Dynamic susceptibility tensor

X,. Xc. Xb shorthand notation for the diagonal components of the generalized dynamic susceptibility tensor along specific crystalline directions tUp plasma frequency of a metal... 

Here x (q, co) and the single ion dynamical susceptibility u( >) are tensors in both sublattice (A, B) and transverse xy) Cartesian coordinates. The poles of eq. (84) determine the collective excitations of 5f-local moments. In the paramagnetic phase they are given by the magnetic exciton dispersion... 

As in the case of molecular orientation at the interface of neat liquids, solute molecular orientation can provide insight into the local intermolecular interactions at the interface, which, in turn, is useful for interpreting dynamics, spectroscopy, and reactivity. The simple picture that the hydrophilic part of an asymmetric solute molecule tends to point toward the bulk aqueous phase, while the hydrophobic part points toward the opposite direction, has been confirmed in both simulations and experiments. Polarization-dependent SHG and SFG nonlinear spectroscopy can be used to determine relative as well as absolute orientations of solute molecules with significant nonlinear hyperpolarizability. The technique is based on the fact that the SFG and the SHG signals coming from an interface depend on the polarization of the two input and one output lasers. Because an interface with a cylindrical symmetry has only four elements of the 27-element second-order susceptibility tensor being nonzero, these elements (which depend on the molecular orientation) can be measured. This enables the determination of different moments of the orientational distribution ... 

The optically accessible d5mamical information on the material is contained in the susceptibility tensor, %y(q,f), and the correlation of its fluctuation. This tensor is a very complex material property. We have to deal with two problems. The first one concerns the proper definition of the tensor on the basis of the fundamental physical parameters of the material. The second challenge is the construction of a theoretical model able to describe the dynamics of such physical parameters. Both are typical many-body problems that can be undertaken only using strong approximations. Here we just want to introduce some... 

The basic physical models of the susceptibility tensor, which are indispensable to connect the response function with the dynamics of simple and complex liquids, will be described here. 

To connect the susceptibility tensor with the molecular dynamics, it is useful to separate the intramolecular from the intermolecular dynamics. This separation is applicable when these two dynamics are decoupled. 

In the previous equations, the intramolecular dynamics is defined by the vibrational coordinates, Vv t), whereas the intermolecular dynamics is contained in several parameters defining the susceptibility tensor. It appears explicitly in the molecule translational coordinates, and impUcitly in the rigid-molecule polarizability, and mode expansion coefficients, 6 . In fact both dependent by the molecule orientational coordinates and hence on the rotational dynamics. 

In the previous section we connected the susceptibility tensor directly to the microscopic parameters, the molecular polarizabilities. Indeed, this point of view clarifies the molecular aspects of the x tensor, but the connection with the dynamical model is complicated by the extremely large number of variables that must be considered. An alternative possibility is represented by a coarsegrained definition of the susceptibility function, in fact if we disregard to the microscopic information we gain a more direct link to the dynamic model. The polarizability of a rigid symmetric-top molecule can be described by (2.29), hence the dielectric tensor becomes... 

In conclusion, we have suggested that the linear response law and the response experiments can be applied to the study of dynamic behavior of complex chemical systems. We have shown that the response experiments make it possible to evaluate the susceptibility functions from transient as well as frequeney response experiments. We have shown that the susceptibility functions bear important information about the mechanism and kinetics of complex chemical processes. We have suggested a method, based on the use of tensor invariants, which may be used for extracting information about reaction mechanism and kinetics from susceptibility measurements in time-invariant systems. 

Although simple models are essential for a fundamental understanding, in the future more realistic models are required, in particular in view of possible applications in cell and tissue engineering. Here we have used cable networks as a first step towards more realistic models for both cell and the matrix. Anisotropic force contraction dipoles are only the first order approximation for the complex mechanical activity of cells and might be extended to more general tensors for mechanical activity and susceptibility. A more sophisticated model would be to replace the force dipoles by whole cell models incorporating the focal adhesion dynamics and stress fibers evolution. 

Table 8.4 Main diagonal tensor components of the dynamic (i = 1907 nm) first hyperpolarizability [p, in a.u.) and of the quadratic susceptibility (/ , in pm V ), as well as keto/enol contrasts, as determined at the LC-BLYP/6-311H-G(d) level of approximation for crystals 10 and 12.

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