Extended equation frequency shift

In the G(w)-approximation, introduced by Bigeleisen and Mayer (8) and later extended to higher orders by Bigeleisen (4), the reduced partition function ratio (Equation 1) is expanded in terms of the isotope frequency shifts, Au, = — Ui. The first three terms are... 

The second difference between Equation 13 and Equation 5 is the fact that there is now a frequency shift in the spectrum which is caused by the extended source. This frequency shift depends on the unshifted frequency o-and the solid angle ii. The spectrum is spread to lower frequencies and the shifted frequency is given by... 

An extended equation for the shift in frequency of an EQCM may be written as... 

In practice one will be dealing with a more complex spin system but it is easy to extend these equations to the AX and AMX cases and hence calculate the frequencies at which the selective proton rf field, B3, must be applied to perturb the Bloch-Siegert shifted transitions. (199)... 

To validate the extended Cole-Cole equation, Li et al. measured the frequency effect using a JNC JC-BPOIM polymer-stabilized BPLC. The employed IPS cell has 10 pm electrode width and 10 pm electrode gap, and 7.5 pm cell gap. Results are plotted in Figure 14.8, where the VT curves gradually shift to the right side and V increases as frequency increases. At 5 kHz, the transmittance at 60 V ns is only -10% of that of the peak transmittance. These results indicate that frequency has a tremendous impact on the electro-optic properties of this BPLC cell. 

In this case, an apparent activation energy is determined, and it has higher values than secondary relaxations 100-300 kJ/mol for urethane-soybean oil networks (Cristea et al. 2013), 200-300 kJ/mol for polyurethane-epoxy interpenetrating polymer networks (Cristea et al. 2009), more than 400 kJ/mol for semicrystalline poly(ethylene terephtalate) (Cristea et al. 2010), and more than 600 kJ/mol for polyimides (Cristea et al. 2008, 2011). The glass transition temperature is the most appropriate reference temperature when applying the time-temperature correspondence in a multifrequency experiment. The procedure allows estimation of the viscoelastic behavior of a polymer in time, in certain conditions, and is based on the fact that the viscoelastic properties at a certain tanperature can be shifted along the frequency scale to obtain the variation on an extended time scale (Brostow 2007 Williams et al. 1955). The shift factor is described by the Williams-Landell-Ferry (WLF) equation ... 

In a perfectly ordered solid or a perfectly ordered macromolecule excitons move according to quantum-mechanical kinetics, i.e., like wave packets. Because this requires strict phase relationships in space and time, this mode of motion, addressed in the literature as coherent motion of excitons, is perturbed by all deviations from regularity. Chains of conjugated polymers always include various defects such as kinks and torsions. These break the conjugation. Typically, regular sequences extend only over five to ten repeat units. The results of measurements for a series of oligomers like the ones displayed in Fig. 7.4 can be used for the estimate. The red shift of the exciton frequency to with increasing monomer number, n, can be described by the equation... 

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