Frequency dependence defined

Similarly, we can define the corresponding frequency-dependent second-order, and third-order,... 

Drops coalesce because of coUisions and drainage of Hquid trapped between colliding drops. Therefore, coalescence frequency can be defined as the product of coUision frequency and efficiency per coUision. The coUision frequency depends on number of drops and flow parameters such as shear rate and fluid forces. The coUision efficiency is a function of Hquid drainage rate, surface forces, and attractive forces such as van der Waal s. Because dispersed phase drop size depends on physical properties which are sometimes difficult to measure, it becomes necessary to carry out laboratory experiments to define the process mixing requirements. A suitable mixing system can then be designed based on satisfying these requirements. 

As the wetting front advances at speed U, the solid undergoes a strain cycle at a variety of frequencies, /, the local frequency depending on the distance of the element of solid from the contact line at the moment under consideration. The solid the furthest from the contact line, yet still perturbed by the presence of the three-phase line, is at a distance of ca. to and thus feels a strain cycle at frequency [//to. At the other extreme, near the lower cutoff at x = 8, the frequency is ca. [7/8. The latter frequency will be dominant, since it is in the direct vicinity of the three-phase line that the solid is strained the most. As a consequence, and using Eq. (10), we can define the rate at which work is being done as ... 

Equations (35) and (36) define the entanglement friction function in the generalized Rouse equation (34) which now can be solved by Fourier transformation, yielding the frequency-dependent correlators . In order to calculate the dynamic structure factor following Eq. (32), the time-dependent correlators are needed. 

The final 2D spectra obtained by shearing or by using the split-/, pulse sequence can benefit from a proper scaling of the isotropic and anisotropic dimensions in order to facilitate comparisons between various experiments [171, 176-179]. In our opinion, the most convenient way to reference such 2D spectra is to define a ppm scale using, in the isotropic dimension, an apparent Larmor frequency depending on the given experiment and defined as... 

Figure 8. Frequency dependent orientation TCFs for HOD/H2O at room temperature. Sub ensembles are defined according to the value of the OD stretch frequency at t 0, and the curves correspond to five sub ensembles as labeled in the graph. See color insert.

Instead of recording separately the decays of the two polarized components, we measure the differential polarized phase angle A (co) = — i between these two components and the polarized modulation ratio A (co) = mfm . It is interesting to define the frequency-dependent anisotropy as follows ... 

In the next section, we will develop a simple model to predict the frequency dependence of the relative dielectric constants si and 2 of a given material. At that point, we will be able to determine the measurable optical magnitudes defined in Chapter 1 at any particular wavelength (or frequency) if the relative dielectric constants (and thus n and k) are known at that wavelength. 

Thus, the Drude model predicts that ideal metals are 100 % reflectors for frequencies up to cop and highly transparent for higher frequencies. This result is in rather good agreement with the experimental spectra observed for several metals. In fact, the plasma frequency cop defines the region of transparency of a metal. It is important to realize that, according to Equation (4.20), this frequency only depends on the density of the conduction electrons N, which is equal to the density of the metal atoms multiplied by their valency. This allows us to determine the region of transparency of a metal provided that N is known, as in the next example. 

In the response function terminology [47] the i, j component of the frequency-dependent dipole polarizability tensor — w) (or the ij, kl component of the traceless quadrupole polarizability tensor is defined through... 

Second-order molecular properties can be defined as second derivatives of the (time-averaged) quasienergy Q with respect to frequency-dependent perturbation strengths b (wb) at zero perturbation (s=0)... 

Frequently, the number of active sites is expressed in mole units (the number of active sites divided by the Avogadro number) and thus, turnover frequency is found in s"1 units. For a specific reaction, the turnover frequency depends on the nature of the catalytic active site, the temperature, and the reactants concentration. The above-defined catalytic rate could be described as an active-site level rate. 

For most treatments, the spectral density, J(a>), Eq. 2.86, also referred to as the spectral profile or line shape, is considered, since it is more directly related to physical quantities than the absorption coefficient a. The latter contains frequency-dependent factors that account for stimulated emission. For absorption, the transition frequencies ojp are positive. The spectral density may also be defined for negative frequencies which correspond to emission. 

All the system response curves in frequency and time domains were calculated numerically from equations that are much too involved to reproduce in detail here. Transfer functions in Laplace transform notation are easily defined for the potentiostat and cell of Figure 7.1. Appropriate combinations of these functions then yield system transfer functions that may be cast into time- or frequency-dependent equations by inverse Laplace transformation or by using complex number manipulation techniques. These methods have become rather common in electrochemical literature and are not described here. The interested reader will find several citations in the bibliography to be helpful in clarifying details. 

In the expression of the eigenvalues, the wavevector- and frequency-dependent transport coefficients are present. As mentioned before, these are defined by the U matrix elements. Thus in the definition of all these transport coefficients there appears a structure of the form... 

As noted before, the polarizability of a material is frequency dependent so that the wave surfaces are frequency dependent. If one of the wave surfaces at the second harmonic frequency intersects one of the wave surfaces at the fundamental frequency, phase-matched SHG can occur. The ray passing through the origin of the ellipsoids and their point of intersection defines the direction of propagation for phased-matched SHG. 

Finally, the circuit is solved. It can be done in two ways. Either a corresponding physical circuit is actually built from these electrical components and its transfer function (H) is measured, or the transfer functions of the individual elements and of the entire circuit are represented by explicit mathematical equations and the transfer function is calculated. There are always certain eigenvalues in these solutions that aid in the assignment of physical meaning to the calculated parameters. The transfer function is defined as the frequency-dependent ratio of the output voltage to the input voltage. It is a complex variable for an AC excitation signal. 

Powell et al. give an excellent review of several approaches to interpret the frequency dependence of Tle and T2e in these systems [71]. One convenient approach is that developed by Hudson and Lewis [72], who showed that the eigenvalues of the relaxation matrix R as defined in Bloch-Wangsness-Red-field (BWR) theory [73] are functions of rv and the experimental frequency co, and are related to the relaxation time T2ei of the i-th allowed electron spin transition by the expression ... 

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