Susceptibility critical exponents

Zhang, K.C., Briggs, M.E., Gammon, R.W., and Levelt Sengers, J.M.H. The susceptibility critical exponent for a nonaqueous ionic binary mixture near a consolute point. 

In the (PB PS) mixture a crossover from Ising to isotropic critical Lifshitz behavior was observed at about 4.8% diblock concentration as indicated by the dashed area in the phase diagram of Fig. 23. On the other hand, a crossover to a renormalized Lifshitz critical behavior was not observed in this system. The susceptibility critical exponent y of both systems has been depicted in Fig. 26 versus diblock concentration. A crossover of the exponent y at about 4.8% and 6.2% from the Ising 1.24 to larger values is visible for the (PB PS) and (PEE PDMS), respectively. A constant y = (1.62 0.01) and... 

Figure A2.3.29 Calculation of the critical temperature and the critical exponent y for the magnetic susceptibility of Ising lattices in different dimensions from high-temperature expansions.

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... 

Tc. The two power-law exponents are not independent but depend on a single parameter, the so-called critical exponent X, which is specific for a given interaction potential (e.g., hard spheres). Actually, the interaction potential enters the MCT equations only indirectly via the structure factor S(q), which fixes the nonlinear coupling in the generalized oscillator equation. It is important to note that the MCT exponents are not universal in contrast to those of second-order phase transitions. In the case of hard spheres, for example, S(q) can be calculated via the Percus-Yevick approximation [26], and the full time and -dependence of < >(q. f) were obtained. As an example, Fig. 10 shows the susceptibility spectra of the hard-sphere system at a particular q. Note that temperature cannot be defined in the hard-sphere system instead, the packing fraction cp is used as a parameter. Above the critical packing fraction 0), which corresponds to T < Tc in systems where T exists, the a-process is absent (frozen) and only the fast dynamics is present. At cp < tpc the a-peak and the concomitant susceptibility minimum shift to lower frequencies with increasing cp, so that the closer cp is to the critical value fast dynamics can be identified (curve c in Fig. 10). 

The critical isotherm is characterized by t = 0 and H = 2brj. This immediately shows that the critical exponent for the magnetic field variation with r = M is 5 = 3, in conformity with the standard mean field value encountered in the literature. Returning to Eq. (7.5.10) and differentiating with respect to M one obtains the magnetic susceptibility as... 

More generally, for other lattices and dimensions, numerical analysis of the high-temperature expansion provides information on the critical exponents and temperature. The high-temperature expansion of the susceptibility may be written in powers of AT = p J as... 

All other critical exponents may be obtained from familiar scaling relations. For example, the susceptibility exponent 7 is given by ... 

A sensitive test of the crossover behavior is obtained from an analysis of the effective critical exponent of the susceptibility (the third derivative of the free energy), defined as... 

Experimentally determined susceptibility, 1(0) versus T, like that shown in Figure 7.11, can effectively be analyzed by applying a single function describing 1(0) within the whole one-phase regime. Based on an e-expansion model, one may develop a function that describes the experimental data very well [13], as seen in Figure 7.11. The parameters of the crossover Junction are the Ginzburg number, the critical temperature, and the critical exponents. 

The straightforward consequence of this analogy is that the SmA-SmC transition may be continuous at a temperature Tsmc-smA with X Y critical exponents. Below Tsmc-smA t e tilt angle 9 for instance should vary as 0 = 0qUI with P=035. Above T mc-SmA external magnetic field can induce a tilt 9 proportional to the susceptibility if r with 7= 1.33. 

Linear and non-linear effects of thermal composition fluctuations become visible in a scattering experiment. Within the mean field and Ising regimes the susceptibility S(0) and the correlation length are described by simple scaling laws as functions of the reduced temperature r according to Cr ) with the critical amplitudes C( o) and the critical exponents y(v). The critical exponents y(v) are known to be equal to y = 1 and 1.239 0.003 and V = 0.5 and 0.634 0.001 in the mean field and Ising cases, respectively [66]. The mean field case has already been discussed in the context with Eq. 11. 

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