizability measured by hyper-Rayleigh scattering. 

The intensities I can be identified with the intensities of the nth spinning side bands of the chemical-shielding difference tensor [74]. In the time-dependent interaction frame the Hamiltonian is given by... 

This molecular response tensor is a more general form of three apparently different tensors which have featured in previous work on multiphoton processes one is the Xy2 tensor arising in the single-frequency distributive two-photon absorption (Andrews and Harlow 1984a), and another is the tensor which appears in the theory of three-photon absorption (Andrews and Wilkes 1985). It is also exactly identical to the two-frequency hyper-Raman transition tensor (Andrews 1984). 

Let us now derive phenomenological equations of the kind (5.193) corresponding to the expression (5.205). As has been mentioned before, each flux is a linear function of all thermodynamic forces. However the fluxes and thermodynamic forces that are included in the expression (5.205) for the dissipative function, have different tensor properties. Some fluxes are scalars, others are vectors, and the third one represents a second rank tensor. This means that their components transform in different ways under the coordinate transformations. As a result, it can be proven that if a given material possesses some symmetry, the flux components cannot depend on all components of thermodynamic forces. This fact is known as Curie s symmetry principle. The most widespread and simple medium is isotropic medium, that is, a medium, whose properties in the equilibrium conditions are identical for all directions. For such a medium the fluxes and thermodynamic forces represented by tensors of different ranks, cannot be linearly related to each other. Rather, a vector flux should be linearly expressed only through vectors of thermodynamic forces, a tensor flux can be a liner function only of tensor forces, and a scalar flux - only a scalar function of thermodynamic forces. The said allows us to write phenomenological equations in general form... 

The ambiguity gives rise to just two different tensors and can be resolved by measurements in a fourth skew plane [9]. As a rule only one of the two tensors reproduces the experimentally determined -factor variation in the skew plane. A simpler procedure is applicable when the principal values can be read from the experimental powder spectrum. Only one of the two sets of principal values obtained from the analysis usually matches the experimental data. 

Fig. 3.37 Symbols used in formulae for axial (3.2.1) and rhombic (3.2.2) symmetry. Parameters g, A and D occur along the Z axis, while gj, and Aj, are in the XY plane. Due to the axial symmetry angle 4> does not enter the equations. The g-, A- and D-tensors with rhombic symmetry can be thought of as the lengths and the directions of the x, y, and z axes of an eUip-soid. The principal axes of the different tensors need not coincide with each other or with the X, Y, Z axes...

Energies are given to 1st and 2nd order assuming that the electronic Zeeman energy is the dominating one, i.e. the high field approximation holds. The principal axes of the coupling tensors need not coincide with the (X, Y, Z) axes. The principal axes of the different tensors need not coincide. 

Then the velocity of the medium at rj, is not simply given by an equation of the type of eq 1.63 because now the fluid is moving with respect to the coordinate system at r as a consequence of the motion of Let us denote by V/(r) the velocity of the fluid at position r and Vr(a) the relative velocity of S with respect to the fluid at Tn- The two spheres are not necessarily of equal radii Kais the radius of Sa and Ht, that of St,. Hence the actions of the spheres on the fluid are different. Different tensors 4 (eq 1.63) must be considered for Sa and St,... 

Again the V p, V pys and G pys can be calculated from the CEF-potential. It turns out and is important that the V p and V pys contain the same CEF-parameters and thus are related to each other. The same holds true for G p and Gapys- These relations are necessary to ensure rotational invariance. Until now we have shown how the different (strain and rotational) interactions can be derived. It has been left open what combinations of J-operators go with the different combinations of e p and o)ap- This information is of course contained in the different tensors G p, F pys etc. which can be calculated from the CEF-potential. However one would prefer a simple prescription how to construct the proper combinations of J-operators for a given symmetry. Such a prescription can indeed be given but for the price that the connection is lost between the different coupling constants which was pointed out above (see Thalmeier and Fulde 1975, the conventional linear strain interaction was derived this way in the pioneering work of Callen and Callen 1965). To demonstrate this we consider cubic symmetry and form the irreducible combinations of Sap and

Table 8 shows experimental second-order nonlinear optical susceptibilities for different tensor components dll and various fundamental wavelengths. The quantities dn are defined as follows ... 

At first glance this appears to be an odd choice since the above expression contains a combination of four different tensor elements and does not appear to contain any intuitive meaning. Tbe key to tbe tensor element lies in tbe cascaded signals generated by this response. Tbe following expression gives the third-order response for a mixed polarization state [50] ... 

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