Critical power laws

The basic Equation 3 becomes explicit as soon as the parameters P(PT) and A(Pr) are specified. This specification is the result of a lengthy trial procedure, based on the following factors (a) consistency with known near-critical power laws, (b) approximate consistency with the law of the rectilinear diameter (c) the tendency of the low-pressure vapor curve to form a straight line in logarithmic coordinates, as predicted by Equation 5 (d) imposition of a definite form of the corresponding-states principle and (e) consistency with a large collection of experimental data. 

Fig. 23. Left Zero field spectra of polycrystalline DyAg at two temperatures within the paramagnetic regime. The Neel temperature is 60 K. The two spectra are shifted vertically for clarity. Right Temperature dependence of the relaxation rate A derived from fits of an exponential decay of polarization to spectra of the type shown in the left hand panel. Typical is the sharp rise on approach to the magnetic transition temperature. The line is a guide to the eye, but fits to a critical power law are often possible. After Kalvius et al. (1986).

Fig. 77. [iSR studies of RGa mtermetallics. Left Temperature dependence of the fast relaxation rate in GdGa below and above the Neel temperature (12.5 K). Right Temperature dependences of the fast and slow relaxation rates in paramagnetic TbGa. The solid lines are fits to a critical power law (see text) using 20.25 K as the Neel temperature. From Lidstrom et al. (1996c). 

Fig. 83, Temperature dependence of the pSR relaxation rate in NdRhjSij. Left A single e>q>onentially relaxing signal is seen for the c-axis parallel to muon spin polarization. The rate below the Neel temperature is fitted to a two-magnon process. Right Critical behavior of the paramagnetic relaxation rate for the c-axis perpendicular to the muon spin polarization (longitudinal spin fluctuations). The insert shows a fit to a critical power law.

Asymptotic Critical Power Laws, Critical Exponents, ical Amplitudes and Crit-... 

Experimentally, it is well established that asymptotically close to the critical point all physical properties obey simple power laws. The universal powers in these laws are called critical exponents, the values of which can be calculated from RG recursion relations. The phenomenological approach that interrelates the critical power laws is called scaling theory. In particular, the isochoric heat capacity diverges at the vapour-liquid critical point of one-component fluids along the critical isochore as... 

Table 10.6 Critical power laws for thermodynamic properties.

The second contribution to g(r) in Eq. (3.2) is called the correlation hole effect by deGennes and is associated with the longer wavelength universal aspects of chain connectivity and interchain repulsive forces. On intermediate length scales it has a critical power law form due to chain conformation self-similarity, and this simple analytic form remains an excellent representation even for chemically realistic models when intersite separations exceed an intrinsic (/V-independent) distance of the order of 3-5 site diameters... 

It should be noted that the validity of the critical power laws (6.10)-(6.11) is restricted to a very small range of temperatures near the critical point (Levelt Sengers Sengers 1981). Equations of state that incorporate these critical power laws but which remain valid in an appreciable range of densities and temperatures in the critical region have been developed (Sengers 1994). 

Figure 67 Probability distribution tis of clusters with S water molecules at planar surfaces of various sizes at surface coverages close to the percolation thresholds C = 0.078 (circles), 0.070 A (squares), and 0.078 (triangles). The critical power law ns is shown by a solid line.

Figure 68 Probability distribution ns of clusters with S water molecules for several surface coverages C (in A below and above the percolation threshold (Cp 0.078 A ) at the plane with L — 80 A. The critical power law ns -2.05 shown by the solid lines. The distributions are shifted vertically by one order of magnitude consecutively. Reprinted, with permission, from [394],...

The critical power-law exponents (t and t ) are expected to be "universal," independent of the system details [42,48,54], and dependent only on fundamental properties such as symmetries, dimensions, interaction ranges, etc. [42,55]. (For more on self-organized criticality, see Chapter 1 by Franklin.)... 

Table 1. Critical power-law parameters at the rheological gel point for incipient i-carrageenan and t-carrageenan /chitosan NPEC gels in 0.25 mol-L NaCl.

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