Order parameter, susceptibilities

The structure factor S(q as defined in Eq. (54) in terms of the Ising pseudospins Si, in the framework of the first Bom approximation describes elastic scattering of X-rays, neutrons, or electrons, from the adsorbed layer. SCq) is particularly interesting, since in the thermodynamic limit it allows to estimate both the order parameter amplitude tj/, the order parameter susceptibility X4, and correlati length since for q near the superstructure Bragg reflection q we have (k = q— q%)... 

The order parameter susceptibility, which diverges at the transition temperature, is the nonlinear susceptibility Xa. defined as M/h = Xo + + s where Xo is... 

Some insight into the mechanism by which Pc is changed is provided by the frequency of the soft mode responsible for the phase transition. In the stability field of the high symmetry (tetragonal) phase, the inverse order parameter susceptibility, x of the order parameter varies as... 

Expressions for the elastic constant variations due to the tetragonal orthorhombic transition in stishovite derived by applying Equation (5) to Equation (23) are listed in Table 4. The most important factors in determining the form of evolution of the elastic constants are, self-evidently, the strength of coupling between the order parameter and strain ( ii, ) the order parameter susceptibility, x given by... 

There is a marked curvature to all the elastic constants below the transition point, which reflects the variation of the order parameter and the order parameter susceptibility. No individual elastic constant or symmetry-adapted combination of elastic constants is expected to tend to zero at the transition point. In this case the agreement between observed and calculated values shown in Figure 18 implies that the model represented by Equation (28) provides a good description of the phase transition. Agreement is not as close for C33 as it is for the other elastic constants, however, suggesting that the causes of strain parallel to [001] of a-quartz have not yet been fully explained. 

R. L. C. Vink, J. Horbach, and K. Binder (2005) Critical phenomena in coUoid-polymer mixtures Interfacial tension, order parameter, susceptibility, and coexistence diameter. Phys. Rev. E 71, 011401... 

Figure 60. Scaling plot (5.9) of the head-tail order parameter susceptibility obtained from Monte Carlo simulations of complete monolayer (-J3 x yfi)R30° CO on graphite (6 = 0.13 A) with two-dimensional Ising exponents y = 7/4 and y = 1, and = 11.9 K from the cumulant intersection in Fig. 59. Only the scaling regime 1 - T/T L " 0 is shown, and the data above (asterisks) and below (circles) the transition are superimposed for all system sizes I = 18. .. 60 solid and dotted lines are the amplitude fits (5.11) of the data above and below the transition, respectively. (Adapted from Fig. 4c of Ref. 215.)...

In eq. (1) the ETT order parameter z = s(p-Pc) measures, in a convenient direction, the chemical potential from that corresponding to the ETT. From the values given in Table I for the above s and q, we readily see that the occurrence of the ETTs discussed in this paper always implies an increase of the alloy free energy. Thus, CuPt random alloys, that just below and above the equiatomic concentration present both the relevant ETT s, are less stable than CuPd or AgPd and, thus more likely to be destabilised. Moreover, the proximity to both the critical concentrations implies large contributions to the BSE from the X and L points. Now, the concentration wave susceptibility, Xcc(q). as observed by Gyorffy and Stocks, is proportional to... 

For example, 0 describes the temperature dependence of composition near the upper critical solution temperature for binary (liquid + liquid) equilibrium, of the susceptibility in some magnetic phase transitions, and of the order parameter in (order + disorder) phase transitions. 

On the other hand, the nonlinear optical properties of nanometer-sized materials are also known to be different from the bulk, and such properties are strongly dependent on size and shape [11]. In 1992, Wang and Herron reported that the third-order nonlinear susceptibility, of silicon nanocrystals increased with decreasing size [12]. In contrast to silicon nanocrystals, of CdS nanocrystals decreased with decreasing size [ 13 ]. These results stimulated the investigation of the nonlinear optical properties of other semiconductor QDs. For the CdTe QDs that we are concentrating on, there have been few studies of nonresonant third-order nonlinear parameters. 

