Spatial dispersion effects

Spatial dispersion effects are usually considered separately from time dependences and correspond to static limit to = 0. Consequently s(k, 0) = s(k) and x(k, 0) = x(k) are basic susceptibility functions. Within the LRA the relation similar to Equation (1.131) is valid. It formally represents a solution to the nonlocal Poisson equation with a -dependent susceptibility. 

The nature of the dependences n (u>) and nn(w) are known to be dependent on the signs of the coefficients fi and n2, respectively (1). Therefore, the experimental results for the ratio (5.53) can, in principle, be used to identify the spatial dispersion effects of the crystal matrix. However, the decay of the excited states of both the impurity and the matrix makes this difficult. If the levels of these states are wide enough then it becomes practically impossible to sneak up on the frequency fli(O) and to distinguish the impurity absorption from the matrix absorption. 

SPATIAL DISPERSION EFFECTS IN GYROTROPIC CRYSTALS. //ENGLISH TRANSLATION OF FIZ. TVERD. TELA 9 /3/... 

In 1.1.3°-1.1.7°, we have assumed the medium to be nonabsorbing and, thus, the parameters e and n to be constant and o =0. However, if the medium absorbs electromagnetic radiation, these quantities become dependent on the frequency of incident radiation, the function e((o) termed the dielectric function. Below we will neglect the so-called spatial dispersion effects [18] connected with the dependence of the dielectric function on the wave vector (k). This is permissible for the IR range (the limit k 0). 

Thus, a susceptibility that depends on frequency and wave vector implies that the relation between P(x, t) and E(x, t) is nonlocal in time and space. Such spatially dispersive media lie outside our considerations. However, spatial dispersion can be important when the wavelength is comparable to some characteristic length in the medium (e.g., mean free path), and it is well at least to be aware of its existence it can have an effect on absorption and scattering by small particles (Yildiz, 1963 Foley and Pattanayak, 1974 Ruppin, 1975, 1981). 

Nevertheless, the concept of spatial dispersion provides a general background for a qualitative understanding of those solvation effects which are beyond the scope of local continuum models. The nonlocal theory creates a bridge between conventional and well developed local approaches and explicit molecular level treatments such as integral equation theory, MC or MD simulations. The future will reveal whether it can survive as a computational tool competitive with these popular and more familiar computational schemes. 

The OntoCODE system effectively combines the advantages of both the spatially dispersed and split and recombine strategies and allows the chemist to build large archiv-able combinatorial libraries with milligram quantities of each compound and without the need for chemical tagging. 

III. Spatially Averaged Models for Describing Dispersion Effects in Tubes and Packed Beds... 

In this section, we illustrate the spatial averaging procedure by considering several simple examples of non-reacting systems that describe dispersion effects in tubes, packed beds, and monoliths. 

Thus, we recover the Danckwerts model only if no distinction is made between the cup-mixing and spatial average concentrations (with this assumption, the effective axial dispersion coefficient is given by the Taylor-Aris theory). This derivation also shows that the concept of an effective axial dispersion coefficient and lumping the macro- and micromixing effects into one parameter is valid only at steady-state, constant inlet conditions and when the deviation from plug flow is small. [Remark Even with all these constraints, the error in the model because of the assumption (cj) — cym is of the same order of magnitude as the dispersion effect ]... 

We would like to present somewhat more extensively the results of the work of Dzhavakhidze et al. [Ill], who studied the role of the spatial dispersion of the solvent dielectric permittivity and field penetration into a metal in determining the kinetics of electrode reactions. Considering the particular case of the field penetration effect on the reorganization energy, they found [111] that the AG value obtained is greater than predicted by the Marcus theory. Moreover, under some eonditions the dependence of AG on the reactant-electrode distance d) exhibits an anti-Mar-cusian behavior. 

The theory of nonlinear optical processes in crystals is based on the phenomenological Maxwell equations, supplemented by nonlinear material equations. The latter connect the electric induction vector D(r,t) with the electric field vector E(r, t). In general, the relations are both nonlocal and nonlinear. The property of nonlocality leads to the so-called spatial dispersion of the dielectric tensor. The presence of nonlinearity leads to the interaction between normal electromagnetic waves in crystals, i.e. makes conditions for the appearance of nonlinear optical effects. 

Solvents with vanishing molecular dipole moments but finite higher order multipoles, such as benzene, toluene, or dioxane, can exhibit much higher polarity, as reflected by its influence on the ET energetics, than predicted by the local dielectric theory [228], Full spatially dispersive solvent response formulation is required in this case [27-29, 104, 229-233], There are different approaches to the problem of spatial dispersion. The original formulation by Kornyshev and co-workers [27c, 28] introduces the frequency-dependent screening effect on the basis of heuristic arguments. More recent approaches are based on the density-function theory [104,197],... 

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