Equilibrium linear susceptibility

The equilibrium linear susceptibility is, in the absence of an external bias field, given by... 

Figure 3.2. Equilibrium linear susceptibility in reduced units X = x Hi[/m) versus temperature for three different ellipsoidal systems with equation x ja +y lb + jc < I, resulting in a system of N dipoles arranged on a simple cubic lattice. The points shown are the projection of the spins to the xz plane. The probing field is applied along the anisotropy axes, which are parallel to the z axis. The thick lines indicate the equilibrium susceptibility of the corresponding noninteracting system (which does not depend on the shape of the system and is the same in the three panels) thin lines show the susceptibility including the corrections due to the dipolar interaction obtained by thermodynamic perturbation theory [Eq. (3.22)] the symbols represent the susceptibility obtained with a Monte Carlo method. The dipolar interaction strength is itj = d/ 2o = 0.02.

Figure 3.3. Equilibrium linear susceptibility (x/Xiso) versus temperature for an infinite spherical sample on a simple cubic lattice. The dotted lines are the results for independent spins, while the solid lines show the results for parallel and random anisotropy calculated with thermodynamic perturbation theory, as well as for Ising spins calculated with an ordinary high-temperature expansions. We notice in this case that the linear susceptibility for systems with random anisotropy is the same as for isotropic spins calculated with an ordinary high-temperature expansion. The dipolar interaction strength is hj = a/2a = 0.004.

The general expression for the equilibrium linear susceptibility is given by Eq. (3.22) with the following expressions for the coefficients... 

Cpo is the equilibrium specific heat and C oo includes the faster ones. When T t) stays close a certain value, Cpo and Cpoo will be constant. As in linear susceptibility, Cp can be measured in both the t and co domains, hence the integration and the Fourier transform of previous equation gives... 

The existence of the plateau appears as an effect of the non-linear susceptibility. Since the model has a zero-temperature transition, the linear equilibrium susceptibility should follow a Curie law (°=1/T) for all temperatures. Magnetizations measured after ZFC deviate from the plateau-value at low temperatures (fig. 14c). For a given field H and observation time t those ZFC curves merge with the equilibrium-plateaus at a temperature Tf(H, t). This defines a line Tf(//) below which for a given time scale t irreversible behavior is observed. Tf H) is shown in fig. 65 for different times t, and one can see in fig. 66 that the scaled curves, Tj(H)IT,(H = 0), are similar to the AT-line in mean-field theory. However, this line should vanish for r— oo (since Tj = 0). 

Most of the four above-mentioned properties for Raman spectra can be explained by using a simple classical model. When the crystal is subjected to the oscillating electric field = fioc " of the incident electromagnetic radiation, it becomes polarized. In the linear approximation, the induced electric polarization in any specific direction is given by Pj = XjkEk, where Xjk is the susceptibility tensor. As for other physical properties of the crystal, the susceptibility becomes altered because the atoms in the solid are vibrating periodically around equilibrium positions. Thus, for a particular... 

Structural Effects and Solvent. The effect of solvent on the equilibrium of Reaction 4 can be first discussed in terms of effects on the susceptibility to substituent effects. The values of pK2, characterizing this equilibrium, are a satisfactorily linear function of the Hammett constants correlation coefficient r (Table VI). The values of reaction constant p are practically independent of the ethanol concentration (Table VI), as was already indicated by the almost constant value of the difference (A) between pK2(H20) and p 2 (mixed solvent) for a given composition of the mixed solvent (Table I). The same situation is indicated for DMSO mixtures (Table II) by the small variations in A for any given solvent composition. In this case, the number of accessible p 2 values was too small to allow a meaningful determination of reaction constants p. The structural dependence for various water-ethanol mixtures is thus represented by a set of parallel lines. The shifts between these lines are given by the differences between the pK2H values (p 2 of Reaction 4 for the unsubstituted benzaldehyde) in the different solvent mixtures. 

Because of the bulk of comparable material available, it has been possible to use half-wave potentials for some types of linear free energy relationships that have not been used in connection with rate and equilibrium constants. For example, it has been shown (7, 777) that the effects of substituents on quinone rings on their reactivity towards oxidation-reduction reactions, can be approximately expressed by Hammett substituent constants a. The susceptibility of the reactivity of a cyclic system to substitution in various positions can be expressed quantitatively (7). The numbers on formulae XIII—XV give the reaction constants Qn, r for the given position (values in brackets only very approximate) ... 

Spectroscopic methods are very useful for determining molecular properties. Time-resolved spectroscopic methods are useful for monitoring the evolution of the molecular properties in real time. Moreover, time-resolved spectroscopic techniques have the best time resolution available among all kinds of time-resolved experimental techniques. Thus, very often time-resolved spectroscopic methods reveal the dynamics of a molecular system in the non-equilibrium regime. In this section, the density matrix method is applied to calculate the spectroscopic properties of molecular systems. These include the linear and non-linear optical processes, in equilibrium or non-equilibrium cases. The approach is based on the susceptibility theory. 

Linear response theory, applied to the particle velocity, considered as a dynamic variable of the isolated particle-plus-bath system, allows to express the mobility in terms of the equilibrium velocity correlation function. Since the mobility p(co) is simply the generalized susceptibility %vx(o ), one has the Kubo formula... 

Let us now come back to the specific problem of the diffusion of a particle in an out-of-equilibrium environment. In a quasi-stationary regime, the particle velocity obeys the generalized Langevin equation (22). The generalized susceptibilities of interest are the particle mobility p(co) = Xvxi03) and the generalized friction coefficient y(co) = — (l/mm)x ( ) [the latter formula deriving from the relation (170) between y(f) and Xj> (f))- The results of linear response theory as applied to the particle velocity, namely the Kubo formula (156) and the Einstein relation (159), are not valid out-of-equilibrium. The same... 

SEC was also used to study the transesterification reaction by following the rate of equilibration of PET samples with non-equilibrium molecular mass distributions. It was found that the cyclic trimer was more susceptable to alcoholysis than were linear oligomers. SEC was particularly useful in determining the cyclic trimer content in PET. 

Examples of linear response functions (susceptibilities) include the frequency dependent electrical conductivity (the Fourier transform of an equilibrium current autocorrelation function), dielectric susceptibility, which is the transform of a dipole moment autocorrelation function, along with stress, heat flux, and an assortment of velocity correlation functions. 

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