Cross-over region

Within the cross-over region there is more complicated mixing between doublet and quartet states and the luminescence band is broad with additional sub-bands (11). Because of the importance of ruby lasers and related Cr3+ activated systems, the physics of the doublet-quartet transition has been worked out in considerable detail as has the mechanism of non-radiative transfer and the temperature dependence of the luminescence(12). Although the Cr3+ narrow band emission is essentially independent of crystal field and thus of site size and symmetry, the energy of the doublet levels does depend on Racah parameters B and C. The B and C parameters are dependent on the covalency of the metal-ligand bond and thus there is some variability in the Cr3+ emission from host to host. 

For weakly entangled monodisperse and polydisperse polymer melts, J. des Cloizeavuc [26] proposed a theory based on time-dependent diffusion and double reptation. He combines reptation and Rouse modes in an expression of the relaxation modulus where a fraction of the relaxation spectrum is transferred from the Rouse to the reptation modes. Furthermore, he introduces an intermediate time Xj, proportional to M2, which can be considered as the Rouse time of an entangled polymer movii in its tube. But, in the cross-over region, the best fit of the experimental data is obtained by replaced Xj by an empirical combination of... 

Isolation Process in the Cross-Over Region. Chimowitz and Pennisi (13) developed a process for the separation and isolation of components from mixtures by operating in the multicomponent temperature-solubility cross-over region. The cross-over point of a pure component (dy/dT)p = 0 represents the pressure at which the dependence of solubility on temperature reverses itself. At lower pressures, the solubility is principally dependent on solvent density - raising temperature decreases density and thus solubility decreases. At higher pressures, solubility is principally dependent on solute sublimation pressure raising the temperature increases sublimation pressure and thus solubility increases. The cross-over point is therefore unique for each solute-solvent system. When there are two solutes, cross-over points occur at different pressures. At an intermediate pressure, the temperature can therefore be manipulated to deposit either component. 

Our results indicate that single-stage supercritical extraction processes are limited in their ability to separate trace components from complex mixtures even with the use of co-solvents. However, two-stage processes such as the ones developed here can significantly improve the degree of separation. Utilization of the multicomponent temperature-solubility cross-over region is a promising isolation technique. However, when the substrate contains a wide variety of components, multiple separations may be necessary to complete the isolation and the yields may be low. 

Cross-Over Region Purification. The multicomponent cross-over region can be used to isolate components from complex mixtures. However, clear-cut separations are difficult to achieve and multiple stages may be necessary. 

Cross-over Region (L R). As the temperature distance from T increases, interfacial thickness becomes smaller, and at a certain temperature T it is comparable to polymer dimension R. Adopting the expression eq.lO and R near T, we have... 

In this cross-over region, our theory based on the square gradient theory predicts a first order transition from a diffuse interface to a sharp one, which has been supported later by the calculation of Vrij et.al. based on the same model, If it would take place, polymer chains at the interface might collapse at the transition. This situation is similar to that of the adsorption theory by de Gennes. ... 

Tanaka et and by Weill and co-workers. In the recent modified blob theory, the change from Gaussian to excluded volume statistics is not as sudden as was at first surmized, and an estimate of the width of the cross-over region can be made and compared with experimental data. 

The value of these approaches to the problem is that they demonstrate the possibility of evaluating the memory function through the intermolecular correlation functions and structural dynamic factor of the system of interacting Brownian particles. Theoretical evaluations of the memory function are based on some approximations which, apparently, do not allow the calculation of quantities for the limiting case c M, but they are good in the cross-over region from Rouse to entanglement dynamics. 

Until recently, the nature of the ordinary phase was also controversely discussed as the strong increase of the diffusion coefficient with increasing polyion concentration at constant salt concentration was interpreted in terms of scaling laws for semi-dilute poiyelectrolyte concentrations. As it is now proven that the onset of the cross-over region does not depend on molar mass, the coupled mode picture unambiguously describes the ordinary phase best. See Sect. 3.2.3... 

If the exchange interaction J(r) is negative then S - T- also becomes possible, however, it is considered less important than the 5 - To mixing because the radical pair distance required is relatively small in comparison to that required in the 5 - To mechanism. Nonetheless, the 5 - r i mechanism can become important in chemical systems in which the radical pairs have a high hyperfine coupling constant or diffusion is hindered by the solvent viscosity, allowing the radical pair to spend more time near the cross-over region. 

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