Use of Scaling Theory to Predict Amplitudes: Verification of Double-Power-Law Behavior for Crossover of Lattice Dimensionality

Scaling with a parameter, in contrast to ordinary scaling theory, makes predictions concerning amplitude functions as well as critical-point exponents. We provide the first test of these predictions, by examining the dependence of the amplitude on the anisotropy parameter $R=\frac{J_z}{J_{xy}}$ for the susceptibility and the second moment of the simple cubic and fcc Ising models with directional anisotropy. The "double-power-law" behavior found strongly supports the scaling predictions for thermodynamic functions and the two-spin-correlation function. Our analysis provides a measure of the ranges of $R$ over which parameter scaling appears to hold for the simple cubic and fcc lattices. The relative domains of validity on the two lattices are interpretable in terms of the respective lattice structures.