Allen-Cahn Critical Wavelength

A homogeneous free-energy density function fhom(rj) that has an order-disorder transition similar to Fig. 18.66 has the form 

Suppose that the system is initially uniform with an unstable disordered structure (i.e., r] = 0). For instance, the system may have been quenched from a high-temperature, disordered state, 77 = 1 represents the two equivalent equilibrium ordered variants. If the system is perturbed a small amount by a one-dimensional perturbation in the 2-direction, 77(f) = 6(t) sin(/32). Substituting this ordering perturbation into Eq. 18.26 and keeping the lowest-order terms in the amplification factor, 6(t), 

Note that the amplification factor is a weakly increasing function of wavelength (asymptotically approaching at long wavelengths). This predicts that the 

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