Ising amplitudes

Upon inverting the expansion of AT one obtains expansions of Ap, AP, and Ap in terms of AT. Using the relationships between the parametric coefficients a and k and the Ising amplitudes Aq, Bq, and rasymptotic power laws specified in Table 10.6 with... 

The mean field critical amplitudes of the correlation length of both blends are the same within experimental uncertainty and increase by about 10% for a 80% polymer content. The Ising amplitudes are smaller than their mean field numbers, and, instead, they decrease with polymer content by about 20%. In addition, the amplitudes are different in both blends PB(1,4)/PS shows slightly more than 10% larger values than PB(1,2 1,4)/PS. The mean field amplitude is proportional to the polymer radius of gyration and the square root of (1 + pr /rc) (Eq. 29) the polymer parameters are very similar, (Table 2) so the result of equal amplitudes is not surprising. On the other hand, the Ising critical amplitudes seem to depend more sensitively... 

Fig. 19 Critical amplitudes C of the susceptibility and o of the correlation length versus total polymer concentration. The Ising C slightly decreases by about 10%, while the mean field amplitude shows a slightly stronger relative increase of 20%. The critical amplitudes of the correlation length in the lower figure exhibits an increase of the mean field amplitudes and a decrease of the Ising amplitudes with polymer concentration...

The mean-field critical amplvitude of polymer blends was already expressed in this section, whereas the 3D-Ising amplitude was derived by Binder for symmetric blends... 

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%)... 

In concluding this section, we draw attention to the amplitudes 0 derived from the scattering experiments. As shown later, 0 enters into theoretical expressions for the crossover temperature. Large 0 favor a small Ising regime. In simple nonelectrolyte mixtures, 0 is of the order of the molecular... 

The previous treatment deals with a one-component order parameter (such as for a commensurate Peierls distortion) but does not apply to situations where the order parameter is complex with an amplitude and a phase (superconductivity, incommensurate Peierls, or spin density wave transitions). The latter situation is analogous to classical moments which can rotate freely in an XY plane. The coherence length of the XY model is less strongly divergent at low temperature than for the Ising model,... 

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... 

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

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 this analysis, the contribution to the scattering from the micelles in the mixtures was neglected, and the total scattered intensity was attributed to critical fluctuations over the entire e range investigated. For each critical point, the values of v, and y, obtained as described above are very close to those found in independent experiments by varying X at constant temperature [114]. These values are found to vary continuously from the Ising values (y, = 1.21 0.06 v,= 0.64 0.03 for X = 5A7) to significantly smaller ones ()y = 0.40 0.04, V, = 0.21 0.02 for X = 1.03) as the critical endpoint P is approached. For the six critical points studied, the relation y = 2v is verified. The sharp decrease in the values of the exponents in the vicinity of P g is accompanied by a rapid increase in the correlation amplitude Co (3-30 nm). 

Relations between critical amplitudes (3-dim. Ising systems)... 

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. 

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