Logarithmic derivatives coefficients

If the incident light is normal to the cylinder axis the scattering coefficients have their simplest form. However, the coefficients (8.30) and (8.32) are not in the form most suitable for computations. If we introduce the logarithmic derivative... 

Dn mx) in the coefficients (4.88) is computed by the downward recurrence relation (4.89) beginning with Z)NMX. Provided that NMX is sufficiently greater than NSTOP and mx, logarithmic derivatives of order less than NSTOP are remarkably insensitive to the choice of Z)NMX this is a consequence of the stability of the downward recurrence scheme for pn. For vastly different choices of Z)NMX, and a range of arguments mx, computed values of DNMX 5 were independent of Z)NMX. Thus, NMX is taken to be Max(NSTOP, mjc ) -I- 15 in BHMIE, and recurrence is begun with Z)NMX = 0.0 + z 0.0. 

During the last 15 years, Abraham and his co-workers have established a set of five descriptors for the general description of logarithmic partition coefficients by linear regression. Their so-called linear free energy relationship (LFER) descriptors E, S, A, B, and V are effective parameters for the polarizability, polarity, hydrogen-bond acidity and basicity, and volume of the solute molecules, respectively [113-116]. They are mainly derived from experimental refraction and partition coefficients of the solutes. 

In order to describe the critical behavior of the galvanomagnetic properties quantitatively, we now define the logarithmic derivatives Xa(p)> Xr(p) °f the conductivity and of HalTs coefficient as... 

In Figs. 38 and 39 the results of calculations for effective conductivity, Hall coefficient and their logarithmic derivatives near the percolation threshold are... 

Figure 39. Dependence of the relative Hall coefficient (a, c, e) and the logarithmic derivative of the Hall coefficient (b, d,f) on concentration near the percolation threshold (a, b) the H — 0 (c, d)...

A number of interesting results have been obtained (due to our fractal model of structure and the iterative averaging method) for the Hall properties of the composite for example, use of a logarithmic derivative allows one to obtain critical exponents for the effective Hall coefficient (Fig. 39) for various values of the magnetic field H. When cti = a2 (Fig. 40) the effective conductivity is a constant if H = 0 and tends to zero if H —> oo near the percolation threshold. On the left of the percolation threshold (p < pc) the rise in the Hall coefficient is more rapid as the magnetic field increases (Fig. 40). On the right of the percolation threshold (p > pc) the Hall coefficient is practically independent of the concentration p. 

Structural disorder in ICP systems introduces localization phenomena and limits conductivity. The extent of disorder determines whether conductivity is metalHc, insulating, or critical. In the classical definition, a metal should have a positive temperature coefficient of the volume resistivity (p), finite conductivity as T —> 0 K, and the logarithmic derivative of the temperature dependence of conductivity, i.e., the reduced activation energy (W), must show a positive coefficient. The reduced activation energy as a function of temperature is determined by... 

No and Kazimi (1982) derived the wall heat transfer coefficient for the forced-convective two-phase flow of sodium by using the momentum-heat transfer analogy and a logarithmic velocity distribution in the liquid film. The final form of their correlation is expressed in terms of the Nusselt number based on the bulk liquid temperature, Nuft ... 

Hydrogen evolution, the other reaction studied, is a classical reaction for electrochemical kinetic studies. It was this reaction that led Tafel (24) to formulate his semi-logarithmic relation between potential and current which is named for him and that later resulted in the derivation of the equation that today is called "Butler-Volmer-equation" (25,26). The influence of the electrode potential is considered to modify the activation barrier for the charge transfer step of the reaction at the interface. This results in an exponential dependence of the reaction rate on the electrode potential, the extent of which is given by the transfer coefficient, a. 

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