Kerr equation

In the quantitative treatment of ion-exchange processes, several authors used the law of mass action. The main difference among these approaches is how the activities and surface concentration of the ions are treated. The first such approach was the Kerr equation, which uses the concentration of the ions on the solid and liquid as well but totally neglected the activity coefficients (Kerr 1928). The Vanselow (1932) equation applied activities in the solution and expressed the concentration of the ions on the solid phase in mole fraction, and in this way, it defined the selectivity coefficient (Equation 1.79). 

For applications where the ionic strength is as high as 6 M, the ion activity coefficients can be calculated using expressions developed by Bromley (4 ). These expressions retain the first term of equation 9 and additional terms are added, to improve the fit. The expressions are much more complex than equation 9 and require the molalities of the dissolved species to calculate the ion activity coefficients. If all of the molalities of dissolved species are used to calculate the ion activity coefficients, then the expressions are quite unwieldy. However, for the applications discussed in this paper many of the dissolved species are of low concentration and only the major dissolved species need be considered in the calculation of ion activity coefficients. For lime or limestone applications with a high chloride coal and a tight water balance, calcium chloride is the dominant dissolved specie. For this situation Kerr (5) has presented these expressions for the calculation of ion activity coefficients. 

Many interesting phenomena can arise in nonlinear periodic structures that possess the Kerr nonlinearity. For analytic description of such effects, the slowly varying amplitude (or envelope) approximation is usually applied. Alternatively, in order to avoid any approximation, we can use various numerical methods that solve Maxwell s equations or the wave equation directly. Examples of these rigorous methods that were applied to the modelling of nonlinear periodical structures are the finite-difference time-domain method, transmission-line modelling and the finite-element frequency-domain method." ... 

A common way to treat the problem of a picosecond pulse propagation in regular dielectric waveguide with Kerr nonlinearity was to solve the nonlinear Schrodinger equation (NLSE) for the slowly varying temporal amplitude of electrical field ... 

Kerr and Pauson successfully employed vinyl ester 47 as an ethylene substitute. The reaction can be carried out at ambient pressure and temperature (Equation (21)). ... 

Coherent External Field Modulated External Field Pulsed External Field Final Remarks Chaos in Kerr Oscillators A. Introduction Basic Equations... 

Generally, if Kerr systems are driven by external time-dependent forces, the equations of motion are nonintegrable and have to be studied numerically. 

In this study we suppose nonlinear organic material shows optical Kerr effect as n = n0+n2lEl2 and n2 = X<3)/(2n0). Moreover for simplification, we suppose the waveguides allow single mode propagations and TE polarization. After appropriate handling we get the following nonlinear coupled mode equations [ 12] ... 

According to Kerr and Webster (279), the radiolysis of TBP leads to alkyl radicals R and OP(OR)2OR . Investigations of the radical intermediates, by ESR examination or by use of electron scavengers, provided clear evidence regarding the formation of R radicals by dissociative electron capture (Equation 8.1) (294). 

Many of the different susceptibilities in Equations (2.165)-(2.167) correspond to important experiments in linear and nonlinear optics. x<(>> describes a possible zero-order (permanent) polarization of the medium j(1)(0 0) is the first-order static susceptibility which is related to the permittivity at zero frequency, e(0), while ft> o>) is the linear optical susceptibility related to the refractive index n" at frequency to. Turning to nonlinear effects, the Pockels susceptibility j(2)(- to, 0) and the Kerr susceptibility X(3 —to to, 0,0) describe the change of the refractive index induced by an externally applied static field. The susceptibility j(2)(—2to to, to) describes frequency doubling usually called second harmonic generation (SHG) and j(3)(-2 to, to, 0) describes the influence of an external field on the SHG process which is of great importance for the characterization of second-order NLO properties in solution in electric field second harmonic generation (EFISHG). 

Table 2.11 Kerr constant (10 26 V 2 m5 moF1) of pure liquids. A = 632.8nm, T = 298.15 K except where noted otherwise. Experimental gas phatse geometries. Experimental densities, see Equation (2.244). See text for definitions and ref. [30] for further details and pertinent references...

In this equation, pc is the concentration of molecules in the air, 7 the efficiency of MPI and a the photon number required to ionize air molecules, i.e. typically 8 to 10 [40]. Numerically integrating the NLSE yields the evolution of the pulse intensity I = e 2 as a function of the propagation distance, as shown in Fig. 14.3. The initial Kerr self-focusing as well as the MPI-generated plasma are well reproduced by such simulation. Here, the filament propagation is simulated over 60 m, limited by the computer capacity. 

Keesom forces, 242 Kelley-Bueche equation, 611 Kerr effect, 299, 349 Kinetic(s)... 

There are many other types of exchange equations, such as the Gaines and Thomas, the Kerr, and the Krishnamoarthy-Overstreet (Table 4.3). Generally, however, the Capon and Vanselow equations are the most widely used. 

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