Dispersion time independent

Most numerical methods for calculating molecular hyperpolarizability use sum over states expressions in either a time-dependent (explicitly including field dependent dispersion terms) or time-independent perturbation theory framework [13,14]. Sum over states methods require an ability to determine the excited states of the system reliably. This can become computationally demanding, especially for high order hyperpolarizabilities [15]. An alternative strategy adds a finite electric field term to the hamiltonian and computes the hyperpolarizability from the derivatives of the field dependent molecular dipole moment. Finite-field calculations use the ground state wave function only and include the influence of the field in a self-consistent manner [16]. 

One of the benefits of electrochemical batch injection analysis is that dilution of the sample with electrolyte is not necessary, see below. A sample of volume =sl00p.L is injected directly from a micropipette, tip diameter 0.5 mm, over the centre of a macroelectrode exactly as in a wall-jet system. This is equivalent to a flow injection system with zero dispersion. During the injection, and after a short initial period to reach steady-state, the hydrodynamics is wall-jet type and a time-independent current is registered. BIA was first devised using amperometric, e.g., [31], and potentiometric, e.g., [34], detection. A typical amperometric trace is shown in Fig. 16.5. By using a programmable, motorised electronic... 

The theory discussed until now is based on the Kramers-Heisenberg-Dirac dispersion relation for the transition polarizability tensor as given in Eq. (6.1-1). The expression shown in this equation describes a steady state scattering process and contains no explicit reference to time. Therefore, the resonance Raman theory which is based on the KHD dispersion relation is sometimes also termed as time-independent theory (Ganz et al., 1990). 

With solid-in-liquid dispersions, such a highly ordered structure - which is close to the maximum packing fraction (q> = 0.74 for hexagonally closed packed array of monodisperse particles) - is referred to as a soHd suspension. In such a system, any particle in the system interacts with many neighbours and the vibrational amplitude is small relative to particle size thus, the properties of the system are essentially time-independent [30-32]. In between the random arrangement of particles in dilute suspensions and the highly ordered structure of solid suspensions, concentrated suspensions may be easily defined. In this case, the particle interactions occur by many body collisions and the translational motion of the particles is restricted. However, this reduced translational motion is less than with solid suspensions - that is, the vibrational motion of the particles is large compared to the particle size. Consequently, a time-dependent system arises in which there will be both spatial and temporal correlation. 

For the most part, PFDs exhibit shear-thinning (pseudoplastic) rheological behavior that is either time-independent or time-dependent (thixotropic). In addition, many PFDs also exhibit yield stresses. The time-independent flow curves are illustrated in Figure 1. The shear-thinning behavior appears to be the result of breakdown of relatively weak structures and it may have important relationship to mouthfeel of the dispersions. Because the viscosity of non-Newtonian foods is not constant but depends on the shear rate, one must deal with apparent viscosity defined as ... 

Figure 2.3 Longitudinal dispersion of a band of contaminant in a time-independent unidirectional shear-flow.

The preceding discussion is based on the hypothesis that Tc> T however, there is no physical reason why this condition must be true. In the opposite limit Tc < T, the density of states is a more rapidly varying function of energy than the Boltzmann factor, and the peak in the electron density stays in the vicinity of the mobility edge, as illustrated in Fig. 16, and does not follow down into the tail states as in the dispersive a < 1 limit. After an initial logarithmic decay, it can be shown with the help of Fig. 16 and ai uments similar to those used above that at long times the drift mobility becomes a time-independent constant given by... 

With increasing time, the slope decreases, until at t it is extremely small. At somewhat higher temperatures (T = 335 K), a slope of 0 is measured. This corresponds to a time-independent current and thus to a dispersion-free transport at t —> tr. Obviously, the charge carriers have come into thermal equilibrium at the higher temperature even before the transit time, that is before the fastest ones have reached the back electrode. At the lower temperature, the relaxation process is evidently not yet completed at the time tr, so that the charge carriers are discharged before they can relax. 

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