For the application of QDs to three-dimensional biological imaging, a large two-photon absorption cross section is required to avoid cell damage by light irradiation. For application to optoelectronics, QDs should have a large nonlinear refractive index as well as fast response. Two-photon absorption and the optical Kerr effect of QDs are third-order nonlinear optical effects, which can be evaluated from the third-order nonlinear susceptibility, or the nonlinear refractive index, y, and the nonlinear absorption coefficient, p. Experimentally, third-order nonlinear optical parameters have been examined by four-wave mixing and Z-scan experiments. 

The values of the order parameter S, which describe the degree of molecular orientation to film normal, and effective second-order susceptibility at 45"... 

Note that for = 2 both Eqs. (17), (18) essentially reduce again to the Ising Hamiltonian, Eq. (9), with nearest neighbor interaction only. The latter model is described by the following critical behavior for its order parameter if/, ordering susceptibility and specific heat C ... 

The prefactor L in Eq. (36) is understood from the normalization condition, i d Pj ,( ) = 1, which must hold at all temperatures. From Eq. (36) one immediately obtains finite-size scaling relations for the order parameter < >T and ordering susceptibility by taking suitable moments of the distribution (note PJl ) is symmetric around ijf — O in the absence of symmetry-breaking fields and thus < > = 0) ... 

Fig. 21. Finite-size scaling plot of (a) order parameter m using Tc= T2 = 4.26 (b) the ordering susceptibility % using...

Risk Assessment. This model successfully described the disposition of chloroform in rats, mice and humans following various exposure scenarios and developed dose surrogates more closely related to toxicity response. With regard to target tissue dosimetry, the Corley model predicts the relative order of susceptibility to chloroform toxicity consequent to binding to macromolecules (MMB) to be mouse > rat > human. Linking the pharmacokinetic parameters of this model to the pharmacodynamic cancer model of Reitz et al. (1990) provides a biologically based risk assessment model for chloroform. 

The soft mode concept can be extended to all distortive phase transitions (transitions with relatively small atomic displacements), even if they are only close to second order. In the case of a ferro-distortive transition, as for example in BaTiOs or KDP, the order parameter is proportional to the spontaneous electric polarization Fj. d F/ dp is not only proportional to co, but also to the dielectric susceptibility. This does not, however, mean that all components of the order parameter eigenvector must contribute to Ps. 

These measures of transition susceptibility, extent of ordering, and transition breadth apply equally well to equilibrium polymerization and self-assembly as to glass formation. Thus, these measures offer a unified description of the characteristics of rounded transitions. The XgL Eq. (A3) is a susceptibility measure where L corresponds to the average cluster relative mass, the order parameter >1) describes the extent of conversion of particles to the cluster state, and the variance in L or >1) reflect the transition broadness [246]. 

Most of the experimental results on CJTE can be explained on the basis of molecular field theory. This is because the interaction between the electron strain and elastic strain is fairly long-range. Employing simple molecular field theory, expressions have been derived for the order parameter, transverse susceptibility, vibronic states, specific heat, and elastic constants. A detailed discussion of the theory and its applications may be found in the excellent review by Gehring Gehring (1975). In Fig. 4.23 various possible situations of different degrees of complexity that can arise in JT systems are presented. 

The magnetic susceptibilities of dimer liquid crystals such as NC-Ph—Ph—O—(CH2>n—Ph—Ph—CN(n 9, 10) are measured by a SQUID magnetometer. The results obtained are interpreted within the framework of the RIS approximation, the effect arising from the conformational anisotropy of the flexible spacer being strictly taken into account. The order parameters of the mesogenic core axis thus estimated are found to be consistent with those directly observed at just below 7N) by the ZH NMR technique using mesogen-deuterated samples. 

Cv Pc 8 a an P Y c f specific heat reduced density critical exponent for the critical isotherm critical exponent for the specific heat critical exponent for the specific heat along isot r critical exponent for the order parameter critical exponent for the susceptibility reduced temperature friction coefficient... 

The use of an electric field is not the only effective way to influence the LC polymer structure, magnetic fields displays a closely similar effect167 168). It is interesting as a method allowing to orient LC polymers, as well as from the viewpoint of determining some parameters, such as the order parameter, values of magnetic susceptibility, rotational viscosity and others. Some relationships established for LC polymer 1 (Table 15), its blends with low-molecular liquid crystals and partially deuterated polyacrylate (polymer 4, Table 15) specially synthesized for NMR studies can be summarized as follows ... 

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