Quadratic phase relation

Since there is a definite phase relation between the fiindamental pump radiation and the nonlinear source tenn, coherent SH radiation is emitted in well-defined directions. From the quadratic variation of P(2cii) with (m), we expect that the SH intensity 12 will also vary quadratically with the pump intensity 1 ... 

We start with a modified version of the gas-phase molar-balance equation (6.99) and a modified version of the liquid-phase molar-balance equation (6.102) with a quadratic equilibrium relation F(Xj) =Yj = aXj + b. This modification affects the term RMT by a nonlinearly, namely... 

Another important category of optical application uses the electro-optic switching character of perovskites. The system (Pb, La)(Zr, Ti)03 has been hot pressed and sintered to transparency by Haertling [26] and Snow [65], respectively. Phase relations are critical to successful fabrication where Pb volatility makes it difficult to retain compositional limits, and compositional variations make possible memory, linear, or quadratic applications. 

An inference of fundamental importance follows from Eqs. (2.3.9) and (2.3.11) When long axes of nonpolar molecules deviate from the surface-normal direction slightly enough, their azimuthal orientational behavior is accounted for by much the same Hamiltonian as that for a two-dimensional dipole system. Indeed, at sin<9 1 the main nonlocal contribution to Eq. (2.3.9) is provided by a term quadratic in which contains the interaction tensor V 2 (r) of much the same structure as dipole-dipole interaction tensor 2B3 > 0, B4 < 0, only differing in values 2B3 and B4. For dipole-dipole interactions, 2B3 = D = flic (p is the dipole moment) and B4 = -3D, whereas, e.g., purely quadrupole-quadrupole interactions are characterized by 2B3 = 3U, B4 = - SU (see Table 2.2). Evidently, it is for this reason that the dipole model applied to the system CO/NaCl(100), with rather small values 0(6 25°), provided an adequate picture for the ground-state orientational structure.81 A contradiction arose only in the estimation of the temperature Tc of the observed orientational phase transition For the experimental value Tc = 25 K to be reproduced, the dipole moment should have been set n = 1.3D, which is ten times as large as the corresponding value n in a gas phase. Section 2.4 will be devoted to a detailed consideration of orientational states and excitation spectra of a model system on a square lattice described by relations (2.3.9)-(2.3.11). 

Table 5.51 Energy terms relating standard state chemical potentials of pnre components (p,°o.) to lictive potentials in the host phase (p-Js) in the Will and Powell (1992) application of Darken s Quadratic Formalism to amphiboles. 

It is therefore not surprising that equation 31 fits a variety of curve shapes since) with the Langmuir term expsinded to n = 2 (i.e. equation 32), the relation in fact contains the ratio of two quadratics. Nevertheless, and particularly in view of the successful quantitative interpretation of the Langmuir forms of isotherm equations in the instances of the BET model (equations 7, et seq.) and retentions with blended stationary phases in gas chromatography (equations 14, et seq.), the LC relations cannot be dismissed as entirely empirical since, in any event, although such a connection has yet to be established, whatever interpretations are placed on the fitting constants must presumably involve at least the solute activity coefficients in the mobile and stationary phaises (see below) and, most likely, the (finite-concentration) activity coefficients pertinent to the mobile- and stationary-phase components as well. 

X=LILq is plotted as a function of the reduced temperature red at constant nominal stress CTn = 2.11xlO N mm . Here Lg is the loaded sample length at Tred l-OS. These results will also be used below to establish a close connection between the strain tensor and the nematic order parameter. It has also been shown that a quadratic stress-strain relation yields in the isotropic phase above the nematic-isotropic phase transition a good description of the data for ele-ongations up to at least 60% [4]. 

Due to the effect of external fields, the order can vary in space and gradient terms have to be added to the Landau expansion (8.9). Usually, only the terms up to the quadratic order are considered. There are many symmetry allowed invariants related to gradients of the tensorial order parameter [29]. However, in the vicinity of the phase transition, one is not interested in elastic deformations of the nematic director but rather in spatial variations of the degree of nematic order. Therefore, the pretransitional nematic system is described adequately within the usual one-elastic-constant approximation. 

Many polymer blends that are used in industrial practice have been found to be partially miscible. Examples are PVC/SAN, PC/SAN at certain AN compositions, PET/PHB, etc. The entropic difference model was developed by taking into account the change in entropy of mixing at glass transition in the Couchman model. Equation (6.15) quadratic expression for blend TgS is obtained upon Taylor approximation of the relation obtained by equating the entropy of the blend in glassy phase with the entropy of blend in rubbery phase at glass transition of the blend. 

The temperature difference AT between the actual and the phase transition temperature is taken into account via the coefficient of the quadratic term of the order parameter in the Landau expansion a T), which is proportional to AT. Ln denotes the elastic coefficient of a nematic liquid crystal. Both coefficients are related via the nematic correlation length, which is defined as = o/AT,... 

First is the form of relation between these three factors. The type of relation between Risk Indicator and distance or phase is relatively clear, as discussed earlier. However the function binding Risk Indicator and relative speed may raise concern. In the model we adopt the quadratic dependency, however there is no strong arguments for such a selection. Basically any other kind of relation is equally probable, this brings the uncertainty to the model, which in future can be addressed by alternative hypotheses testing, and studying the effect of the changes on the levels of Risk Indicators and their spatial distribution. 

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