Ising exponents

There is now agreement between experiment and theory on the Ising exponents. Indeed it is now reasonable to assume that the theoretical values are better, smce their range of uncertainty is less. 

Table A2.5.1 shows the Ising exponents for two and tluee dimensions, as well as the classical exponents. The uncertainties are those reported by Guida and Ziim-Justin [27]. These exponent values satisfy the equalities (as they must, considering the scalmg assumption) which are here reprised as functions of p and y ...

Fig. 31. Log-log plot of mN,/2 at T = Tc vs L/N, for the bond fluctuation model at <)>, = 0.5. Different symbols show various chain lengths N, as indicated. The dashed line illustrates the power law expected in the mean-field regime of finite size scaling M >ocL-3, for L/N 1) and the dash-dotted line in the non-mean-field regime K M >ocL-p/v, with p/v 0.515 for Ising exponents [274]. From Binder and Deutsch [275].

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.)...

Dietler, G. cUid Cannell, D.S. (1988) Observation of 3d-Ising exponents in micellar solutions, Phys. Rev. Lett. 60, 1852-1855. 

A very extensive numerical study has been recently carried out by Singh and co-workers. ° Except for the critical divergence Ising exponent, excellent agreement is found between PRISM theory based on the R-... 

Alternatively, if the experimental uncertainty is interpreted as a coarse graining in a log( m) vs log(e) plot, then the determination of the exponents would be closer to the experimental one, if a limiting quotient definition of the critical exponent is used instead of the logarithmic derivative of equation (9). In this case f can take values in the range (0,1) for a wide set of ep values, thus lowering the Ising exponents. A more thorough examination of this situation is called for. 

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