Critical amplitudes/exponents

Another critical exponent (5) is defined considering the variation of the order parameter at Tc as a function of the conjugate field H, D being the associated critical amplitude,... 

At this point, we mention a further consequence of the universality principle alluded to above. For each universality class (such as that of the Ising model or that of the XY model, etc.) not just the critical exponents are universal, but also the scaling function F(H), apart from non-universal scale factors for the occurring variables (a factor for H we have expressed via the ratio C/B in eq. (84), for instance). A necessary implication then is the universality of certain critical amplitude ratios, where all scale factors for the variables of interest cancel out. In particular, ratios of critical amplitudes of corresponding quantities above and below Tc, A+j A [eq. (7)], C+jC [eq. (6)] and f+/ [eq. (38)] are universal (Privman et al., 1991). A further relation exists between the amplitude D and B and C 1 Writing M H -> oo) = XHl/ cf. eqs. (87) and (91), the universality of M(H) states that X is universal. But since 0 = B tfM H) = B t PXH = B] SC S H] X, a comparison with eq. (45) yields... 

While there is reasonable experimental evidence for the universality of scaling functions, the experimental evidence for the universality of amplitude relations such as eq. (94) is not very convincing. One reason for this problem is that the true critical behavior can be observed only asymptotically close to Tc, and if experiments are carried out not close enough to Tc the results for both critical amplitudes and critical exponents are affected by systematic errors due to corrections to scaling. For example, eq. (6) must be written more generally as... 

For ordinary critical phenomena, such a spatial anisotropy is not very important — it gives rise to an anisotropy of the critical amplitude, of the correlation length in different lattice directions ( i = r[-1, fx = xffl 1 ), while the critical exponent dearly is the same for all spatial directions. Of course, this is no longer necessarily true at Lifshitz points There is no reason to assume that both functions / (p), K (p) in eq. (122) vanish for p = pt Let us rather assume that only ih(pl) — 0 while K (pO > 0 this yields the uniaxial Lifshitz point (Homreich et al., 1975). We then have to add a term f Ki (p)[32 (x)/9x ]2 to eq. (122) to find... 

Fig. 29a. Log-log plot of critical amplitudes B, f + and D versus chain length N, for the model of Fig. 3 and 4> = 0.2. Here f + means the amplitude of S coUfq = 0)/(l — 4> )2 for T > Tc and 6 is the amplitude at the critical isotherm, m = D(Ap/kBT)1/6, where 6 = (y + (3) /p. Points for N = 1 refer to the standard Ising model. From Sariban and Binder [107]. b Log-log plot of fi (denoted as C[ in the figure) vs N. The straight line is the best fit to all data with N g 32, using the theoretical exponent P — l/2 — 0.176, Eq. (122). These data refer to the bond fluctuation model at

Over the last 10 years or so, a great deal of work has been devoted to the study of critical phenomena in binary micellar solutions and multicomponent microemulsions systems [19]. The aim of these investigations in surfactant solutions was to point out differences if they existed between these critical points and the liquid-gas critical points of a pure compound. The main questions to be considered were (1) Why did the observed critical exponents not always follow the universal behavior predicted by the renormalization group theory of critical phenomena and (2) Was the order of magnitude of the critical amplitudes comparable to that found in mixtures of small molecules The systems presented in this chapter exhibit several lines of critical points. Among them, one involves inverse microemulsions and another, sponge phases. The origin of these phase separations and their critical behavior are discussed next. 

In Eqs. (14)-(18) a, / , 7, and S are universal critical exponents, while Fq, Aq, B01 and Do are critical amplitudes. The superscript and correspond to positive or negative AT, respectively. The values of the critical amplitudes are system dependent. However, between the six amplitudes introduced above there exist four universal relations (see Table 1). 

Therefore, only two amplitudes are independent. It has been established theoretically [1, 5] and verified experimentally [6, 7] that all fluids and fluid mixtures, regardless of variety and complexity in their microscopic structure, belong to the same universality class, i.e. they have the same universal values of the critical exponents (Table 2) and of the critical-amplitude ratios (Table 1) as those of the 3-dimensional Ising model. The physical reason of the critical-point universality originates from the divergence of the order-parameter fluctuations near the critical point. 

Critical exponents and critical amplitudes for Ising dipolar ferromagnets... 

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